Discontinuity, Nonlinearity, and Complexity

Volume 4 Issue 4 December 2015

ISSN 2164‐6376 (print) ISSN 2164‐6414 (online)

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

Discontinuity, Nonlinearity, and Complexity Editors Valentin Afraimovich San Luis Potosi University, IICO-UASLP, Av.Karakorum 1470 Lomas 4a Seccion, San Luis Potosi, SLP 78210, Mexico Fax: +52 444 825 0198 Email: [email protected]

Lev Ostrovsky Zel Technologies/NOAA ESRL, Boulder CO 80305, USA Fax: +1 303 497 5862 Email: [email protected]

Xavier Leoncini Centre de Physique Théorique, Aix-Marseille Université, CPT Campus de Luminy, Case 907 13288 Marseille Cedex 9, France Fax: +33 4 91 26 95 53 Email: [email protected]

Dimitry Volchenkov The Center of Excellence Cognitive Interaction Technology Universität Bielefeld, Mathematische Physik Universitätsstraße 25 D-33615 Bielefeld, Germany Fax: +49 521 106 6455 Email: [email protected]

Associate Editors Marat Akhmet Department of Mathematics Middle East Technical University 06531 Ankara, Turkey Fax: +90 312 210 2972 Email: [email protected]

Ranis N. Ibragimov Applied Statistics Lab GE Global Research 1 Research Circle, K1-4A64 Niskayuna, NY 12309 Email: [email protected]

Jose Antonio Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrotechnical Engineering Rua Dr. Antonio Bernardino de Almeida 4200-072 Porto, Portugal Fax: +351 22 8321159 Email: [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University, Balgat 06530 Ankara, Turkey Email: [email protected]

Nikolai A. Kudryashov Department of Applied Mathematics Moscow Engineering and Physics Institute (State University), 31 Kashirskoe Shosse 115409 Moscow, Russia Fax: +7 495 324 11 81 Email: [email protected]

Denis Makarov Pacific Oceanological Institute of the Russian Academy of Sciences,43 Baltiiskaya St., 690041 Vladivostok, RUSSIA Tel/fax: 007-4232-312602. Email: [email protected]

Marian Gidea Department of Mathematics Northeastern Illinois University Chicago, IL 60625, USA Fax: +1 773 442 5770 Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University 198504, Russia Email: [email protected]

Josep J. Masdemont Department of Matematica Aplicada I Universitat Politecnica de Catalunya (UPC) Diagonal 647 ETSEIB 08028 Barcelona, Spain Email: [email protected]

Juan Luis García Guirao Department of Applied Mathematics Technical University of Cartagena Hospital de Marina 30203-Cartagena, SPAIN Fax:+34 968 325694 E-mail: [email protected]

E.N. Macau Lab. Associado de Computção e Mateática Aplicada, Instituto Nacional de Pesquisas Espaciais, Av. dos Astronautas, 1758 12227-010 Sáo José dos Campos –SP Brazil Email: [email protected]

Michael A. Zaks Technische Universität Berlin DFG-Forschungszentrum Matheon Mathematics for key technologies Sekretariat MA 3-1, Straße des 17. Juni 136, 10623 Berlin, Germany Email: [email protected]

Mokhtar Adda-Bedia Laboratoire de Physique Statistique Ecole Normale Supérieure 24 rue Lhomond, 75231 Paris Cedex 05 France Fax: +33 1 44 32 34 33 Email: [email protected]

Ravi P. Agarwal Department of Mathematics Texas A&M University – Kingsville, Kingsville, TX 78363-8202 USA Email: [email protected]

Editorial Board Vadim S. Anishchenko Department of Physics Saratov State University Astrakhanskaya 83, 410026, Saratov, Russia Fax: (845-2)-51-4549 Email: [email protected]

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 4, December 2015

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 381–382

Discontinuity, Nonlinearity, and Complexity https://lhscientificpublishing.com/Journals/DNC-Default.aspx

Fractional Dynamics and Systems with Power-Law Memory M. Edelman1,3†, J.A. Tenreiro Machado2 1 Dept.

of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA; Courant Institute of Mathematical Sciences, New York University,251 Mercer St., New York, NY 10012, USA 2 Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant´ onio Bernardino de Almeida, 431, Porto, Portugal 3 Department of Mathematics, BCC, CUNY, 2155 University Avenue, Bronx, New York 10453, USA Submission Info

Abstract

Keywords

The special issue of DNC “Fractional Dynamics and Systems with PowerLaw Memory” contains papers related to presentations given at the 5th Conference on Nonlinear Science and Complexity, NSC’14, held on August 4-9, 2014 in Xi’an Jiaotong University, Xi’an, P. R. of China. Within this conference the authors proposed two mini-symposia: “Nonlinear Fractional Dynamics and Systems with Memory” and “Fractional Calculus Applications”.

Fractional calculus Power-law memory

©2015 L&H Scientific Publishing, LLC. All rights reserved.

Communicated by Valentin Afraimovich Received 30 April 2015 Accepted 30 April 2015 Available online 1 January 2016

1 Introduction The special issue “Fractional Dynamics and Systems with Power-Law Memory” contains ten papers. In the first paper, “Fractional Calculus: Models, Algorithms, Technology”, one of the editors of the present issue, J. A. Tenreiro Machado, is continuing his effort to make a concise review of the recent developments in physical, engineering, and biological applications of fractional calculus. This paper emphasizes the significance of fractional calculus and serves as a good introduction to the issue. Fractional dynamics is a dynamics with power-law memory and in the following two papers, • “Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α -Families of Maps” by M. Edelman; • “Analysis of Terrorism Data-series by means of Power Law and Pseudo Phase Plane” by A. M. Lopes and J. A. T. Machado, the authors discuss occurrences of systems with power-law memory in natural and social sciences and investigate general properties of nonlinear systems with power- or asymptotically power-law memory. One of the main areas of applications of fractional calculus is control. Use of fractional calculus and, correspondingly, power-law memory provides a more robust control. The fourth paper, “Adaptive Memory † Corresponding

author. Email address: [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.11.001

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Identification of Fractional Order Systems” by Y. Zhao, Y. Li, and F. Zhou, in which the authors propose a new way to estimate the initialization function, serves as a bridge between the papers on power-law memory and papers on fractional control. This paper is followed by three papers on fractional control: • “The optimal control problem for linear systems of non-integer order with lumped and distributed parameters” by V. A. Kubyshkin and S. S. Postnov; • “Sliding Mode Control of Fractional Lorenz-Stenflo Hyperchaotic System” by J. Yuan, B. Shi; • “Hybrid projective synchronization in mixed fractional-order complex networks with different structure” by L.-X. Yang, J. Jiang, and X.-J. Liu; The subjects of the last three papers, • “Nonlinear four-point impulsive fractional differential equations with p-Laplacian operator” by F. T. Fen and I. Y. Karaca; (19 pages) • “About Utility of the Simplified Grünwald-Letnikov Formula Equivalent Horner Form” by D. W. Brzeziński and P. Ostalczyk; (13 pages) • “The Double Exponential Formula as a Gauss Quadratures Replacement for Numerical Integration” by D. W. Brzeziński and P. Ostalczyk, (11 pages) are of the general interest for the applications of fractional calculus. Acknowledgements The invited editors express their gratitude to the editors of DNC and Prof. Albert C.J. Luo for the opportunity to publish the special issue “Fractional Dynamics and Systems with Power-Law Memory” in the journal Discontinuity, Nonlinearity, and Complexity.



Fractional Calculus: Models, Algorithms, Technology J.A. Tenreiro Machado† Institute of Engineering, Polytechnic of Porto, Dept. of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431 4249-015 Porto, Portugal Submission Info Communicated by Mark Edelman Received 15 December 2014 Accepted 19 January 2015 Available online 1 January 2016

Abstract In the last three decades Fractional Calculus (FC) became an area of intense research and development. The accompanying poster illustrates the present day major achievements in the application of FC in physics, engineering and biology.

Keywords Fractional calculus Fractional derivatives Fractional dynamics Applications of fractional calculus


1 Introduction Fractional Calculus (FC) started with the brilliant ideas of Gottfried Wilhelm Leibniz (1646–1716). Nevertheless, only since the middle of the Seventies FC started to be the object of dedicated conferences and books. A considerable interest in FC, stimulating the emergence of applications in new areas of science reflecting the progress in the human knowledge, occurred in the recent three decades; In [1] a list of information about the progress in FC during the period 1966–2010 was collected. In [2, 3] the progress of scientific publications, both in books and conferences, during the last half century was analyzed. In [4,5] two graphical timelines about the old and recent histories of FC were published. This small note intends to accompany a poster illustrating the present day major achievements in the application of FC in physics, engineering, and biology. The application of FC started with the work of Niels Henrik Abel (1802–1829) concerning the “tautochrone problem”. Later we can mention the works of Oliver Heaviside (1850–1925) about electrical lines. For a survey of the pioneers of the application of FC readers can follow [6, 7]. Presently we verify the adoption of FC tools in numerous areas such as Cole-Cole, Davidson-Cole, HavriliakNegami models [8–11], finance [12], power law models [13–15], signal processing [16–22], single trajectory and chaos [23–28], Brownian motion and Lévy flights [29–36], CRONE (Commande Robuste d’Ordre Non Entier) and control [37–47], toolboxes for Matlab (Ninteger, CRONE) [48–50], biological models [10, 51–54], † Corresponding


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biomedicine [55], hydractive suspensions [56, 57], human respiratory system [54, 58–61], complex non-integer differentiation [62–66], lithium-ion batteries [67, 68], ultracapacitors, fuel cells, thermal rods [69–71], genetics [72], complex systems [73, 74], fractional order element [75, 76] and fractance and dielectrics [77–80], electromagnetism [81–85], stochastic processes [86, 87], fractals [88] and self-similarity [89], entropy [90, 91], diffusion to wave propagation [92,93], variational principles [94–97], viscoelasticity [98–101], special functions [102–105], numerical algorithms [106], injection systems [107–112], multimedia streaming [113–116], dynamical systems [73,117–123], trajectory planning [71,124,125], tissue alterations in pathological states [126], fractional pharmacokinetics [126], non-local elasticity [127], and new formulations and interpretations [128–134].

Fig. 1 Achievements in the application of Fractional Calculus in physics, engineering and biology.

A final note. Lists, such as the one presented in this short paper, can never be complete. The author apologizes for all omissions. Acknowledgment The author would like to thank the collaboration of the following contributors: Dumitru Baleanu, Gary Bohannan, Riccardo Caponetto, Piotr Duch, Mark Edelman, Christophe Farges, Hans Haubold, Clara Ionescu, Virginia Kiryakova, Patrick Lanusse, António Lopes, Guido Maione, Pierre Melchior, Francesco Mainardi, Rachid Malti, Xavier Moreau, Raoul Nigmatullin, Manuel Ortigueira, Piotr Ostalczyk, Alain Oustaloup, Ivo Petrásˇ, Jocelyn Sabatier, Dragan Spasić, József Tar, Duarte Valério, Stéphane Victor, Bruce West.


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Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α Families of Maps M. Edelman† Dept. of Physics, Stern College at Yeshiva University, 245 Lexington Ave, New York, NY 10016, USA Courant Institute of Mathematical Sciences, New York University,251 Mercer St., New York, NY 10012, USA Department of Mathematics, BCC, CUNY, 2155 University Avenue, Bronx, New York 10453, USA Submission Info

Abstract

Keywords

In this paper we extend the notion of an α -family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic α Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.

Fractinal derivative Fractional difference Attractors Maps with memory


Communicated by J.A.T. Machado Received 21 November 2014 Accepted 17 December 2014 Available online 1 January 2016

1 Introduction Systems with memory are common in biology, social sciences, physics, and engineering (see reviews [1,2]). The most frequently encountered type of memory in natural and engineering systems is power-law memory. This leads to the possibility of describing them by fractional differential equations, which have power-law kernels. Nonlinear integro-differential fractional equations are difficult to simulate numerically - this is why in [3] the authors introduced fractional maps, which are equivalent to fractional differential equations of nonlinear systems experiencing periodic delta function-kicks, and proposed to use them for the investigation of general properties of nonlinear fractional dynamical systems. Bifurcation diagrams in the fractional Logistic Map related to a scheme of numerical integration of fractional differential equations were considered in [4]. An adequate description of discrete natural systems with memory can be obtained by using fractional difference equations (see [5–12]). In [10–12] the authors demonstrated that in some cases fractional difference equations are equivalent to maps (which we will call fractional difference maps) with falling factorial-law memory, where falling factorial function is defined as t (α ) =

Γ(t + 1) , t = −1, −2, −3, .... Γ(t + 1 − α )

† Corresponding


(1)

392

M. Edelman / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 391–402

Falling factorial-law memory is asymptotically power-law memory lim

t→∞

Γ(t + 1) = 1, α ∈ R, Γ(t + 1 − α )t α

(2)

and we may expect that fractional difference maps have properties similar to the properties of fractional maps. The goal of the present paper is to introduce fractional difference families of maps depending on memory and nonlinearity parameters consistent with the previous research of fractional maps (see [1, 3, 13–20]) in order to prepare a background for an investigation of general properties of systems with asymptotically power-law memory. In the next section (Sec. 2) we will remind the reader how the regular Universal, Standard, and Logistic Maps (see [21–24]) are generalized to obtain fractional Caputo α -Families of Maps (α FM). In Sec. 3 we’ll present some basics on fractional difference/sum operators, which will be used in Sec. 4 to derive the fractional difference Caputo Universal, Standard, and Logistic α FMs. In Sec. 5 we’ll present some results on properties of fractional difference Caputo Standard α FM. 2 Fractional α -Families of Maps Fractional α FM were introduced in [19], further investigated in [20], and reviewed in [1]. The Universal α FM was obtained by integrating the following equation: ∞ dα x + G (x(t − Δ)) δ (t − (k + ε )) = 0, K ∑ dt α k=−∞

(3)

where ε > Δ > 0, α ∈ R, α > 0, ε → 0, with the initial conditions corresponding to the type of fractional derivative to be used. GK (x) is a nonlinear function which depends on the nonlinearity parameter K. It is called Universal because integration of Eq. (3) in the case α = 2 and GK (x) = KG(x) produces the regular Universal Map (see [23]). In what follows the author considers Eq. (3) with the left-sided Caputo fractional derivative (see [25–27]) ˆ t 1 DNτ x(τ )d τ C α N−α N Dt x(t) = , (N = α ), (4) 0 Dt x(t) = 0 It Γ(N − α ) 0 (t − τ )α −N+1 where N ∈ Z, DtN = d N /dt N , 0 Itα is the Riemann-Liouville fractional integral, Γ() is the gamma function, and the initial conditions are (5) (Dtk x)(0+) = bk , k = 0, ..., N − 1. There are two reasons to restrict the consideration in this paper to the Caputo case (the Riemann-Liouville case won’t be considered): a) as in the case of fractional differential equations, in the case of fractional difference equations it is much easier to define initial conditions for Caputo difference equations than for RiemannLiouville difference equations; b) the main goal of this work is to compare fractional and fractional difference maps, and the case of Caputo maps serves the purpose. Comparison of the Riemann-Liouville and Caputo Standard Maps was considered in [17]. The problem Eqs. (3)–(5) is equivalent to the Volterra integral equation of the second kind (t > 0) [26] ˆ t N−1 1 bk k GK (x(τ − Δ)) ∞ dτ (6) x(t) = ∑ t − ∑ δ (τ − (k + ε )). Γ(α ) 0 (t − τ )1−α k=−∞ k=0 k! After the introduction x(s) (t) = Dts x(t) in the limit ε → 0 (recall that ε > Δ > 0) the Caputo Universal α FM can be written as (see [16]) (s) xn+1

=

N−s−1 x(k+s) 0

∑

k=0

k!

(n + 1)k −

n 1 GK (xk )(n − k + 1)α −s−1 , ∑ Γ(α − s) k=0

(7)


393

(k+s)

where s = 0, 1, ..., N − 1 and x0 = bk+s . In the case GK (x) = K sin(x) and α = 2 with p = x(1) Eq. (7) yields the well–known Standard Map (see [21]), which on a torus can be written as pn+1 = pn − K sin(xn ) (mod 2π ),

(8)

xn+1 = xn + pn+1 (mod 2π ).

(9)

This is why the Caputo Universal α FM Eq. (7) with GK (x) = K sin(x)

(10)

is called the Caputo Standard α FM: (s)

xn+1 =

N−s−1 x(k+s) 0

∑

k!

k=0

(n + 1)k −

n K ∑ sin(xk )(n − k + 1)α −s−1, Γ(α − s) k=0

(11)

where s = 0, 1, ..., N − 1. In the case GK (x) = x − Kx(1 − x) and α = 1 Eq. (7) yields the well–known Logistic Map (see [24]) xn+1 = Kxn (1 − xn ).

(12)

This is why the Caputo Universal α FM Eq. (7) with GK (x) = GLK (x) = x − Kx(1 − x)

(13)

is called the Caputo Logistic α FM: (s) xn+1

=

N−s−1 x(k+s) 0

∑

k!

k=0

(n + 1)k −

n 1 xk − Kxk (1 − xk ) , ∑ Γ(α − s) k=0 (n − k + 1)1+s−α

(14)

where s = 0, 1, ..., N − 1. The Caputo Standard and Logistic α FMs were investigated in detail in [1, 19, 20] for the case α ∈ (0, 2] which is important in applications. • For α = 0 the Caputo Standard and Logistic α FMs are identically zeros: xn = 0. • For 0 < α < 1 the Caputo Standard α FM is xn = x0 −

K n−1 sin (xk ) ∑ (n − k)1−α (mod 2π ). Γ(α ) k=0

(15)

1 n−1 xk − Kxk (1 − xk ) ∑ (n − k)1−α . Γ(α ) k=0

(16)

and the Caputo Logistic α FM is xn = x0 −

• For α = 1 the 1D Standard Map is the Circle Map with zero driving phase xn+1 = xn − K sin(xn ) and the 1D Logistic α FM is the Logistic Map Eq. (12).

(mod 2π ).

(17)

394


• For 1 < α < 2 the Caputo Standard α FM is pn+1 = pn −

n−1 K Vα2 (n − i + 1) sin(xi ) + sin(xn ) (mod 2π ), ∑ Γ(α − 1) i=0

xn+1 = xn + p0 −

K n 1 ∑ Vα (n − i + 1) sin(xi ) (mod 2π ), Γ(α ) i=0

(18) (19)

where Vαk (m) = mα −k − (m − 1)α −k and the Caputo Logistic α FM is xn+1 = x0 + p(n + 1)k − pn+1 = p0 −

1 n ∑ [xk − Kxk (1 − xk )](n − k + 1)α −1, Γ(α ) k=0

n 1 ∑ [xk − Kxk (1 − xk )](n − k + 1)α −2. Γ(α − 1) k=0

(20) (21)

• For α = 2 the Caputo Standard Map is the regular Standard Map as in Eqs. (8) and (9) above. The 2D Logistic Map is pn+1 = pn + Kxn (1 − xn ) − xn ,

(22)

xn+1 = xn + pn+1 .

(23)

3 Fractional difference/Sum operators In this paper we will adopt the definition of the fractional sum (α > 0)/difference (α < 0) operator introduced in [5] as 1 t−α −α f (t) = (24) a Δt ∑ (t − s − 1)(α −1) f (s). Γ(α ) s=a Here f is defined on Na and a Δt−α on Na+α , where Nt = {t,t + 1,t + 2, ...}, and falling factorial t (α ) is defined by Eq. (1). As Miller and Ross noticed, their way to introduce the discrete fractional sum operator based on the Green’s function approach is not the only way to do so. In [6] the authors defined the discrete fractional sum operator generalizing the n-fold summation formula in a way similar to the way in which the fractional Riemann–Liouville integral is defined in fractional calculus by extending the Cauchy n-fold integral formula to the real variables. They mentioned the following theorem but didn’t present a proof. Theorem 1. For ∀n ∈ N 0

−n a Δt f (t) =

n−2

t−n t−n s s 1 (n−1) (t − s − 1) f (s) = ... f (sn−1 ), ∑ ∑ ∑ ∑ (n − 1)! s=a s0 =a s1 =a sn−1 =a

(25)

where si , i = 0, 1, ...n − 1 are the summation variables. Proof. Indeed, this formula is obviously true for n = 1. Let’s assume that Eq. (25) is true for n − 1: −(n−1)

a Δt

t−(n−1)

f (t) =

∑

s=a

C(t − s − 1, n − 2) f (s) =

t−(n−1) s1

∑ ∑

...

s1 =a s2 =a

sn−2

∑

f (sn−1 ),

(26)

sn−1 =a

where C(i, j) is the number of j-combinations from a given set of i elements. Then, for t = s0 + n − 1 Eq. (26) gives s0

∑ C(s

s1 =a

0

1

1

− s + n − 2, n − 2) f (s ) =

s0

s1

sn−2

∑ ∑ ... ∑

s1 =a s2 =a

sn−1 =a

f (sn−1 ).

(27)


395

Now Eq. (25) can be obtained from t−n

s0

sn−2

∑ ∑ ... ∑

s0 =a s1 =a

=

t−n

∑

n−1

f (s

sn−1 =a t−n

f (s1 )

s1 =a

∑

)=

t−n

s0

∑ ∑ C(s0 − s1 + n − 2, n − 2) f (s1 )

s0 =a s1 =a

C(s0 − s1 + n − 2, n − 2) =

s0 =s1

t−n

∑ C(t − s1 − 1, n − 1) f (s1 ) =a Δt−n f (t).

(28)

s1 =a

Here we used the identity t−n

∑

C(s0 − s1 + n − 2, n − 2) = C(t − s1 − 1, n − 1),

(29)

s0 =s1

which is true for t = n + s1 and can be proven by induction for any t t+1−n

∑

C(s0 − s1 + n − 2, n − 2) = C(t − s1 − 1, n − 2) +C(t − s1 − 1, n − 1) = C(t − s1 , n − 1).

(30)

s0 =s1

This ends the proof. As we see, this approach is consistent with the definition of the fractioanal sum operator given by Miller and Ross (see also [8]). For α > 0 and m − 1 < α < m Anastassiou [9] defined the fractional (left) Caputo-like difference operator as C α a Δt x(t)

−(m−α ) m

=a Δt

Δ x(t) =

t−(m−α ) 1 ∑ (t − s − 1)(m−α −1)Δm x(s), Γ(m − α ) s=a

(31)

where Δm is the m-th power of the forward difference operator defined as Δx(t) = x(t + 1) − x(t). The proof (see [5], p.146) that 0 Δtλ in the limit λ → 0 approaches the identity operator can be easily extended to the a Δtλ operator. In this case the definition Eq. (31) can be extended to all real α ≥ 0 with Ca Δtm x(t) = Δm x(t) for m ∈ N0 . Then, the Anastassiou’s fractional Taylor difference formula [9] x(t) =

m−1

∑

k=0

t−α 1 (t − a)(k) k Δ x(a) + ∑ (t − s − 1)(α −1)Ca Δtα x(t), k! Γ(α ) s=a+m− α

(32)

where x is defined on Na , m = α , and a ∈ N0 for ∀t ∈ Na+m is valid for any real α > 0 and for integer α = m is identical to the integer discrete Taylor’s formula (see p.28 in [7]) x(t) =

m−1

∑

k=0

t−m 1 (t − a)(k) k Δ x(a) + ∑ (t − s − 1)(m−1)Δmx(t). k! (m − 1)! s=a

(33)

As it was noticed in [11] and [12], Lemma 2.4 from [10] on the equivalency of the fractional Caputo-like difference and sum equations can be extended to all real α > 0 and formulated as follows: Theorem 2. The Caputo-like difference equation C α a Δt x(t)

with the initial conditions

= f (t + α − 1, x(t + α − 1))

Δk x(a) = ck , k = 0, 1, ..., m − 1, m = α

(34) (35)

is equivalent to the fractional sum equation x(t) =

m−1

∑

k=0

where t ∈ Na+m .

t−α 1 (t − a)(k) k Δ x(a) + ∑ (t − s − 1)(α −1) f (s + α − 1, x(s + α − 1)), k! Γ(α ) s=a+m− α

(36)

396


Here we should notice that the authors of [11] and [12] didn’t consider the Caputo difference operator with integer α . As a result, Theorem 2 is not valid for integer values of α with their definition m = [α ] + 1. This theorem in the limiting sense can be extended to all real α ≥ 0. Indeed, taking into account that lim Ca Δtα x(t) = x(t), Eq. (34) for α = 0 turns into α →0

x(t) = f (t − 1, x(t − 1)).

(37)

For α = 0 the first sum on the right in Eq. (36) disappears and in the second sum the only remaining term with s = t − α in the limit α → 0 turns into f (t − 1, x(t − 1)). 4 Fractional difference α -families of maps In the following we assume that f is a nonlinear function f (t, x(t)) = −GK (x(t)) with the nonlinearity parameter K and adopt the Miller and Ross proposition to let a = 0. Now, with xn = x(n), Theorem 2 can be formulated as Theorem 3. For α ∈ R, α ≥ 0 the Caputo-like difference equation C α 0 Δt x(t)

= −GK (x(t + α − 1)),

(38)

where t ∈ Nm , with the initial conditions Δk x(0) = ck , k = 0, 1, ..., m − 1, m = α

(39)

is equivalent to the map with falling factorial-law memory xn+1 =

m−1

∑

k=0

1 n+1−m Δk x(0) (n + 1)(k) − ∑ (n − s − m + α )(α −1)GK (xs+m−1), k! Γ(α ) s=0

(40)

where xk = x(k) which we will call the fractional difference Caputo Universal α -Family of Maps. The fractional difference Caputo Universal α FM is similar to the general form of the Caputo Universal α FM Eq. (7). Both of them can be written as xn = x0 +

m−1

∑

k=1

1 n−1 pk (0) (k) n − ∑ Wα (n − k)GK (xk ), k! Γ(α ) k=M

(41)

where pk (0) are the initial value of momenta defined as ps = Dts x(t) for fractional maps and as ps (t) = Δs x(t) for fractional difference maps; M = 0 for fractional maps and M = m − 1 for fractional difference maps; n(s) = ns for fractional maps and n(s) = Γ(n + 1)/Γ(n + 1 − s) for the fractional difference maps. Wα (s) = sα −1 for fractional maps and Wα (s) = Γ(s + α − 1)/Γ(s) for fractional difference maps. Asymptotically, both expressions for Wα (s) coincide because of Eq. (2). 4.1

Fractional difference universal α FM

Let’s consider the case α = 2. Then the difference Eq. (34) produces Δ2 xn = −GK (xn+1 )

(42)

and the equivalent sum equation is n−1

xn+1 = x0 + Δx0 (n + 1) − ∑ (n − s)GK (xs+1 ). s=0

(43)


397

After the introduction pn = Δxn−1 with the assumption GK (x) = KG(x) the map equations indeed can be written as the well–known 2D Universal Map (44) pn+1 = pn − KG(xn ), xn+1 = xn + pn+1 ,

(45)

which for G(x) = sin(x) produces the Standard Map Eqs. (8) and (9). In the rest of this paper we’ll call Eq. (40) with GK (x) = K sin(x) the fractional difference Caputo Standard α -Family of Maps. In the case α = 1 the fractional difference Caputo Universal α FM is xn+1 = xn − GK (xn ),

(46)

which produces the Logistic Map if GK (x) = x − Kx(1 − x). In the rest of this paper we’ll call Eq. (40) with GK (x) = x − Kx(1 − x) the fractional difference Caputo Logistic α -Family of Maps. 4.2

α = 0 Difference Caputo Standard and Logistic α FMs • In the case α = 0 the 0D Standard Map turns into the Sine Map (see, e.g., [28]) xn+1 = −K sin(xn ) (mod 2π ).

(47)

xn+1 = −xn + Kxn (1 − xn ).

(48)

• The 0D Logistic Map is

4.3

0 < α < 1 Fractional Difference Caputo Standard and Logistic α FMs • For 0 < α < 1 the fractional difference Standard Map is xn+1 = x0 −

K n Γ(n − s + α ) ∑ Γ(n − s + 1) sin(xs ) (mod 2π ), Γ(α ) s=0

(49)

which after the π -shift of the independent variable x → x + π coincides with the “fractional sine map” proposed in [11]. • The fractional difference Logistic Map can be writen as xn+1 = x0 −

1 n Γ(n − s + α ) ∑ Γ(n − s + 1) [xs − Kxs(1 − xs )]. Γ(α ) s=0

(50)

The fractional Logistic Map introduced in [12] does not converge to the Logistic map in the case α = 1. 4.4

α = 1 Difference Caputo Standard and Logistic α FMs • The α = 1 difference Caputo Standard α FM is identical to the Circle Map with zero driven phase Eq. (17). The map considered in [11] xn+1 = xn + K sin(xn ) (mod 2π ) is obtained from this map by the substitution x → x + π . • The α = 1 Difference Caputo Logistic α FM is the regular Logistic Map.

(51)

398

4.5


1 < α < 2 Fractional Difference Caputo Standard and Logistic α FMs • For 1 < α < 2 the fractional difference Standard Map is xn+1 = x0 + Δx0 (n + 1) −

K n−1 Γ(n − s + α − 1) ∑ Γ(n − s) sin(xs+1 ) (mod 2π ). Γ(α ) s=0

(52)

• The 1 < α < 2 fractional difference Logistic Map is xn+1 = x0 + Δx0 (n + 1) −

1 n−1 Γ(n − s + α − 1) ∑ Γ(n − s) [xs+1 − Kxs+1(1 − xs+1)]. Γ(α ) s=0

(53)

Let’s introduce pn = Δxn−1 ; then these maps can be written as 2D maps with memory: • The fractional difference Standard Map is pn = p1 −

n K Γ(n − s + α − 1) sin(xs−1 ) (mod 2π ), ∑ Γ(α − 1) s=2 Γ(n − s + 1)

xn = xn−1 + pn , (mod 2π ), n ≥ 1,

(54) (55)

which in the case x0 = 0 is identical to the ”fractional standard map” introduced in [11] (Eq. (18) with ν = α − 1 there). • The fractional difference Logistic Map is n K Γ(n − s + α − 1) [xs−1 − Kxs−1 (1 − xs−1 )], ∑ Γ(α − 1) s=2 Γ(n − s + 1) xn = xn−1 + pn , n ≥ 1.

pn = p1 −

4.6

(56) (57)

α = 2 Difference Caputo Standard and Logistic α FMs • The α = 2 difference Caputo Standard α FM is the regular Standard Map Eqs. (8) and (9). • From Eqs. (44) and (45) the 2D difference Caputo Logistic α FM is pn+1 = pn − xn + Kxn (1 − xn ),

(58)

xn+1 = xn + pn+1 ,

(59)

which is identical to the 2D Logistic Map Eqs. (22) and (23). 5 Conclusion As we saw in Sec. 3, the fractional difference operator is a natural extension of the difference operator. The simplest fractional difference equations (of the Eq. (34) type), where the fractional difference on the left side is equal to a nonlinear function on the right side, are equivalent to maps with falling factorial-law (asymptotically power-law) memory Eq. (36). Systems with power-law memory play an important role in nature (see [1]) and investigation of their general properties is important for understanding behavior of natural systems. Properties of the fractional difference Caputo Standard α FM were investigated in detail in [29] (see also Sec. 3 in [30]. Qualitatively, properties of the fractional difference and fractional maps (maps with falling factorial-


399

10 6

8

c

K

Kc

6 4 2 0 0

Chaos

Chaos

n Period 2 sinks

2

Stable fixed point (sink) in the origin

1

Stable fixed point (sink) in the origin

0 0

2

α

a

Period 2n sinks

4

b

1

2

α

2

2

0

0

x

x

Fig. 1 α − K (bifurcation) diagrams for the Caputo, Eq. (11), (a) and Fractional Difference Caputo, Eq. (40) with GK (x) = K sin(x), (b) Standard α FMs.

−2

−2 2

a

4

K

6

1

b

2 K

3

Fig. 2 Bifurcation diagrams for the Caputo, Eq. (15), (a) and Fractional Difference Caputo, Eq. (49), (b) Standard α FMs with α = 0.5 obtained after 5000 iterations with the initial condition x0 = 0.1 (regular points) and x0 = −0.1 (bold points).

and power-law memory) are similar. The similarity reveals itself in the dependence of systems’ properties on the memory (α ) and nonlinearity (K) parameters (bifurcation diagrams, see Figs. 1, 2, and 3), power-law convergence to attractors, non-uniqueness of solutions (intersection of trajectories and overlapping of attractors), and cascade of bifurcations and intermittent cascade of bifurcations type behaviors (see Figs. 4 and 5). The differences of the properties of the falling factorial-law memory maps from the power-law memory maps are the results of the differences in weights of the recent (with (n − j)/n 0, if its complementary cumulative distribution function is given by: P (X ≥ x) ∼ C · x−α

(1)

Parameters (C, α ) ∈ R represent the PL scaling coefficient and the exponent, respectively [11]. For many systems, only the tails of the distributions follow a PL, meaning that the empirical phenomena unveil PL behavior just for events greater than some minimum threshold value xmin . In this case, expression (1) may be written as: P (X ≥ x) =

x xmin

− α

, x > xmin

(2)

where C = (xmin )α . Pseudo Phase Space (PPS) reconstruction has been used with dynamical systems in order to mitigate the lack of information about the system [8]. PPS is justified by Takens’ embedding theorem [13], which states that if a time series is one component of an attractor that can be represented by a smooth d-dimensional manifold, then the topological properties of the signal are equivalent to those of the embedding formed by the n-dimensional phase space vectors: u(t) = [s(t)

s(t + τ ) s(t + 2τ ) ...

s(t + (n − 1)τ )]

(3)

where n ≥ 2d + 1, {d, n} ∈ N and τ ∈ R + . Parameters τ and n represent the time delay and embedding dimension, respectively. Usually, we chose n = 3 or n = 2 and represent u(t) in a graph. When the embedding space dimension is two the PPS degenerates in the Pseudo Phase Plane (PPP). PPP has been used to assess the complexity of human genome [14], to study musical sounds [7], to analyze stock markets [5], and to study earthquake data series [9], among others. A key issue with PPS is the choice of τ . One method consists of finding the earliest time at which the autocorrelation function decreases below a certain percentage of its initial value, or has a point of inflection [7]. In this paper we analyze a dataset of terrorist events by means of PL distributions and PPP reconstruction. We consider worldwide events grouped into 13 geographic regions. First, we approximate the empirical data by PL functions and analyze the PL parameters. Second, we model the dataset as time-series and interpret the data as the output of a dynamical system. For each region, we compute the autocorrelation coefficient to find the optimal time delay for reconstructing the PPP. Third, we compare the PPP curves using clustering tools in order to unveil relationships among the data. Bearing these ideas in mind this paper is organized as follows. Section 2 describes the experimental dataset and analyzes the data in the perspective of PL distributions. Section 3 models the empirical data as time-series, applies the PPP reconstruction methodology and compares the PPP curves. Finally, Section 4 outlines the main conclusions. 2 Dataset analysis based on PL In this study, data from the Global Terrorism Database (GTD) are analyzed. The GTD database is available online and includes information on more than 125,000 worldwide terrorist attacks, from 1970 up to 2013 (http://www.start.umd.edu/gtd). Each record consists of the date of the event (year and month), geographic location (region) and total number of deaths and injuries, among other information. The annual evolution of the number of occurrences and number of human casualties (deaths plus injuries) are illustrated in Figures 1 and 2 for the regions of Australasia & Oceania, Central America & Caribbean, Central Asia, East Asia, Eastern Europe, Middle East & North Africa, North America, South America, South Asia, Southeast Asia, Sub-Saharan Africa, Russia & the Newly Independent States (NIS) and Western Europe. It can

António M. Lopes, J.A. Tenreiro Machado / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 403–411

405

be noted that the regions of Middle East & North Africa, South Asia, Sub-Saharan Africa and Southeast Asia have been the most affected. Moreover, for these regions, it is visible an increasing trend over the 21st century. 4

10

Aus+Oce − Australasia & Oceania C Ame − Central America & Caribbean C Asi − Central Asia E Asi − East Asia E Eur − Eastern Europe MEas+NAf − Middle East & North Africa N Ame − North America

S Ame − South America S Asi − South Asia S Asi − Southeast Asia SSah Af − Sub−Saharan Africa Rus+NIS − Russia & the Newly Independent States (NIS) W Eur − Western Europe

MEas+NAf S Asi

3

number of events with casualties

10

SSah Af SE Asi

2

Rus+NIS S Ame

10

W Eur N Ame E Asi

1

10

E Eur C Ame

0

10 1970

1973

1976

1979

1982

1985

1988

1991 year

1994

1997

2000

2003

2006

2009

2012

C Asi Aus + Oce

Fig. 1 Total number of events (with casualties) per year. The time period of analysis is years 1970 - 2013.

5

10

4

10

Aus+Oce − Australasia & Oceania C Ame − Central America & Caribbean C Asi − Central Asia E Asi − East Asia E Eur − Eastern Europe MEas+NAf − Middle East & North Africa N Ame − North America

S Ame − South America S Asi − South Asia S Asi − Southeast Asia SSah Af − Sub−Saharan Africa Rus+NIS − Russia & the Newly Independent States (NIS) W Eur − Western Europe

MEas+NAf S Asi

SSah Af

number of casualties

SE Asi 3

10

Rus+NIS N Ame S Ame E Asi W Eur

2

10

C Ame

1

10

E Eur

0

10 1970

1973

1976

1979

1982

1985

1988

1991 year

1994

1997

2000

2003

2006

2009

2012

C Asi Aus + Oce

Fig. 2 Total number of casualties per year. The time period of analysis is years 1970 - 2013.

We analyze the distributions of the total number of casualties, over the period 1970 up to 2013, for the 13 regions mentioned above. We adopt the maximum likelihood estimator (MLE) and related statistical tests [2,16] to fit PLs into the empirical data of each region, ri (i = 1, ..., 13). This is accomplished by (i) estimating the parameters α and xmin of the PL model; (ii) calculating the goodness-of-fit (GOF) between the empirical data

406


and the PL model and decide whether the PL model is a plausible hypothesis for the data or not; (iii) comparing the PL model with alternative hypotheses and decide if the PL model should be accepted or, on the contrary, the alternative must be favored over the PL [10]. While this is a ‘standard’ and widely used approach to fit a PL model into experimental data, different methods could be adopted, namely the Eigen-Coordinates method proposed by Nigmatullin [17, 18]. First, for each possible choice of xmin , we estimate the PL exponent α , using the MLE, and we calculate the reweighted Kolmogorov-Smirnov (rKS) GOF statistic, D∗ [2]. The adopted xmin value corresponds to the minimum value of D∗ over all tested xmin values. Second, based on the estimated (xmin , α ), we repeatedly generate synthetic data and compare each dataset with the experimental data using the rKS statistic. In this case, we compute 1,000 tests and adopt a significance threshold value equal to 0.05. Based on the p − value we decide if the null hypothesis should be accepted (p > 0.05) or rejected (p < 0.05). Finally, we compare the PL model with the exponential and log-normal alternative hypotheses, above the same threshold value xmin . We use the likelihood ratio test proposed by Vuong [16], which involves computing the logarithm of the likelihood ratio, L R, of the data under two competing distributions and determining the statistical significance of the observed L R sign. A small p − value means that the sign can be adopted as a reliable indicator of which model is the better. On the contrary, a large p − value indicates that the sign is not a reliable indicator and the test does not favor either model over the other. Table 2 presents detailed results for the rKS GOF. Parameter n denotes the total number of events in each region, ri (i = 1, ..., 13), min and max represent the value of extreme events, and ntail is the number of events above xmin . Table 3 compares the PL model with the exponential and log-normal alternative hypotheses. We present the p − value for the PL fit, the L R values for the two alternative hypotheses and the p − values assessing the significance of each likelihood ratio test. Positive values of L R indicate that the PL model is favored over the competing alternative. Comparing the PL and log-normal fits we obtain negative L R values and, for PL versus exponential, the L R sign is positive or negative, depending on the region we are assessing. However, for all cases the calculated p − values are large, which means that the sign of L R has no statistical relevance and the alternatives can not be favored over the PL fit. In Figure 3 we depict the empirical data and PL approximation for all regions, where we can visually assess the GOF. We conclude that for all cases the random variable ‘number of casualties’ due to terrorist attacks follows a PL distribution. 3 Dataset analysis based on PPP reconstruction The PPP technique is primarily intended to ‘mimic’ the phase plane while avoiding the problems of noise that emerge when calculating derivatives. The interpretation is then made by comparing past-to-future, or preferably better, long past-to-recent past series. So, clusters and patterns may not occur or may emerge with very different configurations. As there is little or no knowledge about the distinct types of patterns, or even possible clusters in the data, the PPP locus and its interpretation is a relevant technique of analysis. In this section, for each region ri (i = 1, ..., 13), we first model the data as a time-series. Second, we reconstruct the PPP and depict the corresponding graph. For the PPP, the optimal time delay corresponds to the instant at which the autocorrelation function has its first point of inflection. Third, we compare the PPP curves and adopt a clustering and visualization tool to unveil the relationships between the data. For the ith region we determine the corresponding time-series, xi (t), adopting a sampling period ts = 3 months:

xi (t) =

T

∑ Aik δ (t − kts)

k=0

(4)


407

Table 1 PL parameters corresponding to the 13 regions, ri . max

α

1

53

1

300

1 1

443

1

6

20677

7

739

ri

n

min

xmin

ntail

p

1

88

2

4476

1.39

9

14

0.86

2.22

75

50

0.65

3

175

4

208

800

0.93

10

25

0.20

5513

0.79

16

45

0.96

5

180

1.63

19

23

0.07

1

1000

2.09

155

121

0.18

1

1382

0.60

12

50

0.87

8

7522

1

321

1.78

59

61

0.61

9

19393

1

1370

1.83

118

126

0.10

10

5386

1

414

1.80

56

73

0.57

11

6053

1

4224

1.35

79

116

0.65

12

1522

1

1071

1.24

41

60

0.06

13

5681

1

523

1.19

45

71

0.37

Table 2 Comparison between the PL model and the log-normal and exponential alternative hypothesis for the 13 regions, ri . ri

PL

Log-normal

Exponential

p

LR

p

LR

p

1

0.86

-0.64

0.52

-0.21

0.42

2

0.65

-0.70

0.48

-0.21

0.42

3

0.20

-0.25

0.80

1.84

0.97

4

0.96

-0.16

0.87

2.42

0.99

5

0.07

-0.45

0.65

0.44

0.67

6

0.18

-1.07

0.29

0.05

0.52

7

0.87

-0.88

0.38

2.62

1.00

8

0.61

-1.10

0.27

-0.14

0.45

9

0.10

-0.50

0.62

1.45

0.93

10

0.57

-0.90

0.37

-0.13

0.45

11

0.65

-0.14

0.89

1.70

0.96

12

0.06

-0.76

0.44

0.71

0.76

13

0.37

-1.08

0.28

0.99

0.84

where, Aik represents the total number of casualties over the time interval kts , T is the total time-length of the time-series (in trimesters) and δ (·) denotes the Dirac delta function. We calculate the integral of xi (t): ˆ Xi (t) =

t 0

xi (u)du

(5)

Empirical data PL approximation


0.05

0.01

0.05

0.02

0.10

0.10

0.20

P(X>=x)

0.10

P(X>=x)

0.05

0.20

P(X>=x)

0.20

0.50

0.50

0.50


1.00

1.00


1.00

408

5

10

20

50

1

2

5

10

20

100

200

1

5

10

1.00


100

500

1000


0.02

0.05

0.05

0.10

0.005 0.010 0.020

P(X>=x)

0.20

P(X>=x)

0.20

50 x

0.50

0.50


0.10

P(X>=x)

50

x

1.00

x

0.500 1.000

2

0.050 0.100 0.200

1

50

100

500

5000

1

2

5

10

20

50

100

200

1

1.00

50

100

500

1000

100

500

1000


0.200 P(X>=x)

0.10

P(X>=x)

0.010

0.020

0.05

0.005

0.02 0.01 10

50

0.500


0.20

0.50 0.20

P(X>=x)

0.10 0.05

5

10 x

0.50


1

5

x

1.00

x

1.000

10

0.100

5

0.050

1

1

2

x

5

10

20

50

100

200

1

5

10

x

50

100

500

1000

x

Fig. 3 Empirical data and PL approximation for regions, ri (i = 1, ..., 13).

and we determine the autocorrelation between Xi (t) and its time delayed version Xi(t − τ ): cii (τ ) =

XiT (t) · Xi (t − τ ) ||Xi (t)|| · ||Xi (t − τ )||

(6)

The optimal time delay (to be denoted by τ ∗ ) corresponding to the instant at which cii (τ ) has its first point of inflection is used for the PPP reconstruction. Calculating cii (τ ) for all regions ri (i = 1, ..., 13), we obtain the optimal time delays T = {91, 126, 69, 79, 85, 32, 47, 112, 30, 80, 82, 42, 139} (expressed in trimesters). Figure 4 represents the PPP portrait fol all regions, ri , where the z−axis corresponds to τ ∗ . The reconstructed

5

10

20

50

100

200

500

0.50 0.05 0.02 1

5

10

50

100

500

5000

x

1.00

x

0.01

0.005 2


0.10

P(X>=x)

0.100

P(X>=x)

0.050 0.020 0.010

0.02 0.01 1

409

0.20

0.200

0.50 0.20 0.10 0.05

P(X>=x)


0.500


1.00

1.000

1.00


1

5

10

50

100

500

1000

x

0.10 0.01

0.02

0.05

P(X>=x)

0.20

0.50


1

2

5

10

20

50

100

200

500

x

Fig. 3 Continued.

attractors give a good representation of the system that generated the time-series. To compare the PPP portraits we calculate the Hausdorff distance between each pair of PPP portraits {Pi , Pj }: H(Pi, Pj ) = max {h(Pi , Pj ), h(Pj , Pi )}

(7)

h(Pi , Pj ) = max min ||pi − p j ||

(8)

where, pi ∈Pi p j ∈Pj

For data visualization, we calculate a 13 × 13 matrix D = [H(Pi, Pj )], (i, j) = 1, ..., 13 and we use the multidimensional scaling (MDS) technique [4]. MDS is a statistical tool for visualizing data that explores similarities/dissimilarities between objects. The results are expressed by maps, where objects perceived to be similar to each belong to the same cluster. MDS is feed with a s × s symmetric matrix, D, of inter-object similarities (or, alternatively, of dissimilarities), where s is the number of objects. In a m-dimensional space, MDS assigns a point to each item, and maps the objects in order to reproduce the observed similarities (or, alternatively, dissimilarities). Different configurations are evaluated, while maximizing some goodness-of-fit function. Usually, we chose m = 2 or m = 3 to facilitate the interpretation of the MDS maps. Figure 5 depicts the 3-dimensional MDS map, where we can note the emergence of two main clusters: A = {S Asi, MEas+NAf, W Eur, S Ame} and B = {C Ame, E Asi, E Eur, Aus+Oce, C Asi, Rus+NIS, N Ame, SE Asi, SSah Af}. Furthermore, A comprises sub-clusters A1 = {S Asi, MEas+NAf} and A2 = {W Eur, S

410

António M. Lopes, J.A. Tenreiro Machado / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 403–411 Aus+Oce − Australasia & Oceania C Ame − Central America & Caribbean C Asi − Central Asia E Asi − East Asia E Eur − Eastern Europe MEas+NAf − Middle East & North Africa N Ame − North America S Ame − South America S Asi − South Asia S Asi − Southeast Asia SSah Af − Sub−Saharan Africa Rus+NIS − Russia & the Newly Independent States (NIS) W Eur − Western Europe

140

120

τ*

100

W Eur C Ame S Ame

80

SSah Af SE Asi

Aus+Oce E Asi

60

E Eur

C Asi N Ame

40

MEas+NAf S Asi

Rus+NIS

20 5 10

x(t+τ) 0

10

10

0

10

5

10

4

10

3

10

2

10

1

x(t)

Fig. 4 PPP portrait for regions ri (i = 1, ..., 13). The time period of analysis is years 1970 - 2013. Aus+Oce − Australasia & Oceania C Ame − Central America & Caribbean C Asi − Central Asia E Asi − East Asia E Eur − Eastern Europe MEas+NAf − Middle East & North Africa N Ame − North America S Ame − South America S Asi − South Asia S Asi − Southeast Asia SSah Af − Sub−Saharan Africa Rus+NIS − Russia & the Newly Independent States (NIS) W Eur − Western Europe

SSah Af SE Asi

0.3 0.25 0.2

C Ame

0.15

N Ame S Ame

0.1

W Eur

0.05 0 −0.05

Rus+NIS MEas+NAf E Asi E Eur Aus+Oce C Asi

0.4

S Asi

0.2

−0.1

0

−0.15 −0.3

−0.2 −0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−0.4

Fig. 5 MDS map for m = 3. The time period of analysis is years 1970 - 2013.

Ame}, while B includes B1 = {E Asi, E Eur, Aus+Oce, C Asi, Rus+NIS, N Ame}, B2 = {SE Asi, SSah Af} and B3 = {C Ame}. 4 Conclusion Terrorism data-series were investigated in the perspective of statistical physics and dynamical systems. Global data, collected from the Global Terrorism Database was analyzed for the time period 1970 up to 2013. Firstly,


411

we found that the statistical distributions of the number of causalities have PL behavior. Secondly, we modeled the data as time-series, used the PPP reconstruction and compared the PPP portraits. The adopted methodology and findings proved to be an alternative to more classical approaches, that can further contribute to better understanding the complexity of the phenomena and their ruling principles. Acknowledgments National Consortium for the Study of Terrorism and Responses to Terrorism (START). (2013). Global Terrorism Database [Data file]. Retrieved from http://www.start.umd.edu/gtd. References [1] Bogen, K.T. and Jones, E.D. (2006), Risks of mortality and morbidity from worldwide terrorism: 1968–2004, Risk Analysis, 26, 45–59. [2] Clauset, A., Shalizi, C.R. and Newman, M.E. (2009), Power-law distributions in empirical data, SIAM Review, 51, 661–703. [3] Clauset, A., Young, M. and Gleditsch, K.S. (2007), On the frequency of severe terrorist events, Journal of Conflict Resolution, 51, 58–87. [4] Cox, T.F. and Cox, M.A. (2001), Multidimensional scaling, Chapman & Hall/CRC: New York. [5] Duarte, F.B., Machado, J.T. and Duarte, G.M. (2010), Dynamics of the Dow Jones and the Nasdaq stock indexes, Nonlinear Dynamics, 61, 691–705. [6] Guzzetti, F., Malamud, B.D., Turcotte, D.L. and Reichenbach, P. (2002) Power-law correlations of landslide areas in central Italy, Earth and Planetary Science Letters, 195, 169–183. [7] Lima, M.F., Machado, J. and Costa, A.C. (2012), A multidimensional scaling analysis of musical sounds based on pseudo phase plane, Abstract and Applied Analysis, 2012. [8] Lima, M.F. and Machado, J.T. (2011) Representation of robotic fractional dynamics in the pseudo phase plane, Acta Mechanica Sinica, 27, 28–35. [9] Lopes, A.M and, Machado, J.T. (2013), Dynamic analysis of earthquake phenomena by means of pseudo phase plane, Nonlinear Dynamics, 74, 1191–1202. [10] Lopes, A.M and, Machado, J.T. (2015), Power law behavior and self-similarity in modern industrial accidents, International Journal of Bifurcation and Chaos, 25, 1550004, 12 pages.. [11] Newman, M.E. (2005), Power laws, Pareto distributions and Zipf’s law, Contemporary Physics 46, 323–351. [12] Pinto, C., Mendes Lopes, A. and Machado, J. (2012), A review of power laws in real life phenomena, Communications in Nonlinear Science and Numerical Simulation, 17, 3558–3578. [13] Takens, F. (1981) Detecting strange attractors in turbulence, In: Dynamical systems and turbulence, Springer: Warwick. [14] Tenreiro Machado, J. (2012), Accessing complexity from genome information, Communications in Nonlinear Science and Numerical Simulation, 17, 2237–2243. [15] Tenreiro Machado, J., Pinto, C. and Lopes, A.M. (2013), Power law and entropy analysis of catastrophic phenomena, Mathematical Problems in Engineering, 2013. [16] Vuong, Q.H. (1989), Likelihood ratio tests for model selection and non-nested hypotheses, Econometrica: Journal of the Econometric Society, 57, 307–333. [17] Nigmatullin, R.R. (2000), Recognition of nonextensive statistical distributions by the eigencoordinates method, Physica A: Statistical Mechanics and its Applications, 285, 547–565. [18] Nigmatullin, R.R. (1998), Eigen-coordinates: New method of analytical functions identification in experimental measurements, Applied Magnetic Resonance, 14, 601–633.



Adaptive Memory Identification of Fractional Order Systems Yang Zhao, Yan Li†, Fengyu Zhou School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, PR China Submission Info Communicated by J.A.T. Machado Received 9 November 2014 Accepted 11 January 2015 Available online 1 January 2016 Keywords Fractional calculus Initialization function Initialization response Iterative learning method Identification.

Abstract This paper deals with a previously ignored problem that how to find the memory (initialization function) of fractional order system by using the recent sampled input-output data. A novel and practical strategy is proposed to estimate the initialization function, which adapts to all system parameters but fractional order. To implement this method, a P-type order learning approach is introduced to identify the system order separably and accurately, thanks to the fractional order sensitivity function. The initialization response is computed through an iterative learning identification strategy that guarantees the accuracy and adaptiveness simultaneously. Along with the estimations of order and initialization response, a practical piecewise identification criterion of initialization function is established by using the least squares and instrumental variable methods. The above strategy is available for both Caputo and Riemann-Liouville fractional order systems, where the initial values are applied rather than the initial conditions. Two illustrated examples are provided to support the conclusions. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction As a generalization of traditional calculus, fractional calculus is attracting widespread interests in various fields [1–4]. It also shows some unique characteristics by applying fractional calculus to system identification [5–9], diffusion [10], image processing [11], nonlinear modeling [12, 13] , lead-acid battery [16, 17] and control theory [14, 15] etc. Particularly, numerous investigations have proved that the fractional order model is a powerful tool to describe real dynamic systems more accurately [18–20]. For example, Stéphan et al. [8] showed that a non commensurate fractional order linear model can be accurately used to describe thermal diffusion in an aluminum rod. In order to design state of charge and state of health estimators, [21–23] fractional order approaches are proposed to get a dynamical model of a lithium-ion battery. Besides, the earliest applications of fractional calculus to viscoelasticity [26, 27], continuous time random walk [28], electrical circuit [29], etc also show advantages of using various fractional order models. In view of the present achievements on identification of fractional order systems, the results can be classified into time domain [30, 31] and frequency domain ones [32]. In time domain, the filtering based method is proposed in [33] and [34], where the output and equation errors are applied. Given a model decomposition of non-integer state-space representation, the Marquardt algorithm can be used to identify parameter matrices [35]. † Corresponding

author. Email address: [email protected] (Yang Zhao), [email protected] (Yan Li), [email protected] (Fengyu Zhou)

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.11.005

414

Yang Zhao, Yan Li, Fengyu Zhou / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 413–428

A fractional order instrumental variable algorithm was shown in [8]. Hartley et al. [36] presented the identification of fractional order systems using the concept of continuous order distribution. In [37, 38], the fractional orthogonal basis were synthesized to approximate fractional models with reduced number of parameters. Using fractional state-space models, [39] proposed several subspace identification methods which were applicable to both SISO and MIMO fractional order systems [40]. An improved Luus-Jaakola algorithm has been proposed in [41] basing on the discretization of fractional order systems. Liu et al. [42] extended the method of modulating functions to the estimation of fractional order systems. On the other hand, the identification methods in frequency domain were shown in [43, 44]. In [45], a frequency domain identification technique was used to obtain the fractional piezoelectric actuators model. ValÏrio et al. [46] extended Levy’s identification method to estimate non-commensurable fractional models from frequency response. Malti et al. [47] solved the problem of parameter estimation of fractional models in a bounded error context. A combination of Levenberg-Marquardt optimization and a quadratic logarithmic criterion based on the output error method was presented to model fractional order electrical element networks [48]. The relay feedback approach has been extended to identify fractional order plus dead time model in frequency domain [49]. Although the present identification approaches can be successfully applied to the modeling of fractional order systems, few results have addressed the problem of fractional order system with non-zero history function, which is imitated from many classical fractional order control strategies [50]. This assumption is theoretically right, but obviously unpractical or even impossible in reality. Unlike the integer order derivative, fractional order derivative is a non-local operator that it is related to all the history before the current instant [51], where the left sided definition is applied. Thus different history functions lead to different solutions [52, 53]. That is a striking feature of fractional calculus in modeling memory-dependent process, such as viscoelastic materials [54], thermal diffusion [55] and so on [56]. The fractional order system with non-zero history function belongs to the initialization issue of fractional order differential equations, which is an unmature but high-profile problem [52, 53]. In order to realize truly fractional order system modeling, a novel strategy is proposed in this paper to the identification of initialization function of initialized fractional order systems. The following of this paper is organized as: In the next section, some basic definitions of fractional order derivatives and the background of fractional order initialized systems are introduced. Section 3 presents the problem formulation. Section 4 shows an adaptive learning strategy for the estimation of initialization response, which plays a significant role for the initialization function identification in Section 5. Numerical examples are presented in Section 6 and this paper is concluded in Section 7 . 2 Preliminaries 2.1

Fractional calculus

Fractional calculus is the study of derivatives and integrals with noninteger orders, which plays an important role in modern science [51, 57, 58]. In this paper, the most two popular approaches, Riemann-Liouville and Caputo fractional order derivatives are used as our main tools. The Riemann-Liouville fractional order integral with order α ∈ (0, 1) is defined as ˆ t 1 f (τ ) RL −α f (t) = dτ , t0 Dt Γ(α ) t0 (t − τ )1−α where t0 Dt−α is the fractional integral of order α on [t0 ,t], f (t) is an arbitrary integrable function, and Γ(·) denotes the Gamma function. Moreover, for an arbitrary real positive constant β , the Riemann-Liouville and Caputo fractional derivatives are respectively defined as RL β t0 Dt

f (t) =

d[β ]+1 RL −([β ]−β +1) [ Dt f (t)], dt [β ]+1 t0


and C β t0 Dt

−([β ]−β +1)

f (t) =tRL Dt 0

[

415

d[β ]+1 f (t)], dt [β ]+1

where [β ] stands for the integer part of β , RL D and C D are Riemann-Liouville and Caputo fractional operators, respectively. In this paper f (α ) (t) denotes either the α th order Riemann-Liouville or Caputo fractional order derivative of f (t). Some physical backgrounds of fractional calculus are illustrated in [59, 60]. 2.2

Short-memory principle

Neglecting the behaviour of the function f (t) near the lower terminal (the starting point) t = t0 , the short-memory principle takes into account the behaviour of f (t) only in the interval [t − L,t], where L is the memory length, and α α t0 Dt f (t) ≈t−L Dt f (t), (t > t0 + L). If | f (t)| < M for t0 ≤ t ≤ t f , the estimate for the error introduced by the short-memory principle can be derived that ML−α , (1) |t0 Dtα f (t) −t−L Dtα f (t)| ≤ |Γ(1 − α )| where the terminal time t f ∈ [t, +∞). For t0 ≤ t ≤ t f , the memory length L can be determined by the inequality (1) providing the required accuracy ε: M )1/α . |t0 Dtα f (t) −t−L Dtα f (t)| ≤ ε , if L ≥ ( ε |Γ(1 − α )| 2.3

Fractional order initialized system

History dependence or non-localization is the nature of fractional order operators, and, in this paper, the history denotes what happened to the fractional order system before the initial time t0 , which is named as the initialization function. Moreover, memory signifies that the non-zero history function is necessarily to be considered in fractional order system identification [61, 62]. Besides, the short memory principle can be applied to implement the initialization by using finite data. Let φ (t) be the initialization function defined before the initial time instant ti . For α ∈ (0, 1), the initialized fractional integral of order α − 1 starting at time ti is defined as: α −1 f (t) ti Dt

=

α −1 f (t) + ti Dt

ˆ

ti

t0

(t − τ )−α φ (τ )d τ , Γ(1 − α )

(2)

where ti Dt1−α f (t) is called the uninitialized fractional integral starting at time t = ti , and ti is usually taken as 0. The initialized Riemann-Liouville and Caputo fractional order derivatives are defined as RL α ti Dt

C α ti Dt

f (t) =

RL α ti Dt

f (t) =

C α ti Dt

d f (t) + dt ˆ f (t) +

ti

t0

ˆ

ti

t0

(t − τ )−α φ (τ )d τ , Γ(1 − α )

(t − τ )−α dφ (τ ) dτ , Γ(1 − α ) dτ

(3)

(4)

respectively, where the right most terms in (3) and (4) are named as the initialization response of initialized Riemann-Liouville and Caputo fractional order derivatives. Particularly, we define in this paper that d Ψ(t) = dt

ˆ

ti

t0

(t − τ )−α φ (τ )d τ , Γ(1 − α )

(5)

416


or

ˆ Ψ(t) =

ti

t0

(t − τ )−α dφ (τ ) dτ . Γ(1 − α ) dτ

(6)

is the initialization response defined before the initial time instant ti for Riemann-Liouville and Caputo fractional order operators, respectively. Moreover, the Caputo fractional order initialized nonlinear system C α 0 Dt x(t)

can be rewritten as 2.4

C α 0 Dt x(t)

= f (t, x)

= f (t, x) − Ψ(t).

λ −norm

To analyze the iterative learning process, it is necessary to introduce the following λ -norm of vector-valued function e(·) defined on [0,t f ] e(·) λ = sup {e−λ t e(·) ∞ }, 0≤t≤t f

where e∞ = max | ek |. 1≤k≤N

3 Problem statement 3.1

System statement

In this paper, the following fractional order initialized linear time invariant (LTI) system is first considered: α 0 Dt x(t)

= Ax(t) + Bu(t),

t ≥ 0.

(7)

where t = 0 is the initial time instant, the initial condition is x0 = x(0), and x(t) = φ (t) for t ∈ [−L, 0− ), the double (An×n , Bn×p ) are constant matrices with fixed dimensions (i.e. n, p are constant integers known in prior), the system is controllable and observable. It can be seen from (3)-(6) that α 0 Dt x(t)

= 0 Dtα x(t) + Ψ(t),

where D denotes either Riemann-Liouville or Caputo fractional order operator. Thus equation (7) can be rewritten as α (8) 0 Dt x(t) = Ax(t) + Bu(t) − Ψ(t). where the initialization response Ψ(t) defined in (5) or (6) is only relevant with α and initialization function φ . 3.2

Adaptive order estimation

The identification of fractional order system is a very recent topic mainly regarding the continuous system estimation in frequency domain, and the most popular application is pointing to the modelling and identification of thermal system [8] and lithium-ion battery [21, 24]. Nevertheless, these methods can hardly be used in state-space representations, especially to time varying and nonlinear systems. Besides, the discrete system identification in Z-domain is available to both integral and fractional order systems, and can be further extended to some specific nonlinear cases. Particularly, the iterative learning identification (ILI) technique is hopefully a more general and applicable way to the order identification of fractional systems that is adaptive to other parameters, where the order learning law can be defined as [63, 64]:

αk+1 αk + Γk ek (Tk ), (αk+1 ∈ [0, 1]).

(9)


where denotes

417

⎧ (αk + Γk ek (Tk ) < 0), ⎨ 0, αk+1 = αk + Γk ek (Tk ), (0 ≤ αk + Γk ek (Tk ) ≤ 1), ⎩ 1, (αk + Γk ek (Tk ) > 1).

For system (7) and order learning law (9), we have lim αk = αd , if |1 − Γk Λk | ≤ 1. The selection of Tk plays k→∞

the fundamental role to improve the convergence speed of the above iterative learning algorithm, which can be determined by the order sensitivity function [65]. By applying Laplace transform to (7), it follows from the homogeneous initial conditions that X (s) = (sα − A)−1 B, (10) G(s) = U (s) where X (s) and U (s) denote respectively the Laplace transform of x(t) and u(t). Therefore, the sensitivity function can be denoted as ∂ G(s) = −(sα − A)−2 B sα ln s. ∂α which illustrates how large are the differences among g(t) (L −1 {G(s)}) with different α . Thus Tk can be ∂ g(t) determined according to the maximum of ∂ α = L −1 ∂∂ αG , as shown in Figure 1. Moreover, if the system is minimum phase and with known control direction, a small gain Γ can always guarantee the convergence condition |1 − ΓΛk | ≤ ρ < 1.

2 1.5

∂g(t) ∂α

1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −3

0

Tk

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t Fig. 1 Fractional order sensitivity function, where

∂ g(t) ∂α

= L −1 { ∂ ∂G(s) α }.

Allowing for the identification of fractional order system, this paper focuses on the estimation of initialization function φ (t) along with the estimations of order and initialization response Ψ(t). A practical identification criterion of initialization function is established by using the least squares and instrumental variable methods that are shown in the following of this paper.

418


4 Identification of Initialization Response This section discusses the application of fractional order iterative learning identification (FOILI) [66, 67] to fractional order initialized systems, and how to identify the initialization response, where the fractional order α is firstly assumed to be known from the above Section 3 and any order identification methods [68]. In the kth iteration, the FOLTI state space system (8) becomes α 0 Dt xk (t) =

Axk (t) + Bud (t) − Ψk (t),

t ∈ [0, T ],

(11)

where the desired control input ud (t) comes from either of the following strategies: • Given the desired system output xd (t), ud (t) is acquired from any adaptive FOILC scheme [66, 67] • ud (t)is an arbitrary appropriate control input, and xd (t) is the corresponding (desired) system output. Besides, the referential system is α 0 Dt xd (t)

= Axd (t) + Bud (t) − Ψd (t),

(12)

where xd (t) is the desired system output which is provided in advance. The main target of this section is to estimate the value of Ψd from the following Dβ -type learning law β

Ψk+1 (t) = Ψk (t) + Γ 0 Dt ek (t),

(0 ≤ β ≤ α )

(13)

where lim Ψk = Ψd ,

k→∞

if the learning gain satisfies the convergence condition. Lemma 1. For fractional-order system

α 0 Dt x(t)

= Ax(t) + Bu(t), y(t) = Cx(t),

and the iterative learning control scheme ⎧ ⎨ yk (t) = g1 (t) + g2 (t) ∗ uk (t), ek (t) = yd (t) − yk (t), ⎩ (β ) uk+1 (t) = uk (t) + Λ(t)ek (t), where t ∈ [0, T ], α ∈ (0, 1], k ∈ {0, 1, 2, 3, . . .}, 0 ≤ β ≤ α , g1 (t) = CEα (At α )x(0) and g2 (t) = Ct α −1 Eα ,α (At α )B. β −1 Suppose there exists a constant ρ satisfying I − [ 0 Dτ H(t)]Λ(t − τ )|τ →0+ ∞ ≤ ρ < 1, then there exist positive (β ) (β ) constants λ and ρ0 ∈ [0, 1) such that ek+1 λ ≤ ρ0 ek λ and lim yk (t) = yd (t) [66, 69, 70]. k→∞

Theorem 2. For the Caputo fractional order initialized system (11) and the Dβ −type learning law (13), we have lim Ψk = Ψd (t ∈ [0, T ]). The convergence condition for the fractional-order iterative learning scheme (14) is k→∞

β −1

I − [ 0 Dτ H(t)]Λ(t − τ )|τ →0+ ∞ ≤ ρ < 1, (0 ≤ β < α ), I + Γ∞ ≤ ρ < 1, (α = β ).


Proof. In Laplace domain,

419

X (s) = (s − A)−1 (sα −1 X (0) + BU (s) − Ψ(s)).

By applying inverse Laplace transform to the above equation, it follows that x(t) = g(t) + H(t) ∗ Ψ(t), = {Eα (At α )x0 + t α −1Eα ,α (At α ) ∗ Bu(t)} + t α −1 Eα ,α (At α ) ∗ Ψ(t), where L −1 {(s − A)−1 sα −1 X (0)} = Eα (At α )x0 , L −1 {(s − A)−1 sα −1 BU (s)} = Eα ,α (At α ) ∗ (Bu(t)), L −1 {(s − A)−1 sα −1 Ψ(s)} = Eα (At α ) ∗ Ψ(t). Therefore, the iterative learning control scheme can be expressed as ⎧ ⎨ xk (t) = g(t) + H(t) ∗ Ψ(t), ek (t) = xd (t) − xk (t), ⎩ β Ψk+1 (t) = Ψk (t) + Λ 0 Dt ek (t).

(14)

Moreover, it follows from Lemma 1 that the convergence condition is β −1

I − [ 0 Dτ

H(t)]Λ(t − τ )|τ →0+ ∞ ≤ ρ < 1.

The above result guarantees the all parameter adaptive identification of initialization response. Lastly, when α = β , the convergence condition becomes I + Λ ∞ ≤ ρ < 1. 5 Identification of initialization function Consider the Laplace transform of the fractional Riemann-Liouville derivative [52] of f (t) α L { RL 0 Dt f (t)} = L {

d dt

RL −q 0 D t f (t)},

where α = 1 − q, we have α L { RL 0 Dt f (t)} = L {

= L{

d dt

d dt

RL −q 0 Dt f (t)} + L {

RL −q 0 Dt f (t)}

d Ψ(φ , −q, −L, 0,t)}. dt

(15)

The first term on the right hand side (RHS) of equation (15) can be rewritten as: L{

d dt

RL −q 0 Dt f (t)}

= s1−q L { f (t)}.

(16)

As shown in Figure 2, φ (t) can be estimated by a piecewise function on [−L, 0− ), where the interval [−L, 0− ) is devided into N + 1 parts [−L, −t1 ), [−t1 , −t2 ), . . . , [−tN , 0− ), and the smaller the interval length, the more accurate φ (t) can be identified. The second term on RHS of equation (15) can be rewritten as :

420


ˆ − d d 0 (t − τ )q−1 L { Ψ(φ , −q, −L, 0,t)} = L { φ (τ )dτ } dt dt −L Γ(q) ˆ ˆ d −t2 (t − τ )q−1 d −t1 (t − τ )q−1 φ (τ )d τ } + L { φ (τ )d τ } + · · · =L { dt −L Γ(q) dt −t1 Γ(q) ˆ − d 0 (t − τ )q−1 +L{ φ (τ )d τ }. dt −tN Γ(q)

(17)

Suppose [−tm , −tm+1 ) denotes one piece of the time interval [−L, 0− ) , it follows that d L{ dt ˆ = sL {

ˆ

−tm+1

−tm −tm+1 −tm

(t − τ )q−1 φ (τ )d τ } Γ(q)

ˆ −tm+1 (t − τ )q−1 (t − τ )q−1 φ (τ )d τ } − [ φ (τ )d τ ]t=0+ , Γ(q) Γ(q) −tm

(18)

where τ < 0, and ˆ L{

−tm+1 −tm

(t − τ )q−1 φ (τ )d τ } = Γ(q)

ˆ

+∞

ˆ

−tm+1

(t − τ )q−1 φ (τ )d τ dt Γ(q) −tm 0 ˆ −tm+1 ˆ +∞ 1 φ (τ ) e−st (t − τ )q−1 dtd τ = Γ(q) −tm 0 ˆ −tm+1 1 = φ (τ )L{(t − τ )q−1 }d τ , Γ(q) −tm e−st

(19)

where the time-shifting property doesn’t work for τ < 0. Moreover, L {(t − τ )v } = s−v−1 e−τ s Γ(v + 1, −τ s), for | arg(−τ )| < π , Re(s) > 0,

(20)

where Γ(v + 1, −τ s) is the complementary incomplete gamma function. Moreover, (19) that ˆ L{

−tm+1 −tm

s−q (t − τ )q−1 φ (τ )d τ } = Γ(q) Γ(q)

ˆ

−tm+1 −tm

φ (τ )e−τ s Γ(q, −τ s)d τ .

Substituting the above terms back into equation (18) yields ˆ d −tm+1 (t − τ )q−1 φ (τ )d τ } L{ dt −tm Γ(q) ˆ ˆ −tm+1 (t − τ )q−1 s1−q −tm+1 −τ s φ (τ )e Γ(q, −τ s)d τ − [ φ (τ )d τ ]t=0+ . = Γ(q) −tm Γ(q) −tm Using the integration by parts, we arrive at the following equation [52]: ˆ

−tm+1 −tm

Γ(q, −τ s)e−τ s φ (τ )d τ

ˆ −tm+1 1 = q[ e−τ s Γ(q + 1, −τ s)φ (τ )d τ − etm+1 s Γ(q + 1,tm+1 s)φ (−tm+1 ) + etm s Γ(q + 1,tm s)φ (−tm )]. s −tm

(21)


421

Then an alternative form of equation (21) can be shown as ˆ ˆ −tm+1 (t − τ )q−1 d −tm+1 (t − τ )q−1 φ (τ )d τ } = −[ φ (τ )d τ ]t=0+ (22) L{ dt −tm Γ(q) Γ(q) −tm ˆ −tm+1 s−q [ φ (τ )e−τ s Γ(q + 1, −τ s)d τ ] + Γ(q + 1) −tm s−q [etm s Γ(q + 1,tm s)φ (−tm ) − etm+1 s Γ(q + 1,tm+1 s)φ (−tm+1 )], + Γ(q + 1) As φ (t) is piecewise constant on [−L, 0− ), thus the second integral part in equation (22) can be rewritten as: ˆ −tm+1 φ (τ )e−τ s Γ(q + 1, −τ s)d τ = etm s Γ(q + 1,tm s)[φ (−tm ) − φ (−tm−1 )]. (23) −tm

Substitute the above equation into (22), the Laplace transform for the derivative of integral on [−tm , −tm+1 ) is finally obtained. Thus L

RL 0

Dtα f (t) = s1−q L { f (t)} − Ψ(φ , −q, −L, −t1 ,t)|t=0+ − · · · − Ψ(φ , −q, −tN , 0,t)|t=0+

(24)

s−q

[2eLs Γ(q + 1, Ls) − et1 s Γ(q + 1,t1 s)]φ (−L) Γ(q + 1) s−q [et1 s Γ(q + 1,t1 s) − et2 s Γ(q + 1,t2 s)]φ (−t1 ) + Γ(q + 1) s−q [etN−1 s Γ(q + 1,tN−1 s) − etN s Γ(q + 1,tN s)]φ (−tN−1 ) +··· + Γ(q + 1) s−q etN s Γ(q + 1,tN s)φ (−tN ) − s−q φ (0− ). + Γ(q + 1) +

Remark 1. To avoid the conflict with initial condition, the left continuous is assumed in this case. Equation (23) will be different for various fitting strategies, such as ramp. The above discussions are summarized in the following theorem. Theorem 3. Given the estimation of the fractional order α and the initialization response ψ (t), the identification of the initialization function of the Riemann-Liouville fractional order initialized system (7) can be converted into the following matrix form ⎡ ⎤ ⎤ ⎡ θ (ta ) ρ (ta ) ⎢ ⎥ ⎢ θ (tb ) ⎥ Φ = ⎣ ρ (tb ) ⎦ (25) ⎦ ⎣ .. .. . . Θ(t) Λ(t) where

⎡

s−q [2eLs Γ(q + 1, Ls) − et1 s Γ(q + 1,t1 s)] Γ(q+1) −q s [et1 s Γ(q + 1,t1 s) − et2 s Γ(q + 1,t2 s)] L −1 Γ(q+1)

L −1

⎤T

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . θ (t) = ⎢ ⎥ , −q ⎥ ⎢ −1 s t s t s N−1 N + 1,tN s)] ⎥ ⎢L Γ(q+1) [e Γ(q + 1,tN−1 s) − e Γ(q ⎥ ⎢ ⎥ ⎢ s−q −1 tN s Γ(q + 1,t s) e L ⎦ ⎣ N Γ(q+1) t q−1 /Γ(q) T Φ = φˆ (−L) φˆ (−t1 ) · · · φˆ (−tN−1 ) φˆ (−tN ) φˆ (0− ) , α RL α ρ (t) = RL 0 Dt f (t) −0 Dt f (t) + Ψ(φ , −q, −L, −t1 ,t) |t=0+ + · · · + Ψ(φ , −q, −tN , 0,t) |t=0+ .

422


ɸW

Real Initialization function Identified initialization function

ĂĂ

ɸ(0-)

ɸ(tk)

ɸ(t2) ɸ(t1) /

W W

ĂĂ WN

ĂĂ

W

Fig. 2 A demonstration of the initialization function φ (t) which is piecewise on [−L, 0− ].

where φˆ (−tk ) denotes the estimated value of φ (t) at −tk . Proof. This conclusion can be derived by using (24). Remark 2. Given different t, equation (25) can be calculated according to the following items: • If matrix Φ is a reversible square, we have Φ(t) = Θ−1 Λ. • If matrix Φ is full column rank, employ least-square method we have Φ = (ΘT Θ)−1 ΘT Λ. • If noise properties is unknown, instrumental variable method can be applied for identification to avoid the noise effect [8]. Apply low-pass filter to Θ, the optimal instrumental variable estimate can be obtained as Θ f , then we have Φ = (ΘTf Θ)−1 ΘTf Λ. It should be noted that the uniqueness of the identified initialization is guaranteed by the full column ranks of Φ, Θ or Θ f , and the determination of L. The short memory principle is applied to compute the components of such matrices and to find out an acceptable lower bound of L. Remark 3. In matrix θ (t), the element Γ(q + 1,ts), which is complicated to calculate, can be converted by applying (20) and Laplace transform derivative property L −1 {sF(s)} = f (t) + f (0)δ (t) as follows s−q [etm s Γ(q + 1,tm s)] = s · s−q−1 · etm · Γ(q + 1,tm s)

= [(t + tm )q ] + (t + tm )q |t=0 ·δ (t) = q(t + tm )q−1 + tmq δ (t). Remark 4. For fractional derivative, there exists a link between the Riemann-Liouville and the Caputo approaches to differentiation of arbitrary real order. The exact condition of the equivalence of these two approaches are the following t −α RL α C α f (0). (26) D f (t) = D f (t) + t t 0 0 Γ(1 − α ) Combining the result (24) with the above transform (26), the Laplace transform of the fractional Caputo derivatives of f (t) can be obtained


L

C

α 0 Dt

f (t) = s1−q L { f (t)} − Ψ(φ , −q, −L, −t1 ,t)|t=0+ − · · · − Ψ(φ , −q, −tN , 0,t)|t=0+ s−q t −α f (0) + [2eLs Γ(q + 1, Ls) − et1 s Γ(q + 1,t1 s)]φ (−L) −L Γ(1 − α ) Γ(q + 1) s−q [et1 s Γ(q + 1,t1 s) − et2 s Γ(q + 1,t2 s)]φ (−t1 ) + · · · + s−q φ (0− ). + Γ(q + 1)

423

(27)

Theorem 4. Given the estimation of the fractional order α and the initialization response ψ (t), the identification of the initialization function of the Caputo fractional order initialized system (7) can be converted into the following matrix form ⎡ ⎤ ⎤ ⎡ θ (ta ) ρ˜ (ta ) ⎢ ⎥ ⎢ θ (tb ) ⎥ Φ = ⎣ ρ˜ (tb ) ⎦ ⎦ ⎣ .. .. . . Θ(t) Λ(t) where ⎡

s−q [2eLs Γ(q + 1, Ls) − et1 s Γ(q + 1,t1 s)] Γ(q+1) −q s [et1 s Γ(q + 1,t1 s) − et2 s Γ(q + 1,t2 s)] L −1 Γ(q+1)

L −1

⎤T

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ . θ (t) = ⎢ ⎥ , −q ⎥ ⎢ −1 s t s t s N−1 N + 1,tN s)] ⎥ ⎢L Γ(q+1) [e Γ(q + 1,tN−1 s) − e Γ(q ⎥ ⎢ ⎥ ⎢ s−q −1 tN s Γ(q + 1,t s) e L ⎦ ⎣ N Γ(q+1) t q−1 /Γ(q) Φ = [φˆ (−L) φˆ (−t1 ) · · · φˆ (−tN−1 ) φˆ (−tN ) φˆ (0− )]T , α RL α ρ˜ (t) = [RL 0 Dt f (t) −0 Dt f (t) + Ψ(φ , −q, −L, −t1 ,t) |t=0+ + · · · + Ψ(φ , −q, −tN , 0,t) |t=0+ +

where φˆ (−tk ) denotes the estimated value of φ (t) at −tk . Proof. This proof is the same with the derivative of (24). 6 Illustrated Examples Example 1. For the Riemann-Liouville fractional order initialized system RL α 0 Dt x(t)

= x(t),

where α = 0.5 and t0 = −4. Assume the initialization function φ (t) is a stair function as 1 , t ∈ [−4, −2), φ (t) = 2 , t ∈ [−2, 0− ). Then the initialization response can be calculated by (5) Ψ(t) =

(t + 4)−0.5 + (t + 2)−0.5 − 2t −0.5 . Γ(0.5)

t −α f (0)], Γ(1 − α )

424


It follows from the previous discussions that t0 Ψ(t) = M(t)φˆ (t0 ) + N(t)φˆ ( ), 2 where d M(t) = dt

ˆ

t0 2

(t − τ )−0.5 dτ , Γ(0.5)

t0

and the result is shown as:

φˆ (t) =

0.9937 ≈ 1 2.0015 ≈ 2

d N(t) = dt

ˆ

0− t0 2

(28) (t − τ )−0.5 dτ . Γ(0.5)

, t ∈ [−4, −2), , t ∈ [−2, 0− ),

which is very precise to the assumed values. The comparison of the identified φˆ (t) and the original φ (t) is illustrated in Figure 3.

ɸW Real Initialization function Identified initialization function

W

Fig. 3 The comparison of initialization function between the original values and the identified ones.

Example 2. For the Caputo fractional order initialized system C0 Dtα x(t) = −x(t), where α ∈ (0, 1) and t0 = −4, it can be easily derived that (29) x(t) = Eα (−(t − t0 )α )x(t0 ) − (t − t0 )α −1 Eα ,α (−(t − t0 )α ), ´ 0 τ )−α dφ (t) where the initialization response is given as Ψ(t) = − t0 (t− Γ(1−α ) dt d τ . Assume the initialization function to be identified is a step function φ (t0 ) , t ∈ [t0 , t20 ) φ (t) = φ ( t20 ) , t ∈ [ t20 , 0− ) Hence the same equation as (28) is obtained to be used for identification. As the initial value x(0) is already known, the estimated φˆ (t0 ) can be derived from (29) at t = 0,

φˆ (t0 ) = x(t0 ) =

x(0) Eα (−(−t0 )α )x(t0 ) − (−t0 )α −1 Eα ,α (−(−t0 )α )

where t0 = −4. On the other hand, φˆ ( t20 ) can be determined by Ψ(t) − M(t) ˆ t0 φˆ ( ) = φ (t0 ). 2 N(t)

,


425

7 Conclusions and Future Works This paper proposes a novel algorithm about the initialization function identification of fractional order linear time invariant system, which fills the blank of some aspect of truly fractional order system identification. It should be noted that the memory identification is related to the knowledge of order and initialization response. Consequently, the system order is first determined accurately by employing iterative learning law, and combing with the order sensitivity function. With this premise, a fractional order D-type iterative learning law is applied to identify the initialization response adaptively so that the least square method or instrumental variable method can be introduced to identify the initialization function piecewisely. The numerical examples are illustrated to support the conclusions. Our future works include the filtering, modeling, computational algorithms and parameter identifications of power batteries. Acknowledgements The authors would like to thank all Editors and Reviewers for their organizations and valuable comments. This work is supported by the National Natural Science Foundation of China (61374101,61375084,61104009). References [1] Pritz, T. (2003), Five-parameter fractional derivative model for polymeric damping materials, Joumal of Sound and Vibration, 265(5), 935-952. [2] Lee, H-H. and Tsai, C.S. (1994), Analytieal model of viscoelastic dampers for seismic mitigation of structures, Computers and Struetures, 50(1), 111-121. [3] Dumlu, A. and Erenturk, K. (2014), Trajectory tracking control for a 3-DOF parallel manipulator using fractional order PI λ Dμ control, IEEE Transactions on Industrial Electronics, 61(7), 3417-3426. [4] Gulistan, S., Abbas, M. and Syed, A.A. (2014), Fractional dual fields for a slab placed in unbounded dielectric magnetic medium. International Journal of Applied Electromagnetics and Mechanics, International Journal of Applied Electromagnetics and Mechanics, 46(1), 11-21. [5] Li Y, Zhai L, Chen Y Q, et al. (2014), Fractional-order iterative learning control and identification for fractional-order Hammerstein system, 2014 IEEE 11th World Congress on Intelligent Control and Automation (WCICA), 840-845. [6] Liu Y, Xue D, Chen Z, et al. (2014), Fractional order model identification for Electro-Active smart actuator, 2014 11th World Congress on Intelligent Control and Automation (WCICA), 3192-3195. [7] Liu X, Liang G. (2014), An Algorithm for Fractional Order System Identification, 2014 IEEE 17th International Conference on Computational Science and Engineering (CSE), 66-71. [8] Victor, S., Malti, R., Garnier, H. and Oustaloup, A. (2013), Parameter and differentiation order estimation in fractional models. Automatica, 49(4), 926-935. [9] Battaglia, J.L. , Cois, O., Puigsegur, L. and Oustaloup, A. (2001), Solving an inverse heat conduction problem using a non-integer identified model. International Journal of Heat and Mass Transfer, 44(14), 2671 2680. [10] Sun, H.G., Chen, W. and Chen, Y.Q. (2009), Variable-order fractional differential operators in anomalous diffusion modeling, Physica A: Statistical Mechanics and its Applications, 388(21), 4586 4592. [11] Bai, J. and Feng, X.C. (2007), Fractional-order anisotropic diffusion for image denoising, IEEE Transactions on Image Processing, 16(10), 2492 2502. [12] Yuan, L.G., and Yang, Q.G. (2012), Parameter identification and synchronization of fractional-order chaotic systems, Communications in Nonlinear Science and Numerical Simulation, 17(1), 305-316. [13] Vanbeylen, L. (2014), A fractional approach to identify Wiener Hammerstein systems, Automatica, 50(3), 903 909. [14] Li, Y., Chen, Y.Q., Ahn H.S. and Tian, G.H. (2013), A survey on fractional-order iterative learning control, Journal of Optimization Theory and Applications, 156(1), 127 140. [15] Jesus, I.S., and Machado, J.A.T. (2007). Application of fractional calculus in the control of heat systems, Journal of Advanced Computational Intelligence and Intelligent Informatics, 11(9), 1086-1091. [16] Cugnet, M., Sabatier, J., Laruelle, S., Grugeon, S., Sahut, B., Oustaloup, A. and Tarascon J.M. (2010), On lead-acidbattery resistance and cranking-capability estimation, IEEE Transactions on Industrial Electronics, 57(3), 909 917. [17] Sabatier, J., Cugnet, M., Laruelle, S., Grugeon, S., Sahut, B., Oustaloup, A., and Tarascon, J.M. (2010), A fractional

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The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters V.A. Kubyshkin†, S.S. Postnov V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Profsoyuznaya str., 65, 117997, Russia Submission Info Communicated by Mark Edelman Received 10 November 2014 Accepted 11 January 2015 Available online 1 January 2016 Keywords Optimal control Caputo derivative Moment method Fractional-order systems

Abstract The optimal control problem for linear dynamic systems of fractional order with lumped and distributed parameters is investigated. This problem is reduced to the classical moment problem. This paper validates the conditions making possible to formulate and resolve the obtained moment problem. Some particular cases of fractional-order systems are discussed. The explicit solutions for the problems of optimal control were obtained in case of systems with lumped parameters. In case of system with distributed parameters an approximate solution analyzed for moment problem. In particular, this paper studies the problem to minimize the norm of control for the assigned time interval and the problem of control with the minimal time of the object transition into the desirable state with the given limitation of the norm of control. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction In the recent time the investigations of the optimal control problems (OCP) for dynamic systems of fractional order have been actively progressing (see, for example, [1]– [5] and references therein). The variational approach is often used to study and resolve OCP for fractional-order systems. This approach permits to obtain the results related to the optimal control with such systems, but it faces certain difficulties when it is necessary to find discontinuous controls. Today there are no methods for resolving OCP for such systems similar to the Pontryagin maximum principle. In the past the moment method was developed and successfully applied to the systems both with lumped and distributed parameters of the integer order with control ( [6]). This method has the following advantages: (1) identifies the required and sufficient conditions of optimality; (2) identifies the unified computation procedure of search for optimal control; (3) provides conditions of optimality for discontinuous (generally, Lebesque measurable) of controls; (4) permits to take into account the limitations of the norm of control. In this work the moment method was used to investigate and resolve OCP with fractional-order systems. Similar to the systems of integral order, the method of the problem reduction to the moment problem is based on integral representation of the solution for the dynamic equation of the system in question. OCP for the linear † Corresponding


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V.A. Kubyshkin, S.S. Postnov/ Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 429–443

systems of fractional order is considered when the controls are the functions which p-integrable, 1 < p < ∞ in the finite time interval (for systems with lumped parameters) or in some space-time domain (for systems with distributed parameters). The conditions under which the formulation of OCP in the form of moment problem is possible and the conditions of the moment problem solvability are obtained. Some computational exercises for calculation of optimal controls for systems with lumped parameters are considered. This paper is organized as follows. Section 2 contains the formulation of OCP for the systems described with linear ordinary and partial differential equations of a fractional order. Section 3 contains information about the theory of moment problem and describes reduction of the problems of optimal control of fractional systems to the problem of moments. Also the theorems for setting and resolving the obtained problem of moments considered. Section 4 provides the analytical solutions of some problems of optimal control with the method of moments. Section 5 provides conclusions. 2 Statement of the optimal control problem 2.1

Systems with lumped parameters

Let vector functions q(t) = (q1 (t), . . . , qN (t)) and u(t) = (u1 (t), . . . , uN (t)) are determined for section t ∈ [0, T ]. ´T Control is u(t) ∈ L p [0, T ], 1 < p < ∞ and has the norm of u(t) = [ 0 | ∑Nk=1 uk (t)| p dt]1/p . Let us assume that if p → ∞ the space L p [0, T ] passes into space M[0, T ] (measurable and almost everywhere bounded functions). Let us take that functions qi (t), i = 1, N (or q(t)) possess all properties required for existence of solutions of equations studied further on, are, in particular, differentiable at least once. The following fractional-order system with lumped parameters is considered: C αi 0 Dt qi (t)

= ai j q j (t) + bi j u j (t) + fi (t).

(1)

Here C0 Dtαi is operator of the left-side fractional Caputo derivative (see [7]), αi ∈ (0, 1], t ∈ [0, T ], fi (t) are perturbations (known), ai j and bi j are known coefficients, i, j = 1, N. The repeated indices suppose summation. The initial and final conditions are determined as follows: q(0) = q0 = (q01 , . . . , q0N ),

(2)

q(T ) = qT = (qT1 , . . . , qTN ).

(3)

Let us take the following OCP: determine such control u(t), t ∈ [0, T ] at which the system (1) will pass from the given initial state (2) to the assigned final state (3) and herewith: 1) norm of control will be minimal with the assigned control time T (OCP A), control time T will be minimal provided u ≤ l, l > 0, where l is the assigned constant (OCP B). Remark 1. The system (1) was considered by O.P. Agrawal [1, 2] with performance index ˆ T (u2 (t) + q2 (t))dt. J[u] = 0

In our investigation (as it follows from OCP statement written above) we will consider system (1) with another performance indices which doesn’t depend on system state q(t). 2.2

Systems with distributed parameters

Let the function Q(x,t) are determined for t > 0 and x ∈ [0, L]. This function defines the system state. We will study the one-dimensional linear system described by a transfer-type equation with a fractional derivative in time: ∂ 2 Q(x,t) C α D Q(x,t) = K + f (x,t), (4) 0 t ∂ x2


431

t ≥ 0, x ∈ [0, L], f (x,t) is the known perturbation, K = const is the transfer coefficient. Let us determine the initial and boundary conditions by the following expressions: Q(x, 0+) = Q0 (x), x ∈ [0, L],

(5)

∂ Q(x,t) + a1,2 Q(x,t)]x=0,L = h1,2 (t) + u1,2 (t),t ≥ 0, (6) ∂x where ai and bi , i = 1, 2 are coefficients, functions u1,2 (t) ∈ L p [0, T ], 1 < p < ∞ define boundary controls. We will consider two cases: 1) functions u1,2 (t) are controls itself; 2) controls are fractional derivative of u1,2 (t), u˜1,2 (t) =C0 Dtα u1,2 (t). We will call hereinafter these cases as local and nonlocal control correspondingly. Final condition may be determined as follows: [b1,2

Q(x, T ) = Q∗ (x),

T > 0, x ∈ [0, L].

(7)

(t) = (u1 (t), u2 (t)) ∈ L p [0, T ] (or Controls u1,2 (t) (or u˜1,2 (t)) may be considered as components of vector U ˜ (t) = (u˜ (t), u˜ (t)) ∈ L [0, T ] correspondingly). Let us formulate OCP (as in case of lumped systems) vector U 1 2 p in the following way. OCP A. Given is the time moment t = T , T > 0. We should find controls u1,2 (t) ∈ L p [0, T ] (or u˜1,2 (t)), which pass system (4) with initial condition (5) and boundary conditions (6) into the final state (7) with the ˜ (t)). (t) (or U minimum norm of control U OCP B. Find controls u1,2 (t) ∈ L p [0, T ] (or u˜1,2 (t)) in such way that system (4) with initial condition (5) and boundary conditions (6) is passed into the final state (7) for the minimum time T with the assigned limitation ˜ (t) ≤ l), l > 0. (t) ≤ l (or U for the norm of control U 3 The moment problem 3.1

Preliminaries

As is known [6], OCP for the integer-order systems is reduced to the classical problem of moments (PM). When control vector have only one non-zero component u(t) the PM may be formulated as follows. Let we have a system of functions gi (t) ∈ L p [0, T ], i = 1, ..., N. Let we also have the assigned numbers ci , i = 1, ..., N and l > 0. We should find function u(t) ∈ L p [0, T ] that satisfies the following conditions: ˆ T gi (τ )u(τ )d τ = ci (T ), (8) 0

u(t) ≤ l,

(9)

and

1 1 + = 1. (10) p p From the theory of the moment problem it is known [6] that PM (8) with restriction (9) is equivalent to the following problem for the constrained minimum. It is necessary to find ˆ T N ˆ T N 1 p 1/p | ∑ ξi gi (t)| dt) =( | ∑ ξi∗ gi (t)| p dt)1/p = (11) min ( λN ξ1 ,...,ξN 0 0 i=1 i=1 with additional condition

N

∑ ξici = 1.

i=1

(12)

432


For correct formulation of PM (8) it is necessary and sufficient to determine the norm of functions gi (t) in space L p [0, T ]. For solvability of PM (8) it is necessary and sufficient to satisfy one of two equivalent conditions [6]: (1) λN > 0; (2) functions gi (t) are linearly independent. In case of PM solvability the optimal control may be found in the explicit form [6]. Thus, for resolving OCP A the following formula may be used: N

N

u(t) = λNp | ∑ ξi∗ gi (t)| p −1 sign[ ∑ ξi∗ gi (t)], t ∈ [0, T ]. i=1

(13)

i=1

OCP B may be resolved by the following formula:

N

N

u(t) = l p | ∑ ξi∗ gi (t)| p −1 sign[ ∑ ξi∗ gi (t)], t ∈ [0, T ∗ ], i=1

(14)

i=1

where T ∗ is the minimal non-negative real root of equation

λN (T ∗ ) = l.

(15)

Below we will demonstrate that OCP for the fractional-order systems may be reduced to PM of type (8) with modified (compared to functions and moments given for the integer-order systems) functions gi (t) and moments ci . 3.2

Systems with lumped parameters

Solution of (1) with initial condition (2) for N = 1 may be presented as follows (the subscripts of phase coordinates and controls are omitted) [7]: ˆ t bu(τ ) + f (τ ) Eα ,α [a (t − τ )α ] d τ , (16) q(t) = q0 Eα (at α ) + 1−α (t − τ ) 0 where Eα ,β (t) is two-parameter Mittag-Leffler function, Eα (t) = Eα ,1 (t). If t = T then (16) with regard to (3) may be written over as (8) with the following symbols: Eα ,α [a(T − τ )α ] , (17) (T − τ )1−α ˆ T Eα ,α [a(T − τ )α ] T 0 α f (τ )d τ . (18) c(T ) = q − q Eα (aT ) − (T − τ )1−α 0 PM (8)-(9) with regard to (17)-(18) (and the respective minimization problem (11)-(12)) may be formulated and resolved at b = 0 and p → ∞ for all α ∈ (0, 1) [8]. According to (12), it is also necessary that c(T ) = 0. In case of finite p PM may be formulated not for each α [8]. g(τ ) = b

Theorem 1. Let we have one-dimensional system (1) with initial condition (2) and final condition (3). Let u(t) ∈ L p [0, T ], 1 < p < ∞. In case of b = 0 the problem of moments (8)-(9) for the given system may be formulated and will be solvable if the following condition is satisfied:

α>

1 . p

(19)

Proof. May be received by direct calculation of norm for g(t) ∈ Lp [0, T ] and number λ (in accordance with (17) and (11)). See [8].2 Remark 2. Note that condition (19) is satisfied trivially in case of integer-order system: for such system we have α = 1 and p > 1. Remark 3. The theorem 1 can be generalized for the multi-dimensional system (1) [8].


3.3

433

Systems with distributed parameters

For system (4) the general solution is known [9, exp. (14)]. It may be written down as follows: Q(x,t) = Q∗ (x) = Q0 (x,t) + v1 (x)u1 (t) + v2 (x)u2 (t) ˆ t ∞ Eα ,α [−μn (t − τ )α ] v1n · C0 Dατ u1 (τ ) + v2n · C0 Dατ u2 (τ ) d τ , − ∑ Xn (x) (t − τ )1−α 0 n=1 where v1 (x) =

a2 (x − L) − b2 ; a2 b1 − a1 b2 − a1 a2 L

v2 (x) =

(20)

b1 − a1 x ; a2 b1 − a1 b2 − a1 a2 L

∞

Q0 (x,t) = V (x,t) + ∑ Eα [−μn t α ] [Q0n −Vn (0+) − v1n u1 (0+) − v2n u2 (0+)] Xn (x) n=1

∞

+ ∑ Xn (x)

ˆ

n=1

t 0

Eα ,α [−μn (t − τ )α ] fn (τ ) − C0 Dτα Vn (τ ) dτ , (t − τ )1−α

V (x,t) = v1 (x)h1 (t) + v2 (x)h2 (t). Here μn and Xn (x) are eigenvalues and eigenfunctions of the homogeneous Sturm-Liouville problem for (4), respectively [9]; Q0n , Vn (t), fn (t) and v(1,2)n are coefficients of expansion of functions Q0 (x), V (x,t), f (x,t) and v1,2 (x) for eigenfunctions {Xn (x)}, respectively. Let us consider the solution (20) at t = T . It’s clear that expansion for eigenfunctions {Xn (x)} exists for functions Q∗ (x), Q0 (x, T ) and v(1,2)n entering into (20). As the system of functions Xn (x) is the complete one, then in order to satisfy equation (20) it is necessary and sufficient that similar equations are satisfied for respective expansion coefficients of all values in this equation for each n. So, in case of nonlocal control (see subsec. 2.2) one can obtain from (20) the following infinite-dimensional problem of moments: ˆ

T 0

gñ (t, T ) [v1n u˜1 (t) + v2n u˜2 (t)] dt = cñ (T ), E

(21)

[− μ (T −t)α ]

where cñ (T ) = Q0n (T ) + v1n u1 (T ) + v2n u2 (T ) − Q∗n , gñ (t, T ) = α ,α(T −t)n 1−α . In case of local control an analogous infinite-dimensional problem of moments can be obtained from (20) using the integration by parts formula (see Appendix): ˆ

T 0

gn (t, T ) [v1n u1 (t) + v2n u2 (t)] dt = cn (T ),

cn (T ) = Q0n (T ) + (v1n u1 (0) + v2n u2 (0)) Eα (−μn T α ) − Q∗n , gn (t, T ) = −μn

Eα ,α [− μn (T −t)α ] (T −t)1−α

(22) = −μn gñ (t, T ).

Remark 4. Expression (20) contains Caputo fractional derivatives of functions u1,2 (t). Consequently, these functions must be differentiable [7]. In case of local control it impose additional restrictions on functions u1,2 (t), which firstly be assumed p-integrable. Note also that Caputo fractional derivative can be defined for piecewise continuous functions with denumerable number of discontinuities (such functions appears as moment problem solution). Remark 5. Expression (20) contains the values u1,2 (0) and u1,2 (T ). In general case these values must be defined from additional assumptions or restrictions. In some special cases (for example, in case of Dirichlet boundary conditions) these values can be defined from matching of initial and final conditions with boundary conditions. Also these values can be estimated using iteration technique.

434


Therefore, OCP for the distributed system (4) may be reduced to some generalized PM. It can be shown that functions gn (t, T ) and gñ (t, T ) are both linearly independent ∀α ∈ (0, 1], t ∈ [0, T ], ∀T > 0 at fixed n. Accordingly, PM (21) and (22) is solvable at each n [6]. The possibility to formulate PM for each fixed n may be studied in the way similar to that for lumped systems (and here similar conditions for index α may be obtained). But solvability and possibility of formulating the infinite-dimensional PM are the subject of a separate investigation. Theorem 2. 1. If the condition (19) satisfied, then moment problems (21) and (22) can be stated for each fixed (finite) n. 2. Moment problems (21) and (22) for each fixed n are solvable ∀α ∈ (0, 1], ∀t ∈ [0, T ], ∀T > 0. Proof. As stated above, functions gn (t, T ) and gñ (t, T ) are linearly independent. So, corresponding moment problems are solvable ∀α ∈ (0, 1], t ∈ [0, T ], ∀T > 0 at fixed n. So, it’s needed to justify a possibility of statement for these problems. Functions gñ (t, T ) differ from gn (t, T ) by constant multiplier. Consequently, it’s sufficient to carry out the calculations only for gñ (t, T ). Let us estimate the norm of functions gñ (t, T ). Correspondingly to basic properties of function norm, the following inequality valid: gñ (t, T ) ≤ Eα ,α [−μn (T − τ )α ] · (T − t)α −1 , n = 1, N. The first multiplier is bounded [7, p. 42]. The second multiplier can be calculated explicitly. At p (α − 1) = −1 we will have logarithmic singularity at upper limit:

(T − t)α −1 = (− ln |T − t|)1/p |T0 . At p (α − 1) = −1 the following expression is valid: α −1

(T − t)

(T − t)[p (α −1)+1]/p T = | ,t ∈ [0, T ]. [p (α − 1) + 1]1/p 0

This expression will be bounded at non-negative exponent. Taking into account the logarithmic singularity at upper limit at p (α − 1) = −1, we can conclude that exponent must be positive. Consequently, condition (19) must be satisfied.2 4 Resolving problems of optimal control by the moment method 4.1

One-dimensional system with lumped parameters

Let p → ∞. From (11)-(12) and (13) we may derive solution of OCP A as follows: u(t) =

a c ,t ∈ [0, T ]. b Eα (aT α ) − 1

(23)

Solution of OCP B may be presented in accordance with (11)-(12) and (14) as follows: b u(t) = l sign( ), t ∈ [0, T ∗ ], c

(24)

where T ∗ may be received from (15). It is seen that controls (23)-(24) have no one switching points. Figures 1-3 shows the dependencies between the control norm and index α in case of OCP A for T = 0.5; 1; 5 correspondingly and b = 1, q0 = 1, qT = 0. The dependencies calculated for several values of a: a = 0.1 (solid


435

||u||

2

1.5

1 0

0.2

0.4

0.6

0.8

α

Fig. 1 The dependencies between the control norm and index α at different a for T = 0.5. ||u||

1.4

1.2

1 0

0.2

0.4

0.6

0.8

α

Fig. 2 The dependencies between the control norm and index α at different a for T = 1. ||u|| 0.8

0.6

0.4

0.2 0

0.2

0.4

0.6

0.8

α

Fig. 3 The dependencies between the control norm and index α at different a for T = 5.

line), a = 0.5 (doted line), a = 0.9 (dash-doted line). It is seen that the curves in Figures 1-3 differ from each other qualitatively.

436


||u|| 1.5

1

0.5

0 0.5

0.6

0.7

0.8

0.9

α

Fig. 4 The dependencies between the control norm and index α at different a for T = 0.5. ||u||

1

0.5

0.5

0.6

0.7

0.8

0.9

α

Fig. 5 The dependencies between the control norm and index α at different a for T = 1. ||u||

1

0.5

0.5

0.6

0.7

0.8

0.9

α

Fig. 6 The dependencies between the control norm and index α at different a for T = 5.

In case of u(t) ∈ L p [0, T ] an explicit solution of minimization problem (11)–(12) can not be obtained. But


437

we can find it in quadratures. An expression for λ can be written as:

λ = u(t) = where KαT

1 c | |, KαT b

(25)

ˆ T Eα ,α (a(T − t)α ) p 1/p =[ ( ) dt] . (T − t)1−α 0

Taking into account (13) one can obtain further for OCP A: u(t) = [

b 1 c p b Eα ,α (a(T − t)α ) p −1 | |] | | sign( ). T 1− α Kα b c (T − t) c

(26)

For OCP B the solution have the following form:

u(t) = l p |

b Eα ,α (a(T ∗ − t)α ) p −1 b | sign( ), ∗ 1− α c (T − t) c

(27)

where T ∗ can be calculated from (15) using (25). The dependencies for control norm from α in case of u(t) ∈ L2 [0, T ] illustrated at figures 4-6 (the same values have been used for all parameters as in case of u(t) ∈ M[0, T ]). We can see that these curves differs from each other qualitatively and from analogous curves at figures 1-3. It also can be noted that in case of a = 0 the explicit solutions of OCP can be obtained for arbitrary p. In case of OCP A we can solve the problem (11)–(12) and derive, correspondingly to (13), the following expression for control: (p (α − 1) + 1)Γ(α ) T (q − q0 )(T − t)(p −1)(α −1),t ∈ [0, T ]. (28) u(t) = (α −1)+1 p T Analogously, in case of OCP B we can derive from (11)–(12) and (14):

u(t) =

lp |qT

− q0 | p −1 Γ p −1 (α )

(T ∗ − t)(p −1)(α −1)sign(qT − q0 ), t ∈ [0, T ∗ ],

where

(29)

p 1 |qT − q0 |Γ(α ) p (α −1)+1 ) (p (α − 1) + 1) p (α −1)+1 . (30) l Remark 6. In case of α = 1 the solutions of OCP obtained above reduce to corresponding results for the systems of first order. It can be obtained by independent analytical calculations for the last systems or by comparison with published results, particularly, with results obtained in [6]. For the double integrator of fractional order such correspondence were demonstrated in [8].

T∗ = (

4.2

One-dimensional system with distributed parameters

Let us consider the system (4) at K = 1, f (x,t) = 0 with following initial condition: Q(x, 0) = Q0 . Final state of the system we will define as:

Q(x, T ) = QT .

Let us define boundary conditions (6) in Dirichlet’s form: Q(0,t) = u(t),

438


Q(L,t) = QT . The eigenfunctions and eigenvalues for homogeneous Sturm-Liouville problem in this case can be defined by following expressions: Xn (x) = sin

μn = (

π nx , L

πn 2 ) . L

(31)

(32)

Since the functions gn (t, T ) differs from gñ (t, T ) by constant multiplier, then the moment problem (21) can be considered for both of local and nonlocal control with gñ (t, T ). In case of local control the moments in (21) will be modified: cn (T ) QT − Q0 + (−1)n QT − Q0 Eα (−μn T α ) cn (T ) = − = . μn μn In case of nonlocal control the moments will be expressed by following formula: cñ (T ) = Q0 − u(0) + (−1)n QT − Q0 Eα (−μn T α ) + u(T ) − QT . From matching of initial and boundary conditions at t = 0 one can obtain: u(0) = Q0 . Analogously, from matching of final and boundary conditions results: u(T ) = QT . Let us consider an approximate solution for moment problem (21) at finite N and N = 3, p = p = 2. Then, from (11)-(12) one can derive the following unconstrained minimization problem: 2 ˆ T 1 − ξ1 c1 − ξ2 c2 1 = min ξ g ˜ (t) + ξ g ˜ (t) + g ˜ (t) dt 1 1 2 2 3 c3 Λ23 ξ1 ,ξ2 0 = min(ξ12 I1 + ξ22 I2 + I3 + 2ξ1 ξ2 I4 + 2ξ1 I5 + 2ξ2 I6 ), ξ1 ,ξ2

(33)

where ˆ

ˆ T c1 c2 2 (g˜1 (t) − g˜3 (t)) dt, I2 = (g˜2 (t) − g˜3 (t))2 dt, I1 = c3 c3 0 0 ˆ T ˆ T 1 c1 c2 g˜23 (t)dt, I4 = (g˜1 (t) − g˜3 (t))(g˜2 (t) − g˜3 (t))dt, I3 = 2 c3 c3 c3 0 0 ˆ T ˆ T 1 c1 1 c2 (g˜1 (t) − g˜3 (t))g˜3 (t)dt, I6 = (g˜2 (t) − g˜3 (t))g˜3 (t)dt. I5 = c3 0 c3 c3 0 c3 T

Here and below we will suppose under moments cn (T ) the moments cn (T ) or the moments cñ (T ). In first case we will have optimal local control and in second case – nonlocal. Solution of the problem (33) can be written as: I1 I2 I3 + 2I4 I5 I6 − I3 I42 − I1 I62 − I2 I52 1 = . Λ23 I2 I1 − I42

(34)

From (34) and (13) one can obtain the solution for OCP A: u(t) =

I2 I1 − I42 I1 I2 I3 + 2I4 I5 I6 − I3 I42 − I1 I62 − I2 I52 c1 I4 I5 − I6 I1 c2 1 I4 I6 − I5 I2 (g˜1 (t) − g˜3 (t)) + (g˜2 (t) − g˜3 (t)) + g˜3 (t)]. ×[ 2 2 c3 c3 c3 I2 I1 − I4 I2 I1 − I4

(35)


439

||u||

1

10

0,55

0,6

0,65

0,7

0,75

0,8

0,85

0,9

0,95

α

Fig. 7 The dependencies for norm of local control from α (logarithmic scale used for ordinates). ||u||

0

10

0,55

0,6

0,65

0,7

0,75

0,8

0,85

0,9

0,95

α

Fig. 8 The dependencies for norm of nonlocal control from α (logarithmic scale used for ordinates)

The solution for OCP B can be obtained analogously. At figures 7 and 8 the dependencies of norm of control from index α represented for local and nonlocal control correspondingly. The curves calculated for different values of QT : QT = 30 (solid line), QT = 50 (doted line), QT = 100 (dash-doted line). Other parameters were fixed as following: Q0 = 10, T = 100, L = 1. It’s seen from figures 7 and 8 that curves for local and nonlocal control differ from each other qualitatively. Also it’s clear that control norm increases with value QT grows. At figures 9 and 10 the temporal dependencies shown for local and nonlocal control correspondingly. An approximate solution for (4) in case considered can be written as: Qˆ l,nl (x,t) =

N

T ∑ Qˆ l,nl n (t)Xn (x) + v1 (x)u1 (t) + v2 (x)Q ,

n=1

where 2 [(Q0 − u(0) + (−1)n (QT − Q0 ))Eα (−μn t α ) Qˆ nl n (t) = π nˆ t Eα ,α [−μn (t − τ )α ] u( ˜ τ )d τ ], − (t − τ )1−α 0

(36)

440


u( t ) 1

10

−1

10

−3

10

−5

10

0

20

40

60

80

t

Fig. 9 The temporal dependencies for local control at different α : α = 0.6 (solid line), α = 0.8 (doted line), α = 0.9 (dash-doted line) (logarithmic scale used for ordinates). ~ u( t ) −6

−10

−3

−10

0

−10

0

20

40

60

80

t

Fig. 10 The temporal dependencies for nonlocal control at different α : α = 0.6 (solid line), α = 0.8 (doted line), α = 0.9 (dash-doted line) (logarithmic scale used for ordinates).

2 [(Q0 − u(0) + (−1)n (QT − Q0 ))Eα (−μn t α ) Qˆ ln (t) = πn ˆ t Eα ,α [−μn (t − τ )α ] u( ˜ τ )d τ ]. −u(t)Eα (−μn (T − t)α ) + u(0)Eα (−μn T α ) + μn (t − τ )1−α 0 Indices ”l” and ”nl” correspond to local and nonlocal control. At figures 11-12 the final space distributions Ql,nl (x, T ) shown, calculated for different α . It’s seen that these distributions differs from each other for different values of α . Also it can be seen that nonlocal control allow to reach more accuracy of final state achieving. 5 Conclusions This paper discusses application of the method of moments to investigation of OCP for fractional-order systems with lumped and distributed parameters. The formulation and solution of OCP for linear fractional-order systems are studied. In the general case the possibility to formulate and solve the problem of moments of respective OCP for a rather broad class of problems is justified. In several particular cases the analytical solutions of OCP are


441

Ql(T,x)

30

29,95

29,9 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

x

Fig. 11 Final state for system (4) in case of local control at different α : α = 0.6 (solid line), α = 0.8 (doted line), α = 0.9 (dash-doted line). nl Q (T,x)

30,0005 30,0004 30,0003 30,0002 30,0001 30 29,9999 0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

x

Fig. 12 Final state for system (4) in case of nonlocal control at different α : α = 0.6 (solid line), α = 0.8 (doted line), α = 0.9 (dash-doted line).

obtained and investigated. The obtained results may be useful for design of fractional-order control systems and calculation of optimal controls. References [1] Agrawal, O.P. (2004), A General Formulation and Solution Scheme for Fractional Optimal Control Problems, Nonlinear Dynamics, 38, 323–337. [2] Agrawal, O.P. (2008), A Formulation and Numerical Scheme for Fractional Optimal Control Problems, Journal of Vibrations and Control, 14(9–10), 1291–1299. [3] Caponetto, R., Dongola, G., Fortuna, L. and Petras, I. (2010), Fractional Order Systems. Modelling and Control Applications, World Scientific, Singapore. [4] Frederico, G.S.F. and Torres, D.F.M. (2008), Fractional Optimal Control in the Sense of Caputo and the Fractional Noether’s Theorem, International Mathematical Forum, 3(10), 479–493. [5] Monje C.A., Chen, Y.Q., Vinagre, B.M., Xue, D. and Feliu, V. (2010), Fractional-order Systems and Controls: Fundamentals and Applications, Springer-Verlag, London. [6] Butkovskiy, A.G. (1969), Distributed Control Systems, American Elsevier, New York. [7] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and applications of fractional differential equations,

442


Elsevier, Amsterdam. [8] Kubyshkin, V.A. and Postnov, S.S. (2014), Optimal control problem for linear stationary system of fractional order in form of problem of moments: statement and investigation, Automation and Remote Control, 75(5), 805–817. [9] Tomovski, Z. and Sandev, T. (2013), Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions, Nonlinear Dynamics, 71, 671–683. [10] Agrawal, O.P. (2007), Fractional Variational Calculus in Terms of Riesz Fractional Derivatives, Journal of Physics A: Mathematical and Theoretical, 40, 6287–6303.

APPENDIX

The following integration by parts formula is valid [10]: ˆ

T

0

f (t) · C0 Dtα g(t)dt

ˆ =

T 0

[α ]

{α }+ j−1

g(t) · tRL DαT f (t)dt + ∑ [tRL DT j=0

[α ]− j

f (t) · tRL DT

g(t)]|T0 .

Here tRL DαT is operator of the right-side Riemann-Liouville derivative; t ITα is operator of the right-side RiemannLiouville integral. Using this formula one can obtain from (21): ˆ cñ (T ) = =

T

0 ˆ T 0

ˆ gñ (t, T )[v1n u˜1 (t) + v2n u˜2 (t)]dt =

0

T

gñ (t, T )[v1n · C0 Dtα u1 (t) + v2n · C0 Dtα u2 (t)]dt

[v1n u1 (t) + v2n u2 (t)] · tRL DαT gñ (t, T )dt + [[v1n u1 (t) + v2n u2 (t)] · tRL IT1−α gñ (t, T )]|T0 .

(A1)

Using definitions of right-side operators [7] and expression for gñ (t, T ) the following expression can be written: RL α t DT gñ (t, T ) =

−

ˆ

d 1 Γ(1 − α ) dt

RL 1−α gñ (t, T ) = t IT

1 Γ(1 − α )

t

ˆ t

T

T

(τ − t)−α (T − τ )α −1Eα ,α [−μn (T − τ )α ]d τ ,

(τ − t)−α (T − τ )α −1 Eα ,α [−μn (T − τ )α ]d τ .

Integral in right side of these expressions can be calculated through representation of Mittag-Leffler function in form of power series [7]. Since the series is convergent at number axis, we can permute integration and summation operators. Consequently, following expression is valid: ˆ

T

(τ − t)−α (T − τ )α −1 Eα ,α [−μn (T − τ )α ]d τ ˆ T (−μn )k × = ∑ (τ − t)−α (T − τ )α (k+1)−1d τ Γ[ α (k + 1)] t k=0 t ∞

=

∞

(−μn )k

∑ Γ[α (k + 1)] (T − t)α k B(1 − α , α (k + 1))

k=0

= Γ(1 − α )Eα [−μn (T − t)α ]. Let us calculate the derivative of expression obtained: 1 ∞ [−μn (T − t)α ]k α k d Γ(1 − α )Eα [−μn (T − t)α ] = ∑ Γ(α k + 1) dt T − t k=0 =

∞ [−μn (T − t)α ]k 1 [−μn (T − t)α ]k α k [ |k=0 + ∑ ] T −t Γ(α k + 1) Γ(α k) k=1


443

The first term in square brackets is zero and second one can be re-written in a following way: ∞ [−μn (T − t)α ]k α k [−μn (T − t)α ]m+1 = ∑ Γ(α k + 1) ∑ Γ(α m + α ) = −μn (T − t)α Eα ,α [−μn(T − t)α ] m=0 k=1 ∞

So, we will have:

RL α t DT gñ (t, T )

= −μn (T − τ )α −1Eα ,α [−μn (T − τ )α ] ,

RL 1−α gñ (t, T ) t IT

= Eα [−μn (T − τ )α ] .

Substituting these expressions in (A1), we can obtain the following formula: ˆ

T 0

gn (t, T )[v1n u1 (t) + v2n u2 (t)]dt = cn (T ),

where cn (T ) = Q0n (T ) + (v1n u0 (0) + v2n u1 (0)) Eα (−μn T α ) − Q∗n , gn (t, T ) = −μn

Eα ,α [− μn (T −t)α ] (T −t)1−α

= −μn gñ (t, T ).



Sliding Mode Control of Fractional Lorenz-Stenflo Hyperchaotic System Jian Yuan†, Bao Shi Institute of System Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, 264001, China Submission Info Communicated by Mark Edelman Received 17 September 2014 Accepted 17 April 2015 Available online 1 January 2016 Keywords Fractional calculus Fractional hyperchaotic system Sliding mode control

Abstract This paper proposes sliding mode control for the 4-D fractional order Lorenz-Stenflo hyperchaotic system. Two methods are utilized: one is based on the frequency distributed model of fractional integral operator; and the other is based on the Mittag-Leffler stability theorem and the Caputo operator property. Both of the two methods involve two steps: firstly, constructing a fractional order sliding surface; secondly, designing a single sliding control law for suppression of the nominal plant. Numerical simulations are carried out to verify the efficiency of the theoretical results.


1 Introduction Fractional calculus dates back to the end of 17th century, though the subject only really came to life over the last few decades [1]. Fractional integrals and derivatives have a strong position and bring practical results in providing a powerful instrument of describing memory and hereditary properties of different substances [2] and in permitting a simple and more adequate representation of some high-order complex integer systems [3, 4]. In the last three decades, there have been published broad surveys on applications of fractional calculus in physics, chemistry, material, engineering, finance and even social science [1]. Recently, some fractional nonlinear systems are discovered to exhibit chaos or hyperchaos [3, 5]. Particularly, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. Linear state feedback control is designed in [6-9], nonlinear feedback control in [10-14], fractional PID control method in [15, 16], and active control technique in [17]. To deal with modeling inaccuracies and external noises which are unavoidable in real world application, fractional sliding mode control is proposed in [18-25]. Nevertheless, most of the recent contributions are devoted to 3-D chaotic systems. There’re few results on control of 4-D fractional chaotic systems. In this paper, we are about to investigate suppression of the 4-D fractional Lorenz-Stenflo hyperchaotic system via sliding mode control technique. For this purpose, we utilize two methods: one is based on the frequency distributed model of fractional integral operator; and the other is † Corresponding


446

Jian Yuan, Bao Shi/ Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 445–455

based on the Mittag-Leffler stability theorem and the Caputo operator property. Both of the two methods involve two steps: firstly, constructing a fractional sliding surface; secondly, designing a single sliding control law for suppression of the nominal plant. The rest of the paper is organized as follows: Section 2 presents some definitions of and Lemmas for fractional calculus. Section 3 formulates the mian problem of this paper. Section 4 proposes the sliding mode control design for the fractional Lorenz-Stenflo hyperchaotic system via the frequency distributed model of fractional integral operator. Section 5 proposes another daptive sliding control design via the Mittag-Leffler stability theorem and the Caputo operator property. Numerical simulations are presented to show the effectiveness of the proposed method in section 6. Finally, the paper is concluded in section 7. 2 Basic definitions and preliminaries In this section we recall three basic definitions and four lemmas in the fractional calculus. The most commonly used definitions of fractional derivatives are Grünwald-Letnikov, Riemann-Liouville, and Caputo definitions [2,5]. Definition 1. The Grünwald-Letnikov derivative definition of order α is described as α a Dt

1 f (t) = lim α h→0 h

[ t−a h ]

∑

(−1) j

j=0

α f (t − jh), j

where h is the time step. For binomial coefficients calculation we can use the relation between Euler’s Gamma function and factorial, defined as Γ (α + 1) α! α = = j j! (α − j)! Γ ( j + 1)Γ (α − j + 1) for

α = 1. 0

Definition 2. The Riemann-Liouville derivative of order α is defined as ˆ dn t f (τ ) dτ 1 α D f (t) = a t Γ (n − α ) dt n a (t − τ )α −n+1 for n − 1 < α < n, where Γ (·) is Euler’s Gamma function. Definition 3. The Caputo definition of fractional derivative can be written as α a Dt

1 f (t) = Γ (n − α )

ˆ a

t

f (n) (τ )dτ

(t − τ )α −n+1

for n − 1 < α < n. Lemma 1 (The continuous frequency distributed model [26]). The fractional integrator 0 Dt−α , 0 < α < 1 is a linear frequency distributed system, with input v (t) and output x (t). Its frequency distributed state z (ω ,t) verifies the differential equation (for the elementary frequency ω )

∂ z (ω ,t) = −ω z (ω ,t) + v (t) ∂t

(1)


447

and the output x (t) of the fractional integrator is the weighted integral (with weight μ (ω )) of all the contributions z (ω ,t) ranging from 0 to ∞: ˆ ∞ μ (ω ) z (ω ,t) dω (2) x (t) = 0

sin(απ ) −α ω . π

with μ (ω ) = The relations (1) and (2) define the frequency distributed model of the fractional integrator. m

m

i=1

i=1

Lemma 2 ((see [27])). Let W1 = ∑ W1i , W2 = ∑ aiW2i with W1i =

´∞ 0

μi (ω ) ω z2i d ω , W2i = x2i , i = 1, 2, . . . , m.

Then the quadratic form W = W1 +W2 is positive semidefinite if ai ≥ 0 and is positive definite if ai > 0. Lemma 3 (Mittag-Leffler stability [28]). Let us consider the following nonlinear fractional differential system C q t0 Dt x (t)

= f (x,t) ,

with the initial condition x0 = x (t0 ), where f : [t0 , ∞) × Ω → Rn is piecewise continuous in t and Ω ⊂ Rn is a domain that contains the equilibrium point xeq = 0 and 0 < q < 1. We assume there exists a unique solution x (t) ∈ C1 [t0 , ∞) to the above system with the initial condition x0 = x (t0 ). Let xeq = 0 be an equilibrium point of the above system and D ⊂ Rn be a domain containing the origin. Let V (t, x (t)) : [0, ∞) × D → R+ be a continuously differentiable function and locally Lipschitz with respect to x such that α1 xa ≤ V (t, x (t)) ≤ α2 xab , C β t0 Dt V

(t, x (t)) ≤ −α3 xab ,

where t ≥ t0 , 0 < β < 1, x ∈ D, α1 , α2 , α3 , a and b are arbitrary positive constants. Then xeq = 0 is Mittag-Leffler stable. If the assumptions hold globally on Ω ⊂ Rn ,then xeq = 0 is globally Mittag-Leffler stable. Lemma 4 (Caputo operator property [29]). Let x (t) ∈ R be a continuous and derivable function. Then, for any time instant t ≥ t0 , 1C q 2 D x (t) ≤ x (t) Ct0 Dtq x (t) , ∀q ∈ (0, 1) . 2 t0 t 3 Problem formulation The fractional Lorenz-Stenflo hyperchaotic system is described by fractional differential equations ⎧ C Dq1 x = α (x − x ) + γ x , ⎪ ⎪ 2 1 4 0 t 1 ⎪ ⎪ ⎨C Dq2 x = x (r − x ) − x , 1 3 2 0 t 2 C Dq3 x = x x − β x , ⎪ 1 2 3 ⎪ 0 t 3 ⎪ ⎪ ⎩C Dq4 x = −x − α x , 1 4 0 t 4

(3)

where x1 , x2 , x3 , and x4 are state variables; α , β , γ , and r are positive known constants;q1 , q2 , q3 , andq4 ∈ (0, 1] are the fractional orders of the system. If the values of the fractional orders are all the same, the fractional dynamic system is called a commensurate-order system; otherwise it is an incommensurate-order system. In terms of the initialization method described in [30], the initial conditions for fractional differential equations with order lying in (0, 1) is the constant function of time. So the initial conditions for the fractional LorenzStenflo hyperchaotic system are chosen as x1 (t) = x1 0+ = 2.8,

448


x2 (t) = x2 0+ = 2, x3 (t) = x3 0+ = 3, x4 (t) = x4 0+ = 4, for −∞ ≤ t ≤ 0. With parameters α = 1, β = 0.7, γ = 1.5, r = 26 and fractional orders q1 = q2 = q3 = q4 = 0.96, the fractional Lorenz-Stenflo hyperchaotic system exhibits hyperchaotic behavior, which is shown in Fig.1 and Fig.2.

40

x3(t)

30 20 10 0 40 10

20

5 0

0

−5

−20

x2(t)

−10

x1(t)

Fig. 1 strange attractor of the fractional Lorenz-Stenflo hyperchaotic system (3) projected onto x1 − x2 − x3 plane with parameters α = 1, β = 0.7, γ = 1.5, r = 26, fractional orders q1 = q2 = q3 = q4 = 0.96, and simulation time 200 s.

4 3 2

x4(t)

1 0 −1 −2 −3 −4 −8

−6

−4

−2

0

2

4

6

8

10

x1(t)

Fig. 2 strange attractor of the fractional Lorenz-Stenflo hyperchaotic system (3) projected onto x1 − x4 plane with parameters α = 1, β = 0.7, γ = 1.5, r = 26, fractional orders q1 = q2 = q3 = q4 = 0.96, and simulation time 200 s.


449

In this paper, our main objective is to design a suitable control u (t) to suppress the fractional Lorenz-Stenflo hyperchaotic system (3). A single control input u (t) is added to the first equation of system (3), so it is rewritten as ⎧ C Dq1 x = α (x − x ) + γ x + u (t) , ⎪ ⎪ 2 1 4 0 t 1 ⎪ ⎪ ⎨C Dq2 x = x (r − x ) − x , 1 3 2 0 t 2 q 3 C ⎪ ⎪ 0 Dt x3 = x1 x2 − β x3 , ⎪ ⎪ ⎩C Dq4 x = −x − α x . 1 4 0 t 4 It’s worth noting that the control input u (t) is a scalar function, which will be determined using fractional sliding control technique in the following. 4 Sliding control design via frequency distributed model In this section, we aim at proposing sliding control design for the incommensurate-order Lorenz-Stenflo hyperchaotic system based on the frequency distributed model of the fractional integral. To this end, a fractional order sliding surface will be firstly constructed; then the corresponding sliding dynamics will be proved to be stable; and finally a desired single control will be derived through a Lyapunov candidate. 4.1

Sliding surface design

A fractional order sliding surface is constructed as s(t) = C0 Dtq1 −1 x1 (t) + D−1 ϕ (t),

(4)

where ϕ (t) is a smooth function to be determined later. Taking time derivative of s (t) gives s(t) ˙ = C0 Dtq1 x1 (t) + ϕ (t). ics

(5)

By virtue of the equivalent control method in [31], letting s(t) ˙ = 0 yields the following sliding mode dynam⎧ C Dq1 x (t) = −ϕ (t), ⎪ ⎪ 1 0 ⎪ ⎪ ⎨C Dq2 x = x (r − x ) − x , 1 3 2 0 t 2 C Dq3 x = x x − β x , ⎪ 1 2 3 ⎪ 0 t 3 ⎪ ⎪ ⎩C Dq4 x = −x − α x . 1 4 0 t 4

(6)

In terms of the continuous frequency distributed model of the fractional integrator in Lemma 1, the sliding mode dynamics (6) is exactly equivalent to the following infinite dimensional ODEs: ⎧ ∂ z1 (ω ,t) ⎪ ⎪ = −ω z1 (ω ,t) − ϕ (t) , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎨ ∂ z2 (ω ,t) = −ω z2 (ω ,t) + x1 (r − x3 ) − x2 , ∂t (7) ∂ z3 (ω ,t) ⎪ ⎪ = − ω z ( ω ,t) + x x − β x , ⎪ 3 1 2 3 ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂ z4 (ω ,t) = −ω z4 (ω ,t) − x1 − α x4 , ∂t with ˆ ∞ μi (ω ) zi (ω ,t) d ω , (8) xi (t) = 0

450


and

μi (ω ) =

sin (qi π ) −qi ω , π

i = 1, 2, 3, 4. In the continuous frequency distributed model (7) and (8), zi (ω ,t) are named true state variables, whereas xi (t) are named pseudo state variables, i = 1, 2, 3, 4. In the sequel, we determine ϕ (t)to ensure the stability of the sliding mode dynamics (6) or (7). For this purpose, we choose the Lyapunov function candidate for (7) 1 4 V1 (t) = ∑ 2 i=1

ˆ

∞ 0

μi (ω ) z2i (ω ,t) d ω .

Taking time derivative of V1 (t) gives 4

V˙1 (t) = ∑

ˆ

∞

μi (ω ) zi (ω ,t)

i=1 0

∂ zi (ω ,t) dω . ∂t

(9)

Substituting (7) into (9) yields V˙1 (t) =

ˆ

∞

ˆ

∞

μ2 z2 [−ω z2 + x1 (r − x3 ) − x2 ] d ω ˆ ∞ ˆ ∞ + μ3 z3 (−ω z3 + x1 x2 − β x3 ) d ω + μ4 z4 (−ω z4 − x1 − α x4 ) d ω 0 0 ˆ ∞ ˆ ∞ ˆ ∞ = −ϕ (t) μ1 z1 d ω + [x1 (r − x3 ) − x2 ] μ2 z2 d ω + (x1 x2 − β x3 ) μ3 z3 d ω 0

μ1 z1 [−ω z1 − ϕ ]d ω +

0

+ (−x1 − α x4 )

ˆ 0

∞

0

4

μ4 z4 d ω − ∑

ˆ

0

∞

i=1 0

(10)

0

μi ω z2i d ω .

Substituting (8) into the three integral terms of (10), one derives V˙1 (t) = −ϕ x1 + [x1 (r − x3 ) − x2 ] x2 + (x1 x2 − β x3 ) x3 4 ˆ ∞ + (−x1 − α x4 ) x4 − ∑ μi ω z2i d ω i=1 0

4

= x1 [−ϕ + rx2 − x4 ] − x22 − β x23 − α x24 − ∑

ˆ

i=1 0

∞

μi ω z2i d ω .

(11)

We choose

ϕ (t) = x1 + rx2 − x4 , then (11) becomes

4

V˙1 (t) = −x21 − x22 − β x23 − α x24 − ∑

ˆ

i=1 0

∞

μi ω z2i d ω .

In view of Lemma 2, V˙1 (t) is negative definite providing that α , β are all positive, whereas V˙1 (t) is negative semi-definite providing that α , β are all non-negative. Hence, the sliding mode dynamics (6) is globally asymptotically stable if α , β are all positive and is globally stable if α , β are all non-negative. Up to this point, we determine a stable sliding mode surface (4) for the fractional Lorenz-Stenflo hyperchaotic system (3). In the following section, we propose control design based on this fractional sliding mode surface.


4.2

451

Sliding control design

To investigate suppression of the nominal plant (3) via sliding control design, we add a single sliding control to the first equation of (3) and choose a Lyapunov candidate as 1 V2 (t) = s2 . 2 By taking its derivative with respect to time yields V˙2 (t) = ss˙ = s C0 Dtq1 x1 + ϕ = s [(1 − α ) x1 + (α + r) x2 + (γ − 1) x4 + u] ,

(12)

where s is the fractional sliding mode surface (4). Then the control law is constructed as u (t) = (α − 1) x1 − (α + r) x2 − (γ − 1) x4

(13)

−k1 sgn (s) − k2 s, where k1 , k2 are known strictly positive constants and their values will be chosen in numerical simulations. Substituting the control law (13) into (12), it follows that V˙2 (t) = s [−k1 sgn (s) − k2 s] = −k1 |s| − k2 s2

= −k1 2V2 − 2k2V2 . The strictly positive constants k1 and k2 implies that V˙2 (t) is negative definite and that the Lyapunov candidate V2 (t) tends to zero in a finite time. The same holds for the sliding surface (4). Further, the finite time vanishing of the sliding surface guarantees that all the states of the fractional Lorenz-Stenflo hyperchaotic system (3) will tend globally and asymptotically to zero. Up to this point, it has been proved that the fractional system (3) can be suppressed via the proposed sliding mode control law (13). 5 Sliding control design via Mittag-Leffler stability theorem In this section, we use another approach to propose the sliding control design for the fractional Lorenz-Stenflo hyperchaotic system. We consider the commensurate fractional system,i.e., q1 = q2 = q3 = q4 = q. Firstly , we construct another fractional sliding surface s2 (t) = x1 (t) + C0 Dt−q ϕ (t).

(14)

Taking q − th order derivative of s2 (t) yields C q 0 Dt s2 (t)

q

= C0 Dt x1 (t) + ϕ (t).

By virtue of the equivalent control method in [31], letting C0 Dtq s2 = 0 gives the following sliding dynamics ⎧ C q ⎪ ⎪ ⎪0 Dt x1 (t) = −ϕ (t), ⎪ ⎨C Dq x = x (r − x ) − x 1 3 2 0 t 2 (15) C Dq x = x x − β x , ⎪ 3 1 2 3 ⎪ t 0 ⎪ ⎪ ⎩C Dq x = −x − α x . 1 4 0 t 4

452


To propose the stability analysis for system (15), we choose the Lyapunov candidate as V3 =

1 4 2 ∑ xi . 2 i=1

Taking q − th order derivative of V3 and using the property for Caputo derivative in Lemma 4, one derives C q 0 Dt V3

3

≤ ∑ xiC0 Dtq xi i=1

= −x21 − x22 − β x23 − α x24 ≤ −η V3 ,

(16)

where η = 2 min {1, α , β }. In terms of the Mittag-Leffler stability in Lemma 3, from inequality (16) one derives the Mittag-Leffler stability of the sliding dynamics (15), which also implies that the sliding dynamics (15) is asymptotically stable. Next, we propose the sliding control design. The Lyapunov candidate is selected as 1 V4 (t) = s2 2 . 2 By taking its derivative with respect to time yields C q 0 Dt V4 (t)

q

≤ s2 (t)C0 Dt s2 (t) = s2 [(1 − α ) x1 + (α + r) x2 + (γ − 1) x4 + u] .

(17)

Then the control law is constructed as u (t) = (α − 1) x1 − (α + r) x2 − (γ − 1) x4 − k1 sgn (s2 ) − k2 s2 .

(18)

Then inequality (17) becomes C q 0 Dt V4 (t)

≤ s2 [−k1 sgn (s2 ) − k2 s2 ]

= −k1 2V4 − 2k2V4 .

(19)

In virtue of the Mittag-Leffler stability theorem of lemma 3, the above inequality (19) indicates that the fractional system (3) can be suppressed via the proposed sliding mode control law (18). Remark: In Section 4, the control law (13) was proposed for the incommensurate-order Lorenz-Stenflo hyperchaotic system; whereas in Section 5, the control law (18) was designed for the commensurate-order one. It is found that the two control laws are the same to each other, though they are obtained through two different methods. 6 Numerical simulations In this section, we apply the fractional sliding mode control method proposed in Section 2 to deal with the problem of suppression of the fractional Lorenz-Stenflo hyperchaotic system. To carry out numerical simulations, we utilize the algorithm for numerical calculation of fractional derivatives introduced in [5]. This method is derived from the Grünwald-Letnikov definition based on the fact that the three definitions– the GrünwaldLetnikov definition, the Riemann-Liouville definition and the Caputo definition –are equivalent for a wide class of functions. Numerical simulations of suppression of the fractional Lorenz-Stenflo hyperchaotic system (3) are shown in Fig. 3 and Fig. 4, with k1 = 0.05 and k2 = 0.1. Fig. 3 displays the state responses, whereas Fig. 4 shows the dynamics of the sliding surface (4) and the control input (13). One can see from Fig. 3 and Fig. 4 that the control laws (13) and (18) are efficient for suppressing the fractional Lorenz-Stenflo hyperchaotic system.


4

4 2 x2(t)

x1(t)

2 0

0

−2 −4

0

2

4 Time(sec)

6

8

−2

3

0

2

4 Time(sec)

6

8

0

2

4 Time(sec)

6

8

4 3 x4(t)

x3(t)

2 1 0

453

2 1

0

2

4 Time(sec)

6

0

8

Fig. 3 state responses of the controlled fractional Lorenz-Stenflo hyperchaotic system.

40

s(t) u(t)

20

0

−20

−40

−60

−80

0

0.5

1 Time(sec)

1.5

2

Fig. 4 the dynamics of the sliding surface (4) and the control input (13).

7 Conclusions In this paper, we have proposed two kinds of sliding mode control to suppress the 4-D fractional Lorenz-Stenflo hyperchaotic system. One is based on the frequency distributed model of fractional integral operator; and the other is based on the Mittag-Leffler stability theorem and the Caputo operator property. Numerical simulations have also been carried out to verify the efficiency of the theoretical results.

454


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Hybrid Projective Synchronization in Mixed Fractional-order Complex Networks with Different Structure Li-xin Yang, Jun Jiang†, Xiao-jun Liu State Key Laboratory for Strength and Vibration, Xi’an Jiaotong University, Xi’an, 710049, China Submission Info Communicated by Mark Edelman Received 10 November 2014 Accepted 29 December 2014 Available online 1 January 2016 Keywords Fractional-order nodes Hybrid projective synchronization Different orders Stability theory

Abstract In this paper, a fractional-order drive-response complex network model with different order nodes is proposed for the first time. To achieve the hybrid projective synchronization (HPS) of drive-response complex network with different orders, a general strategy is proposed and effective controllers for hybrid projective synchronization are designed. The fractional operators are introduced into the controller to transform problem into synchronization problem between drive-response complex network with identical orders. Numerical simulation results which are carried show that the method is easy to implement and reliable for synchronizing the driveresponse fractional-order complex networks. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction As is well known, complex networks permeate every aspect of our daily lives, existing in various fields of the real world, such as social networks, biological networks, organizational networks, neural networks, and many others [1-6]. In fact, most networks can be modeled by complex dynamical networks, in which each node is a nonlinear dynamical system. Synchronization is an important dynamical process on complex networks with wide applications. It is a fundamental phenomenon that enables coherent behaviors in networks as a result of interactions. Up to now, and many types of synchronization phenomena have been reported, such as complete synchronization [7], antisynchronization [8], impulsive synchronization [9], projective synchronization (PS) means that the drive and response systems can be synchronized up to a scaling factor. PS was first reported by Mainieri and Rehacek [10] in partially linear systems. Later, Wu et al. put forward another new PS called hybrid projective synchronization (HPS) [11]. HPS is characterized by master and slave systems that can be synchronized up to a scaling matrix. It should be mentioned that, like the studies on the synchronization of integer-order complex network, the HPS in fractional-order complex networks is still the dominant one among the synchronization research. Although there are many results about synchronization of complex networks, most work have been devoted to the synchronization research complex networks of same orders, namely, the drive-response complex network are both of † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.11.008

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Li-xin Yang, Jun. Jiang, Xiao-jun Liu / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 457–465

integer order or both of identical fractional orders [12]. Therefore, the synchronization study of fractional-order complex networks with non-identical orders is a new subject in the research field of chaos synchronization, no reports relevant to this topic have been present to the best of our knowledge. The organization of this paper is as follows: The definitions and some properties for the fractional calculus and the stability of fractional-order systems are given in Sect.2. In Sect.3, based on the stability theory of the fractional-order system, appropriate controllers are developed for synchronize drive-response complex network by using a scaling matrix. Sect.4 shows the effectiveness of the approach for the extensive simulation studies. Conclusions are drawn in Sect.5. 2 Fractional-order derivative and preliminaries 2.1

Definition

There exist many kinds of definitions for the fractional derivative [13]. Two of which most frequently used ones are: the Riemann-Liouville and the Caputo definitions. The difference between the two definitions is in the order of evaluation, where Riemann–Liouville fractional derivative of order α ≥ 0 of the function f (t) is defined as ˆ t 1 (t − τ ) f (τ )d τ , t > 0. (1) J q f (t) = Γ(q) 0 Caupto definition of the fractional derivative of the function f (t) is defined as: ˆ t 1 f (m) (τ ) α f (τ )d τ , t > 0. D∗ f (t) = Γ(m − α ) 0 (t − τ )α −m+1

(2)

Formula (2) will be used in this paper where α is the fractional order, m is an integer that satisfies m − 1 < α ≤ m, m ∈ N,t > 0, f (m) is the ordinary mth integer derivative of f , Γ(·) is the Gamma function. ˆ ∞ t s−1 e−t dt. (3) Γ(s) = 0

It is well known that the initial conditions for the fractional differential equations with Caputo derivatives take on the same form as those for the integer-order ones, which is very suitable for practical problems. Comparing these two formulas, one easily arrives at a fact that Caputo derivative of a constant is equal to zero,which is not the case for the Riemann-Liouville derivative. Therefore, the Caputo definition for the fractional derivatives is used in this paper. 2.2

Basic properties of fractional calculus (1) Fractional calculus is linear operation: Dq [a f (t) + bg(t)] = aDq f (t) + bDq g(t)

(4)

(2) If q ≥ 0, the following equation holds: Dq D−q f (t) = D0 f (t) = f (t)

(5)

(3) Suppose f has a continuous lth derivative on [0,t](l ∈ N,t > 0), and let α , β > 0 be such that there exists p ∈ N with p ≤ l and α + β ∈ [p − 1, p] .Then Dα Dβ f = Dα +β f In this paper, we consider the case that α , β ∈ (0, 1] and α + β ∈ (0, 1].

(6)


2.3

459

Stability theorems of fractional-order system

In this subsection, some important results of the stability theorems for fractional-order systems are given . Lemma 1. (Ref.16) For a given autonomous fractional-order differential equation Dq x = Ax,

with x(0) = x0

(7)

where x ∈ Rn is the state vector, A ∈ Rn×n is a constant matrix with eigenvalues λ1 , λ2 , . . . , λn , the fractional order q ∈ (0, 1), system (7) is asymptotically stable, if and only if, |arg(λi )| > q2π , i = 1, 2, . . . , n. As a consequence, lim x(t) = 0 for all x0 ∈ Rn .

t→∞

3 Synchronization analysis Before presenting the main results of this paper, the following lemmas are introduced. Lemma 2. (Ref. 14) Let A is an n × n matrix and B is an m × m matrix, x1 = [x11 , x12 , . . . , x1n ] ∈ Rn×n and y1 = [y11 , y12 , . . . , y1m ] ∈ Rm×m be the matrices whose columns are the eigenvectors of A and B, respectively. Let T = A ⊗ Im + In ⊗ B is a nm × nm matrix. The eigenvectors of the T have the form zi = ψi (x1 ⊗ y1 )i = 1, 2, . . . , mn, where ψi (M) is the operator that selects the ith column of matrix M. The eigenvalues of T have the form τi = νk + μ j , where i = 1, 2, . . . , mn, k = [i/n] + 1 and j = i + n − kn. (where [·] is the floor operator) Lemma 3. (Ref. 15) All the eigenvalues of a diffusive matrix P have non-positive real parts, and 0 is an eigenvalue of matrix P. If P is irreducible, 0 is an eigenvalue with multiplicity one. Lemma 4. For any two complex number ε1 , ε2 such that π ≥ arg(ε1 ) > α1 , π ≥ arg(ε2 ) > α2 , the inequality |arg(ε1 + ε2 )| > min(α1 , α2 ) holds true. The proof of Lemma 4 is straightforward and omitted here. A complex dynamical network consisting of N identical fractional-order coupled nodes can be described by: N

Dα xi (t) = f (xi (t)) + ∑ ci j Γx j (t)

i = 1, . . . , N.

(8)

j=1

where 0 < q < 1 is the fractional order, xi = (xi1 , xi2 , . . . , xin )T ∈ Rn denotes the state vector of the ith node, and fi : Rn → Rn is the given continuously differentiable nonlinear function describing the local dynamics of node i. Γ = diag {ω1 , ω2 , . . . , ωn } is an inner-coupling matrix determining the interaction of variables, C = (ci j )N×N indicates the outer coupling matrix of the network. If there exists a link between ith node and jth node (i = j), then ci j = 0, and the positive entry ci j represents the strength of the connection between nodes i and j; otherwise, ci j = c ji = 0, and note that all the diagonal elements are cii = 0. Let us consider the following drive-response dynamical complex network Dα s = f (s) N

Dβ xi = g(xi ) + ∑ ci j Γx j (t)+Ui (s, xi ,t)

(9) (10)

i=1

where s = (s1 , s2 , . . . , sn ), x = (xi1 , xi2 , . . . , xin )T denote the state vectors of drive system and response network, respectively. f : Rn → Rn , g : Rn → Rn are continuous vector functions, α , β denote the fractional orders for each state of the drive system and response complex network, respectively. Ui (s, xi ,t) is the controller which will be designed.

460


Definition 1. For the drive system (9) and the response complex network (10), there is said to be hybrid projective synchronization (HPS), if there exists H = diag(λ1 , λ2 , . . . , λn ), where λl ∈ {1, −1, 0} such that lim ei = lim xi (t) − Hs(t) = 0. ei = (ei1 , ei2 , . . . , ein )T ∈ Rn , 1 ≤ i ≤ N is the error system.

t→∞

t→∞

For designing appropriate controller to realize hybrid projective synchronization behavior, we decompose the nonlinear function the drive system (9) and response network (10) into two parts. Dα s = As + F(s)

(11)

N

Dβ xi = Bxi + G(xi ) + ∑ ci j Γx j (t)+Ui

(12)

i=1

Remark 1. It is easy to see that the definition of the hybrid projective synchronization encompasses the projective synchronization, complete synchronization, anti-synchronization and when the scaling factors are selected to take the corresponding specific values, respectively. In order to illustrate the general method in detail, we separate the controller function Ui into two subcontrollers Ui1 and Ui2 . One want to transform the states of the response complex network into an equivalent fractional-order system with fractional orders being equal to the orders of the corresponding states in the drive system. Therefore, the sub-controller Ui1 is designed as following: N

Ui1 (s, xi ) = (D−(α −β ) − I)[g(xi ) + ∑ ci j Γx j (t)] j=1

N

N

j=1

j=1

= D−(α −β ) g(xi ) − g(xi ) − ∑ ci j Γx j (t) + D−(α −β ) ∑ , ci j Γx j (t)

(13)

where I is the identity operator. By submitting sub-controller (13) into system (10), The response complex network can be rewritten as follows: N

Dβ xi = D−(α −β ) g(xi ) + D−(α −β ) ∑ ci j Γx j (t)+Ui2

(14)

j=1

By applying the factional derivative of order α − β to both the left and right sides of (14), one can obtain N

D(α −β ) [Dβ xi ] = Dα xi = Dα −β [D−(α −β ) g(xi ) + D−(α −β ) ∑ ci j Γx j (t)+Ui2 ] j=1

N

= g(xi ) + ∑ ci j Γx j (t)+D(α −β ) (Ui2 )

(15)

j=1

According to property (3) of fractional calculus in Section 2.2, the above statements holds. Then the synchronization of drive-response factional-order complex network with different orders is reduced to the synchronization of drive-response factional-order complex network with identical orders. Thus, we design the nonlinear controller of the following form is proposed for the sub-controller Ui2 Ui2 = D−(α −β ) [H(A − B)s − G(xi) + HF(s) − Kei ],

(16)

where K = [k1 , k2 , . . . , kn ]T ∈ Rn×n is a matrix for the control parameters. To illustrate the effectiveness of the above controller (13) and (16) , we give the following theoretical analysis. Substituting Eq. (14) into system (13) yields the following form.


461

Based on the properties on fractional calculus, the error dynamic system can be described as: Dα ei = Dα (xi − Hs) = Dα xi − Dα (Hs) N

= Bxi + G(xi ) + ∑ ci j Γx j + [H(A − B)s − G(xi) + HF(s) − Kei ] − H[As + F(s)] j=1

N

= (B − K)ei + ∑ ci j Γe j

(17)

j=1

Since we can pick some appropriate control matrix K to make all the eigenvalues λi (i = 1, 2, . . . , n) of B − K satisfy the stability theorems of fractional-order system, Define an nN dimensional error vector e(t) = [eT1 (t), eT2 (t), . . . , eTN (t)] ∈ RnN , we can obtain Dq e(t) = (P ⊗ IN +C ⊗ Γ)e(t),

(18)

where ⊗ denotes the Kronecker product, P = B − K. Let ε1 , ε2 , . . . , εN be the eigenvalues of P, such that the eigenvalues of P satisfy the stability condition of Lemma 1. Let λ1 , λ2 , . . . , λN be the eigenvalues of C and then λi is real and nonnegative. Moreover, from Lemma 2, all eigenvalues of P ⊗ IN + C ⊗ Γ have the form ε j + λi for some i ∈ {1, . . . , N} and j ∈ {1, . . . , n}. From Lemma 3, ε j + λi is also satisfy the stability condition of Lemma 1. Then lim e(t) = 0. t→∞

4 Numerical simulations In this section, several numerical examples are provided to illustrate the proposed synchronization methods. 4.1

Synchronization of the network with chaotic system nodes

The fractional-order Chen system is used as drive system. Chen and Ueta introduced, in 1999, the Chen system which is similar but not topologically equivalent to Lorenz system. It is a chaotic system with a double scroll attractor. The fractional version of Chen system reads as ⎧ α ⎨ D s1 = a(s2 − s1 ) (19) Dα s = (b − a)s1 − s1 s3 + cs2 ⎩ α 2 D s3 = s1 s2 − cs3 , where the values for each parameters are a = 35, b = 3, c = 28 and α = (0.95, 0.95, 0.95), fractional order Chen system can display chaotic attractors as shown in Fig. 1(a). Consider a complex network consisting of 10 nodes. Rossler system is taken as the local node dynamics of the response complex network. The single fractional-order Rossler system in the ith node is described as below: ⎧ β ⎨ D xi1 = −(xi2 + xi3 ) (20) Dβ x = xi1 + dxi2 ⎩ β i2 D xi3 = xi3 (xi1 − f ) + e, Chaotic attractors are found when β = (0.9, 0.9, 0.9) and (d, e, f ) = (0.4, 0.2, 10). The attractor is described in Fig.1(b) Compared with Eqs.(11) and (12), the Eqs.(19) and (20) can be rewritten as: ⎤⎡ ⎤ ⎡ ⎤ ⎡ α ⎤ ⎡ −a a 0 s1 0 D s1 ⎣ Dα s2 ⎦ = ⎣ c − a 0 c ⎦ ⎣ s2 ⎦ + ⎣ −s1 s3 ⎦ = As + F(s). (21) α D s3 s3 s1 s2 0 0 −b

462


Fig. 1 Three-dimensional plot of the trajectory of the fractional-order (a) Chen system (b) Rossler system.

⎡

⎤ ⎡ Dβ xi1 0 ⎣ Dβ xi2 ⎦ = ⎣ 1 0 Dβ xi3

⎤⎡ ⎤ ⎡ ⎤ −1 −1 xi1 0 d 0 ⎦ ⎣ xi2 ⎦ + ⎣ 0 ⎦ = Bxi + G(xi ). xi3 0 −f xi1 xi3

So the fractional-order drive-response complex dynamical networks with 10 nodes can be described as ⎧ a ⎪ ⎨ D∗ s = As + F(s), β

10

⎪ ⎩ D∗ xi = Bxi + G(xi ) + ∑ ci j Γx j +Ui

i = 1, 2, . . . , 10.

(22)

(23)

j=1

The inner-coupling matrix is taken as Γ = diag(1, 1, 1) and the out-coupling matrix is taken as: ⎡

⎤ −3 0 1 0 1 0 0 0 0 1 ⎢ 0 −5 1 1 0 0 1 0 1 1 ⎥ ⎢ ⎥ ⎢ 1 1 −2 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ 1 0 −3 0 0 1 0 1 0 ⎢ ⎥ ⎢ 1 0 0 0 −2 0 0 1 0 0 ⎥ ⎢ ⎥. C=⎢ 0 0 0 0 −4 1 1 1 1 ⎥ ⎢ 0 ⎥ ⎢ 0 1 0 1 0 1 −6 1 1 1 ⎥ ⎢ ⎥ ⎢ 0 ⎥ 0 0 0 1 1 1 −5 1 1 ⎢ ⎥ ⎣ 0 1 0 1 0 1 1 1 −5 0 ⎦ 1 1 0 0 0 1 1 1 0 −5

(24)

The initial values of the state variables of the network are chosen randomly. We choose the scaling factor as λ1 = 1, λ2 = 1, λ3 = 1. From Fig.2 (a), it can be seen that the state variables of the nodes of drive-response complex network (21)-(22) get closer and closer to the targeted trajectory with the evolution of time, and finally coincide with it as shown. The synchronization of the fractional-order drive-response complex network (21)-(22) is achieved, as can be seen from Fig.2 (b), when all the synchronization errors converge to zero asymptotically. 4.2

Synchronizaiton of the network with hyper-chaotic system nodes

In this subsection, we consider a network with N = 5 nodes. The fractional-order hyper-chaotic Lorenz system is used as the drive system in the drive-response networks. The single hyper-chaotic fractional-order Lorenz system is given by: ⎧ α D s1 = d(s2 − s1 ) + s4 ⎪ ⎪ ⎨ α D s2 = hs1 − s2 − s1 s3 (25) ⎪ Dα s3 = s1 s2 − f s3 ⎪ ⎩ α D s4 = −s2 s3 + rs4 ,


463

Fig. 2 (a) The time evolution of state variables the drive-response complex network (22)-(23). (b) the synchronization error of the drive-response complex network (22)-(23).

when α = 0.99 and (d, h, f , r) = (10, 8/3, 28, −1), hyper-chaotic attractor is shown in Fig.3(a). In the following, fractional-order hyper-chaotic Chen system is taken as the local node dynamics of the response complex network. The single fractional-order hyper-chaotic Chen system in the ith node is described as below: ⎧ β D xi1 = a1 (xi2 − xi1 ) + xi4 ⎪ ⎪ ⎨ β D xi2 = d1 xi1 + c1 xi2 − xi1 xi3 (26) ⎪ Dβ xi3 = xi1 xi2 − b1 xi3 ⎪ ⎩ β D xi4 = xi2 xi3 + r1 xi4 , The default system parameters are chosen (a1 , b1 , c1 , d1 , r1 ) = (35, 3, 12, 7, 0.5) and β = 0.95, respectively. The fractional order hyper-chaotic Chen system shows chaotic attractors as shown in Fig.3(b).

Fig. 3 The attractor of Fractional-order system (a) Lorenz with q = 0.99 (b) Chen system with q = 0.95.

Also, the Eqs.(25) and (26) should be modified as: ⎤ ⎡ −d d 0 Dα s1 ⎢ Dα s2 ⎥ ⎢ h −1 0 ⎥ ⎢ ⎢ ⎣ Dα s3 ⎦ = ⎣ 0 0 −f Dα s4 0 0 0 ⎡ β ⎤ ⎡ D xi1 0 −a1 a1 ⎢ Dβ xi2 ⎥ ⎢ d1 c1 0 ⎥ ⎢ ⎢ ⎣ Dβ xi3 ⎦ = ⎣ 0 0 −b1 0 0 0 Dβ xi4 ⎡

⎤⎡ ⎤ ⎡ ⎤ 0 s1 0 ⎢ ⎥ ⎢ ⎥ 0⎥ ⎥ ⎢ s2 ⎥ + ⎢ −s1 s3 ⎥ = As + F(s) ⎦ ⎣ ⎦ ⎣ s3 s1 s2 ⎦ 0 s4 −s3 s2 r ⎤⎡ ⎤ ⎡ ⎤ 1 xi1 0 ⎢ ⎥ ⎢ ⎥ 0⎥ ⎥ ⎢ xi2 ⎥ + ⎢ −xi1 xi3 ⎥ = Bxi + G(xi ) 0 ⎦ ⎣ xi3 ⎦ ⎣ xi1 xi2 ⎦ xi4 xi2 xi3 r1

(27)

(28)

464


Moreover, the drive-response complex dynamical network with 5 nodes can be described by ⎧ α ⎪ ⎨ D s = As + F(s), 5

β ⎪ ⎩ D xi = Bxi + G(xi ) + ∑ ci j Γx j +Ui

i = 1, 2, . . . , 5.

(29)

j=1

Furthermore, the numerical simulations are given to verify the effectiveness of the strategy. The initial values of the state variables of the network are chosen randomly. Here we choose the scaling factor as λ1 = 1, λ2 = −1, λ3 = −1, λ4 = 1. Note that with such a scaling matrix the HPS contain the anti-synchronization and projective synchronization. From Fig.4 (a), it can be seen that the state variables of the nodes of complex network (22) get closer and closer to the targeted trajectory with the evolution of time, and finally coincide with. The synchronization of the fractional-order edge-colored network (23) is achieved, as can be seen from Fig.4 (b), when all the synchronization errors converge to zero asymptotically.

Fig. 4 (a) The trajectories of nodes of the drive-response complex network (28)-(29). (b) the synchronization errors of the drive-response complex network (28)-(29).

5 Conclusions We investigated the hybrid projective synchronization of fractional-order drive-response complex network with different derivative orders. Based on the properties on fractional calculus and the stability theorems of the fractional-order system, an effective method is proposed to achieve the HPS. The designed controller transform different derivative orders with identical derivative orders between chaotic systems via the idea of active control method. Numerical simulations show that the effectiveness and feasibility of the controllers . References [1] Strogatz,S.H. (2001), Exploring complex networks, Nature, 410, 268–276. [2] Latora,V. and Marchiori, M. (2004), How the science of complex networks can help developing strategies against terrorism, Chaos Solitons Fractals, 20, 69-75. [3] Nakagawa, N. and YKuramoto. (1993), Collective chaos in a population of globally coupled oscillators, Progress Theor Phys, 89, 313-323. [4] Kumpula, J.M., Onnela, J.P., and Saramäki, J. (2007),Emergence of communities in weighted networks.” Physical Review Letter, 99, 22870. [5] Chavez, M., Hwang, D.U., and Amann, A. (2005), Synchronization is enhanced in weighted complex networks, Physical Review Letter, 94, 218701.


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[6] Lu, J.Q., Ho, DWC., and Kurths, J. (2009), Consensus over directed static networks with arbitrary finite communication delays, Physical Review E, 80, 066121. [7] Yang, M.L., Liu, Y.G.,You, Z.S., and Sheng, P. (2010), Global synchronization for directed complex networks.” Nonlinear Analysis: Real World Applications, 11, 2127–2135. [8] Wu,Y.Q. and Li, C.P. (2012), Pinning adaptive anti-synchronization between two general complex dynamical networks with non-delayed and delayed coupling, Appl Math Comput, 218, 7445–7452, [9] Zheng,S., Dong, G.G., and Bi,Q.S. (2009), Impulsive synchronization of complex networks with non-delayed and delayed coupling, Phyical Letter A, 373, 4255-4259. [10] Mainieri, R. and Rehacek, J. (1999), Projective synchronization in three-dimensional chaotic systems,” Physical Review Letter, 82, 3042–3046. [11] Wu, Z.Y.and Fu, X.C. (2012), Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes, Commun Nonlinear Sci Numer Simul , 17, 1628–1636. [12] Guo, W., Chen, S., and Austin, F. (2010), Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling, Commun Nonlinear Sci Numer Simul, 15, 1631–1639. [13] Matignon, D. (1996), Stability results for fractional differential equations with applications to control processing, in: Proceeeding of IMACS, IEEE-SMC, Lille, France, 963-968. [14] Bapat, R.B. (2010), Graphs and Matrices, Spring, New york. [15] Wu,C.W. and Chua, L.O. (1995), Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuit Syst. I. 42, 430–447. [16] Diethelm, K., Ford, N.J., and Freed,A.D. (2002), A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3–22.



Nonlinear Four-point Impulsive Fractional Differential Equations with p-Laplacian Operator Fatma Tokmak Fen1†, Ilkay Yaslan Karaca2 1 2

Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey Submission Info Communicated by Mark Edelman Received 24 October 2014 Accepted 24 December 2014 Available online 1 January 2016

Abstract In this paper, we investigate the existence of solutions for a four-point nonlocal boundary value problem of nonlinear impulsive differential equations of fractional order α ∈ (2, 3]. By using some well known fixed point theorems, sufficient conditions for the existence of solutions are established. Some illustrative examples are also discussed.

Keywords Nonlinear fractional differential equations p-Laplacian operator Four-point boundary value problem Fixed-point theorems ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional calculus has many applications in mechanics, control theory, biology, economics, etc. See [1] and [2]. Differential equations of fractional order naturally appear in mathematical models when they are based on the properties of systems and processes in terms of fractional derivatives. This type of differential equations have been widely investigated in the literature [3–14]. On the other hand, the necessity of the introduction of impulses in the mathematical models can arise, if one considers the application of differential equations in real world problems, see [15–20]. Therefore, it is worth studying fractional differential equations with impulsive effects. The recent results on impulsive fractional differential equations can be found in [21–29]. The research of boundary value problems for p-Laplacian equations of fractional order has just begun in recent years. See [30–36]. Investigations on fractional p-Laplacian impulsive differential equations have not been appreciated well enough. To the best of our knowledge, nonlinear fractional impulsive differential equation with p-Laplacian operator was only considered in the study [37]. In the paper [38], we studied the existence of solutions for a two-point boundary value problem of impulsive fractional differential equations with p-Laplacian † Corresponding

author. Email addresses: [email protected], [email protected] ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.11.009

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Fatma Tokmak Fen, Ilkay Yaslan Karaca / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 467–486

operator based on some standard fixed point theorems. In the present paper, we consider a four-point boundary value problem, and this is the main difference compared to [38]. Moreover, in the papers [38] and [39], derivatives with smaller order are considered. The main subject of investigation in the present paper is the following system ⎧ ⎪ φ p C Dα0+ u(t) = f (t, u(t)), 2 < α ≤ 3, t ∈ J , ⎪ ⎪ ⎪ ⎪ ⎨ Δu(tk ) = Qk (u(tk )), Δu (tk ) = Ik (u(tk )), (1) ⎪ Δu (tk ) = Jk (u(tk )), k = 1, 2, . . . , l, ⎪ ⎪ ⎪ ⎪ ⎩ au(0) − bu (ξ ) = 0, cu(1) + du (η ) = 0, C Dα0+ u(0) = 0, u (0) = 0, where C Dα0+ is the Caputo fractional derivative, φ p is a p-Laplacian operator, φ p (s) = |s| p−2 s, p > 1, φ p is 1 1 invertible and (φ p )−1 = φq where + = 1, f ∈ C ([0, 1] × R, R) , Ik , Jk ∈ C (R, R) , a, b, c, d ∈ [0, ∞) with p q ac + ad + bc > 0, J = [0, 1], 0 = t0 < t1 < · · · < tk < · · · < tl < tl+1 = 1, J = J \ {t1 ,t2 , . . . ,tl } , 0 < ξ < η < 1, ξ = tk , η = tk (k = 1, 2, . . . , l), Δu(tk ) = u(tk+ ) − u(tk− ), where u(tk+ ) and u(tk− ) denote the right and the left limits of u(t) at t = tk (k = 1, 2, . . . , l), respectively. Δu (tk ) and Δu (tk ) have a similar meaning for u (t) and u (t), respectively. 2 Preliminaries In this section, we present auxiliary lemmas which will be used later. Let J0 = [0,t1 ], J1 = (t1 ,t2 ], · · · , Jl−1 = (tl−1 ,tl ], Jl = (tl , 1], and we introduce the spaces PC(J, R) = u : J → R|u ∈ C (Jk ), k = 0, 1, · · · , l and u(tk+ ) exist, k = 1, 2, · · · , l , with the norm u = sup |u(t)| , t∈J

PC1 (J, R) = u : J → R|u ∈ C 1 (Jk ), k = 0, 1, · · · , l, and u(tk+ ), u (tk+ ) exist, k = 1, 2, · · · , l , with the norm uPC1 = max u , u and PC2 (J, R) = u : J → R|u ∈ C 2 (Jk ), k = 0, 1, · · · , l, and u(tk+ ), u (tk+ ), u (tk+ ) exist, k = 1, 2, · · · , l , with the norm uPC2 = max u , u , u . Obviously, PC(J, R), PC1 (J, R) and PC2 (J, R) are Banach spaces. Definition 1. For a continuous function u ∈ C n [0, 1], the Caputo derivative of fractional order α is defined as ˆ t 1 C α D0+ u(t) = (t − s)n−α −1 u(n) (s)ds, n = [α ] + 1, Γ(n − α ) 0 where [α ] denotes the integer part of real number α . Definition 2. The Riemann-Liouville fractional integral of order α is defined as ˆ t 1 α (t − s)α −1 u(s)ds, α > 0, I0+ u(t) = Γ(α ) 0 provided the integral exists.


469

Lemma 1. ( [1]) For α > 0 and u ∈ C n [0, 1], the general solution of the fractional differential equation C

Dα0+ u(t) = 0

is given by u(t) = c0 + c1t + c2t 2 + · · · + cn−1t n−1 , ci ∈ R, i = 0, 1, 2, . . . , n − 1, n = [α ] + 1. where [α ] denotes the integer part of real number α . In view of Lemma 1, for u ∈ C n [0, 1] it follows that I0α+ C Dα0+ u(t) = u(t) + c0 + c1t + c2t 2 + · · · + cn−1t n−1

(2)

for some ci ∈ R, i = 0, 1, 2, . . . , n − 1, n = [α ] + 1. Lemma 2. ( [1]) Let α > 0, f ∈ L∞ (0, 1), then C

Dα0+ I0α+ f (t) = f (t).

Definition 3. A function u ∈ PC2 (J, R) with its Caputo derivative of order α existing on J is a solution of (1) if it satisfies (1). We need the following known results to prove the existence of solutions for (1). Theorem 3. ( [40]) Let E be a Banach space. Assume that Ω is an open bounded subset of E with θ ∈ Ω and ¯ → E be a completely continuous operator such that let T : Ω Tu ≤ u , ∀u ∈ ∂ Ω. ¯ Then T has a fixed point in Ω. Theorem 4. ( [40]) Let E be a Banach space. Assume that T : E → E is a completely continuous operator and the set V = {u ∈ E|u = μ Tu, 0 < μ < 1} is bounded. Then T has a fixed point in E. Lemma 5. Let y ∈ C [0, 1] and ξ , η ∈ (tm ,tm+1 ), m is a nonnegative integer, 0 ≤ m ≤ l. A function u is a solution of the following impulsive boundary value problem of fractional order ⎧ ⎪ φ p C Dα0+ u(t) = y(t), 2 < α ≤ 3, t ∈ J , ⎪ ⎪ ⎪ ⎪ ⎨ Δu(tk ) = Qk (u(tk )), Δu (tk ) = Ik (u(tk )), ⎪ Δu (tk ) = Jk (u(tk )), k = 1, 2, ..., l, ⎪ ⎪ ⎪ ⎪ ⎩ au(0) − bu (ξ ) = 0, cu(1) + du (η ) = 0, C Dα0+ u(0) = 0, u (0) = 0, if and only if u is a solution of the impulsive fractional integral equation

(3)

470


ˆ t ˆ s ⎧ 1 ⎪ α −1 ⎪ (t − s) φq ( y(r)dr)ds − c0 − c1t, t ∈ J0 ; ⎪ ⎪ Γ(α ) 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ˆ s ˆ s ˆ ti ⎪ ⎪ 1 ˆ t ⎪ 1 k α −1 α −1 ⎪ ⎪ (t − s) φ ( y(r)dr)ds + (t − s) φ ( y(r)dr)ds q q ⎪ ∑ t i ⎪ Γ(α ) tk Γ(α ) i=1 ⎪ 0 0 i−1 ⎪ ⎪ ˆ ti ˆ s ⎪ k−1 ⎪ (tk − ti ) ⎪ α −2 ⎪ (ti − s) φq ( y(r)dr)ds +∑ ⎪ ⎪ Γ(α − 1) ti−1 ⎪ 0 ⎪ i=1 ⎪ k−1 ˆ s ˆ ti ⎪ 2 ⎪ (tk − ti ) ⎪ ⎪ (ti − s)α −3 φq ( y(r)dr)ds ⎪+ ∑ ⎪ ⎪ 2Γ(α − 2) ti−1 0 ⎪ i=1 ⎨ ˆ ti ˆ s k (t − t ) k u(t) = + ⎪ ∑ Γ(α − 1) t (ti − s)α −2φq ( 0 y(r)dr)ds ⎪ ⎪ i−1 i=1 ⎪ ⎪ ˆ ˆ s ⎪ k−1 ⎪ (t − t )(t − ti ) ti k k ⎪ α −3 ⎪ (ti − s) φq ( y(r)dr)ds +∑ ⎪ ⎪ Γ(α − 2) ⎪ 0 ti−1 ⎪ i=1 ⎪ ˆ ti ˆ s ⎪ k k k−1 2 ⎪ (t − tk ) ⎪ ⎪ (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tk − ti )Ii (u(ti )) ⎪+ ∑ ⎪ ⎪ 2Γ(α − 2) ti−1 0 ⎪ i=1 i=1 ⎪ ⎪ i=1 k−1 k k−1 2 ⎪ ⎪ (t − t ) k i ⎪ ⎪ Ji (u(ti )) + ∑ (t − tk )Ii (u(ti )) + ∑ (t − tk )(tk − ti )Ji (u(ti )) +∑ ⎪ ⎪ 2 ⎪ i=1 i=1 i=1 ⎪ ⎪ ⎪ k 2 ⎪ (t − t ) k ⎪ ⎪ Ji (u(ti )) − c0 − c1t, t ∈ Jk , k = 1, 2, · · · , l, ⎩+ ∑ 2 i=1 where

ˆ 1 ˆ s ˆ s ˆ 1 1 1 p ti α −1 α −1 (bc[ (1 − s) φq ( y(r)dr)ds + c0 = ∑ t (ti − s) φq( 0 y(r)dr)ds ac + ad + bc Γ(α ) tl Γ(α ) i=1 0 i−1 ˆ ti ˆ ˆ s ˆ s t l−1 l−1 i (tl − ti ) (tl − ti )2 (ti − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ 0 0 i=1 Γ(α − 1) ti−1 i=1 2Γ(α − 2) ti−1 ˆ ˆ ti ˆ ˆ s ti s l l−1 (1 − tl ) (1 − tl )(tl − ti ) α −2 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +∑ Γ(α − 2) 0 0 ti−1 i=1 Γ(α − 1) ti−1 i=1 ˆ ˆ ti s l l l−1 (1 − tl )2 (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) +∑ 0 i=1 2Γ(α − 2) ti−1 i=1 i=1 l−1

l l−1 l (tl − ti )2 (1 − tl )2 Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) + ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ Ji (u(ti ))] 2 2 i=1 i=1 i=1 i=1 ˆ η ˆ ti ˆ s ˆ s m 1 1 (η − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −2 φq ( y(r)dr)ds +bd[ Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti ˆ ˆ s ti s m−1 m (tm − ti ) (η − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) t t 0 0 i−1 i−1 i=1 i=1

+∑

m

m−1

i=1

i=1

+ ∑ Ii (u(ti )) +

m

∑ (tm − ti)Ji (u(ti )) + ∑ (η − tm )Ji (u(ti ))] i=1

ˆ ti ˆ s ˆ s m 1 1 (ξ − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −2 φq ( y(r)dr)ds Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti ˆ ˆ s ti s m−1 m (tm − ti ) (ξ − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) 0 0 t t i−1 i−1 i=1 i=1 −(c + d)b[

ˆ

ξ

(4)

Fatma Tokmak Fen, Ilkay Yaslan Karaca / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 467–486 m

m−1

m

i=1

i=1

i=1

+ ∑ Ii (u(ti )) +

471

∑ (tm − ti)Ji (u(ti )) + ∑ (ξ − tm)Ji (u(ti ))]),

and

ˆ 1 ˆ s ˆ s ˆ ti 1 1 1 l α −1 α −1 (ac[ (1 − s) φq ( y(r)dr)ds + c1 = ∑ t (ti − s) φq ( 0 y(r)dr)ds ac + ad + bc Γ(α ) tl Γ(α ) i=1 0 i−1 ˆ s ˆ s ˆ ti ˆ t l−1 l−1 i (tl − ti ) (tl − ti )2 (ti − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ 0 0 i=1 Γ(α − 1) ti−1 i=1 2Γ(α − 2) ti−1 ˆ ˆ ti ˆ ˆ s ti s l l−1 (1 − tl ) (1 − tl )(tl − ti ) α −2 α −3 +∑ (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds Γ(α − 2) ti−1 0 0 i=1 Γ(α − 1) ti−1 i=1 ˆ ˆ ti s l l l−1 (1 − tl )2 (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) +∑ 0 i=1 2Γ(α − 2) ti−1 i=1 i=1 l−1

l l−1 l (tl − ti)2 (1 − tl )2 Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) + ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ Ji (u(ti ))] 2 2 i=1 i=1 i=1 i=1 ˆ η ˆ ti ˆ s ˆ s m 1 1 (η − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −2 φq ( y(r)dr)ds +ad[ Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti ˆ ˆ s ti s m−1 m (tm − ti ) (η − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) 0 0 t t i−1 i−1 i=1 i=1

+∑

m

+ ∑ Ii (u(ti )) + i=1

1 +cb[ Γ(α − 1) +

m−1

∑

i=1 m

m−1

m

i=1 ξ

i=1

∑ (tm − ti)Ji (u(ti )) + ∑ (η − tm )Ji (u(ti ))]

ˆ

tm

1 i=1 Γ(α − 1)

0

ˆ

ti

ti−1

α −2

(ti − s)

ˆ s φq ( y(r)dr)ds 0

ˆ ˆ s ˆ s m (tm − ti ) (ξ − tm ) ti α −3 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds Γ(α − 2) ti−1 0 0 i=1 Γ(α − 2) ti−1

+ ∑ Ii (u(ti )) + i=1

ˆ

ˆ s m α −2 (ξ − s) φq ( y(r)dr)ds + ∑

ti

m−1

m

i=1

i=1

∑ (tm − ti)Ji (u(ti )) + ∑ (ξ − tm )Ji (u(ti ))]).

Proof. Let u be a solution of (3). Then, by (2), we have ˆ t α u(t) = I0+ φq ( y(s)ds) − c0 − c1t − c2t 2 ˆ t 0 ˆ s 1 α −1 = (t − s) φq ( y(r)dr)ds − c0 − c1t − c2t 2 , t ∈ J0 , Γ(α ) 0 0 for some c0 , c1 , c2 ∈ R. Furthermore

ˆ t ˆ s 1 (t − s)α −2 φq ( y(r)dr)ds − c1 − 2tc2 , Γ(α − 1) 0 0 ˆ t ˆ s 1 (t − s)α −3 φq ( y(r)dr)ds − 2c2 . u (t) = Γ(α − 2) 0 0 u (t) =

If t ∈ J1 , then

ˆ t ˆ s 1 (t − s)α −1 φq ( y(r)dr)ds − d0 − d1 (t − t1 ) − d2 (t − t1 )2 , Γ(α ) t1 0 ˆ t ˆ s 1 α −2 (t − s) φq ( y(r)dr)ds − d1 − 2d2 (t − t1 ), u (t) = Γ(α − 1) t1 0 u(t) =

(5)

472


u (t) =

1 Γ(α − 2)

ˆ

t

t1

ˆ s (t − s)α −3 φq ( y(r)dr)ds − 2d2 , 0

for some d0 , d1 , d2 ∈ R. Thus, we have ˆ t1 ˆ s 1 α −1 = (t1 − s) φq ( y(r)dr)ds − c0 − c1t1 − c2t12 , Γ(α ) 0 0 u(t1+ ) = −d0 , ˆ t1 ˆ s 1 (t1 − s)α −2 φq ( y(r)dr)ds − c1 − 2t1 c2 , u (t1− ) = Γ(α − 1) 0 0 + u (t1 ) = −d1 , ˆ s ˆ t1 1 − α −3 (t1 − s) φq ( y(r)dr)ds − 2c2 , u (t1 ) = Γ(α − 2) 0 0 + u (t1 ) = −2d2 . u(t1− )

In view of Δu(t1 ) = u(t1+ ) − u(t1− ) = Q1 (u(t1 )), Δu (t1 ) = u (t1+ ) − u (t1− ) = I1 (u(t1 )) and Δu (t1 ) = = J1 (u(t1 )), we have ˆ t1 ˆ s 1 α −1 (t1 − s) φq ( y(r)dr)ds − c0 − c1t1 − c2t12 + Q1 (u(t1 )), −d0 = Γ(α ) 0 0 ˆ t1 ˆ s 1 α −2 (t1 − s) φq ( y(r)dr)ds − c1 − 2t1 c2 + I1 (u(t1 )), −d1 = Γ(α − 1) 0 0 ˆ t1 ˆ s 1 (t1 − s)α −3 φq ( y(r)dr)ds − 2c2 + J1 (u(t1 )). −2d2 = Γ(α − 2) 0 0

u (t1+ ) − u (t1− )

Consequently, 1 u(t) = Γ(α )

ˆ

t

t1

α −1

(t − s) ˆ

ˆ s φq ( y(r)dr)ds + 0

1 Γ(α )

ˆ 0

t1

α −1

(t1 − s)

ˆ s φq ( y(r)dr)ds 0

ˆ t1 ˆ s ˆ s (t − t1 )2 (t − t1 ) α −2 α −3 (t1 − s) φq ( y(r)dr)ds + (t1 − s) φq ( y(r)dr)ds + Γ(α − 1) 0 2Γ(α − 2) 0 0 0 1 +Q1 (u(t1 )) + (t − t1 )I1 (u(t1 )) + (t − t1 )2 J1 (u(t1 )) − c0 − c1t − c2t 2 , t ∈ J1 . 2 t1

By a similar process, we get ˆ t ˆ s ˆ s ˆ ti 1 k 1 α −1 α −1 (t − s) φq ( y(r)dr)ds + u(t) = ∑ t (ti − s) φq( 0 y(r)dr)ds Γ(α ) tk Γ(α ) i=1 0 i−1 ˆ ti ˆ ti ˆ s ˆ s k−1 k−1 (tk − ti ) (tk − ti )2 α −2 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ 0 0 i=1 Γ(α − 1) ti−1 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ ti ˆ s ˆ s k k−1 (t − tk ) (t − t )(t − t ) k k i α −2 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +∑ Γ(α − 2) 0 0 ti−1 i=1 Γ(α − 1) ti−1 i=1 ˆ ti ˆ s k k k−1 (t − tk )2 (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tk − ti )Ii (u(ti )) +∑ 2Γ( α − 2) 0 t i−1 i=1 i=1 i=1 k−1 k k−1 k 2 (tk − ti ) (t − tk )2 Ji (u(ti )) + ∑ (t − tk )Ii (u(ti )) + ∑ (t − tk )(tk − ti )Ji (u(ti )) + ∑ Ji (u(ti )) +∑ 2 2 i=1 i=1 i=1 i=1 −c0 − c1t − c2t 2 , t ∈ Jk , k = 1, 2, · · · , l.

(6)


By conditions au(0) − bu (ξ ) = 0, cu(1) − du (η ) = 0 and u (0) = 0, we have ˆ 1 ˆ s ˆ s ˆ ti 1 1 1 l α −1 α −1 (bc[ (1 − s) φq ( y(r)dr)ds + c0 = ∑ t (ti − s) φq( 0 y(r)dr)ds ac + ad + bc Γ(α ) tl Γ(α ) i=1 0 i−1 ˆ ti ˆ ti ˆ s ˆ s l−1 l−1 2 (tl − ti ) (tl − ti ) (ti − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ 0 0 i=1 Γ(α − 1) ti−1 i=1 2Γ(α − 2) ti−1 ˆ ˆ ti ˆ ˆ s ti s l l−1 (1 − tl ) (1 − tl )(tl − ti ) α −2 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +∑ Γ(α − 2) ti−1 0 0 i=1 Γ(α − 1) ti−1 i=1 ˆ ti ˆ s l l l−1 2 (1 − tl ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) +∑ 2Γ( α − 2) t 0 i−1 i=1 i=1 i=1 l−1

l l−1 l (tl − ti )2 (1 − tl )2 Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) + ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ Ji (u(ti ))] 2 2 i=1 i=1 i=1 i=1 ˆ η ˆ ti ˆ s ˆ s m 1 1 α −2 α −2 (η − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +bd[ Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ ti ˆ s ˆ s m−1 m (tm − ti ) (η − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) 0 0 ti−1 ti−1 i=1 i=1

+∑

m

m−1

m

i=1

i=1

i=1

m

m−1

m

i=1

i=1

i=1

+ ∑ Ii (u(ti )) +


ˆ ξ ˆ ti ˆ s ˆ s m 1 1 (ξ − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −2 φq ( y(r)dr)ds −(c + d)b[ Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti ˆ ˆ s ti s m−1 m (tm − ti ) (ξ − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) 0 0 t t i−1 i−1 i=1 i=1 + ∑ Ii (u(ti )) +


ˆ 1 ˆ s ˆ s ˆ ti 1 1 1 l α −1 α −1 (ac[ (1 − s) φq ( y(r)dr)ds + c1 = ∑ t (ti − s) φq ( 0 y(r)dr)ds ac + ad + bc Γ(α ) tl Γ(α ) i=1 0 i−1 ˆ ti ˆ ti ˆ s ˆ s l−1 l−1 2 (tl − ti ) (tl − ti ) α −2 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +∑ 0 0 i=1 Γ(α − 1) ti−1 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ ti ˆ s ˆ s l l−1 (1 − tl ) (1 − tl )(tl − ti ) (ti − s)α −2 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ(α − 2) 0 0 ti−1 i=1 Γ(α − 1) ti−1 i=1 ˆ ˆ ti s l l l−1 2 (1 − tl ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) +∑ 2Γ( α − 2) 0 t i−1 i=1 i=1 i=1 l−1

l l−1 l (tl − ti)2 (1 − tl )2 Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) + ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ Ji (u(ti ))] 2 2 i=1 i=1 i=1 i=1 ˆ η ˆ ti ˆ s ˆ s m 1 1 α −2 α −2 (η − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds +ad[ Γ(α − 1) tm 0 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ ti ˆ s ˆ s m−1 m (tm − ti ) (η − tm ) (ti − s)α −3 φq ( y(r)dr)ds + ∑ (ti − s)α −3 φq ( y(r)dr)ds +∑ Γ( α − 2) Γ( α − 2) 0 0 ti−1 ti−1 i=1 i=1

+∑

473

474


+ ∑ Ii (u(ti )) + i=1

1 +cb[ Γ(α − 1) +

m−1

∑

i=1 m

m

i=1 ξ

i=1

∑ (tm − ti)Ji(u(ti )) + ∑ (η − tm )Ji (u(ti ))]

ˆ

tm

ˆ

ˆ s m (ξ − s)α −2 φq ( y(r)dr)ds + ∑

1 i=1 Γ(α − 1)

0

ˆ

ti

ti−1

ˆ s (ti − s)α −2 φq ( y(r)dr)ds 0

ˆ ˆ s ˆ s m (tm − ti ) (ξ − tm ) ti α −3 α −3 (ti − s) φq ( y(r)dr)ds + ∑ (ti − s) φq ( y(r)dr)ds Γ(α − 2) ti−1 0 0 i=1 Γ(α − 2) ti−1

+ ∑ Ii (u(ti )) + i=1

m−1

ti

m−1

m

i=1

i=1

∑ (tm − ti)Ji(u(ti )) + ∑ (ξ − tm)Ji (u(ti ))]),

and c2 = 0. Substituting the value of ci (i = 0, 1, 2) in (5) and (6), we can get (4). Conversely, assume that u is a solution of the impulsive fractional integral equation (4), then by a direct computation, it follows that the solution given by (4) satisfies (3). This completes the proof. 3 Main Results For the sake of convenience, we set

λ1 =

2ac + ab + ad + bc ad + 2bc + 2bd , λ2 = , ac + ad + bc ac + ad + bc λ3 = ac + 2ad + 2bc.

Define the operator T : PC(J, R) → PC(J, R) as ˆ t ˆ s 1 α −1 (t − s) φq ( f (r, u(r))dr)ds Tu(t) = Γ(α ) tk 0 ˆ s ˆ ti 1 k α −1 + ∑ t (ti − s) φq ( 0 f (r, u(r))dr)ds Γ(α ) i=1 i−1 ˆ ti ˆ s k−1 (tk − ti ) α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti s k−1 (tk − ti )2 (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ s k (t − tk ) α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s k−1 (t − tk )(tk − ti) ti (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ti ˆ s k (t − tk )2 α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 k k−1 k−1 (tk − ti )2 Ji (u(ti )) + ∑ Qi (u(ti )) + ∑ (tk − ti )Ii (u(ti )) + ∑ 2 i=1 i=1 i=1 k k−1 k (t − tk )2 Ji (u(ti )) + ∑ (t − tk )Ii (u(ti )) + ∑ (t − tk )(tk − ti )Ji (u(ti )) + ∑ 2 i=1 i=1 i=1 −m0 − m1t,

(7)


where ˆ 1 ˆ s 1 1 α −1 (bc[ (1 − s) φq ( f (r, u(r))dr)ds m0 = ac + ad + bc Γ(α ) tl 0 ˆ s ˆ ti 1 l α −1 (t − s) φ ( f (r, u(r))dr)ds + q ∑ t i Γ(α ) i=1 0 i−1 ˆ ti ˆ s l−1 (tl − ti ) α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ s l−1 (tl − ti )2 α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 ˆ ˆ s l−1 (1 − tl )(tl − ti ) ti (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ti ˆ s l (1 − tl )2 α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 l

l−1

+ ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) i=1 l−1

i=1

l (tl − ti Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) 2 i=1 i=1

)2

+∑

l−1

l

(1 − tl )2 Ji (u(ti ))] 2 i=1 ˆ η ˆ s α −2 (η − s) φq ( f (r, u(r))dr)ds

+ ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ i=1

+bd[

1 Γ(α − 1)

m

1 i=1 Γ(α − 1)

+∑

tm ti

ˆ

ti−1

ˆ α −2 (ti − s) φq (

0

s 0

f (r, u(r))dr)ds

ˆ ˆ s (tm − ti ) ti α −3 (t − s) φ ( f (r, u(r))dr)ds i q ∑ 0 i=1 Γ(α − 2) ti−1 ˆ ˆ s m (η − tm ) ti (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 +

m−1

m

m−1

i=1

i=1

+ ∑ Ii (u(ti )) +

m

∑ (tm − ti)Ji (u(ti )) + ∑ (η − tm )Ji (u(ti ))] ˆ

i=1

ˆ s 1 α −2 (ξ − s) φq ( f (r, u(r))dr)ds −(c + d)b[ Γ(α − 1) tm 0 ˆ ti ˆ s m 1 (ti − s)α −2 φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s m−1 (tm − ti ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 ˆ ˆ s m (ξ − tm ) ti (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 ξ

m

m−1

m

i=1

i=1

i=1

+ ∑ Ii (u(ti )) +


475

476

and


ˆ 1 ˆ s 1 1 α −1 (ac[ (1 − s) φq ( f (r, u(r))dr)ds m1 = ac + ad + bc Γ(α ) tl 0 ˆ s ˆ ti 1 l α −1 + ∑ t (ti − s) φq ( 0 f (r, u(r))dr)ds Γ(α ) i=1 i−1 ˆ ti ˆ s l−1 (tl − ti ) α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ s l−1 (tl − ti )2 (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ s l (1 − tl ) α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s l−1 (1 − tl )(tl − ti ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ti ˆ s l (1 − tl )2 (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 l

l−1

+ ∑ Qi (u(ti )) + ∑ (tl − ti )Ii (u(ti )) i=1 l−1

i=1

l (tl − ti Ji (u(ti )) + ∑ (1 − tl )Ii (u(ti )) 2 i=1 i=1

+∑

l−1

)2

l

(1 − tl )2 Ji (u(ti ))] 2 i=1 ˆ η ˆ s (η − s)α −2 φq ( f (r, u(r))dr)ds

+ ∑ (1 − tl )(tl − ti )Ji (u(ti )) + ∑ i=1

1 Γ(α − 1) tm 0 ˆ ti ˆ s m 1 (ti − s)α −2 φq ( f (r, u(r))dr)ds +∑ Γ( α − 1) ti−1 0 i=1 ˆ ˆ s m−1 (tm − ti ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 ˆ ˆ s m (η − tm ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1

+ad[

m

m−1

m

i=1 ξ

i=1

m

m−1

m

i=1

i=1

i=1

+ ∑ Ii (u(ti )) + i=1


ˆ

ˆ s 1 α −2 (ξ − s) φq ( f (r, u(r))dr)ds +cb[ Γ(α − 1) tm 0 ˆ ti ˆ s m 1 α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s m−1 (tm − ti ) ti (ti − s)α −3 φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 ˆ ˆ s m (ξ − tm ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 + ∑ Ii (u(ti )) +

∑ (tm − ti)Ji (u(ti )) + ∑ (ξ − tm)Ji (u(ti ))]).


477

Using Lemma 5 with y(t) = f (t, u(t)), problem (1) reduces to a fixed point problem u = Tu, where T is given by (7). Thus problem (1) has a solution if and only if the operator T has a fixed point. Lemma 6. The operator T : PC(J, R) → PC(J, R) defined by (7) is completely continuous. Proof. It is obvious that T is continuous in view of the continuity of f , Qk , Ik and Jk . Let Ω ⊂ PC(J, R) be bounded. Then, there exist positive constants Li > 0 (i = 1, 2, 3, 4) such that | f (t, u)| ≤ φ p (L1 ), |Qk (u)| ≤ L2 , |Ik (u)| ≤ L3 and |Jk (u)| ≤ L4 , ∀u ∈ Ω. Thus, ∀u ∈ Ω, we have ˆ s ˆ 1 1 1 α −1 (1 − s) φq ( | f (r, u(r))| dr)ds {bc[ |m0 | ≤ ac + ad + bc Γ(α ) tl 0 ˆ s l ˆ ti 1 + ∑ t (ti − s)α −1φq ( 0 | f (r, u(r))| dr)ds Γ(α ) i=1 i−1 ˆ ti ˆ s l−1 (tl − ti ) α −2 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ s l−1 2 (tl − ti) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 2Γ( α − 2) 0 t i−1 i=1 ˆ ˆ t s l i (1 − tl ) (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) 0 ti−1 i=1 ˆ ti ˆ s l−1 (1 − tl )(tl − ti ) α −3 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ti ˆ s l 2 (1 − tl ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 2Γ( α − 2) 0 t i−1 i=1 l

l−1

+ ∑ |Qi (u(ti ))| + ∑ (tl − ti ) |Ii (u(ti ))| i=1 l−1

i=1

(tl − ti 2 i=1

+∑

)2

l

|Ji (u(ti ))| + ∑ (1 − tl ) |Ii (u(ti ))| i=1

l−1

l

(1 − tl )2 |Ji (u(ti ))|] 2 i=1 i=1 ˆ η ˆ s 1 α −2 (η − s) φq ( | f (r, u(r))| dr)ds +bd[ Γ(α − 1) tm 0 ˆ ti ˆ s m 1 (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) 0 ti−1 i=1 ˆ ti ˆ s m−1 (tm − ti ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) ti−1 0 i=1 ˆ ˆ s m (η − tm ) ti (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) 0 ti−1 i=1 + ∑ (1 − tl )(tl − ti ) |Ji (u(ti ))| + ∑

m

m−1

i=1

i=1

+ ∑ |Ii (u(ti ))| + +(c + d)b[

m

∑ (tm − ti) |Ji (u(ti ))| + ∑ (η − tm) |Ji(u(ti ))|]

1 Γ(α − 1)

ˆ

i=1

ξ

tm

ˆ s (ξ − s)α −2 φq ( | f (r, u(r))| dr)ds 0

478


1 i=1 Γ(α − 1)

+∑

m−1

ˆ

ti

ti−1

ˆ

ti

ˆ s (ti − s)α −2 φq ( | f (r, u(r))| dr)ds 0

ˆ s α −3 (ti − s) φq ( | f (r, u(r))| dr)ds

(tm − ti ) 0 i=1 Γ(α − 2) ti−1 ˆ ti ˆ s m (ξ − tm ) +∑ (ti − s)α −3 φq ( | f (r, u(r))| dr)ds Γ( α − 2) 0 ti−1 i=1 +

∑

m

m−1

m

i=1

i=1

i=1

+ ∑ |Ii (u(ti ))| +

∑ (tm − ti) |Ji (u(ti ))| + ∑ (ξ − tm ) |Ji (u(ti ))|]}

ˆ ti ˆ 1 L1 L1 l 1 bc[ (1 − s)α −1 ds + ≤ ∑ t (ti − s)α −1ds ac + ad + bc Γ(α ) tl Γ(α ) i=1 i−1 ˆ ti ˆ ti l−1 l−1 L1 L1 (ti − s)α −2 ds + ∑ (ti − s)α −3 ds +∑ Γ( α − 1) 2Γ( α − 2) t t i−1 i−1 i=1 i=1 ˆ ti ˆ ti l l−1 L1 L1 (ti − s)α −2 ds + ∑ (ti − s)α −3 ds +∑ Γ( α − 1) Γ( α − 2) ti−1 ti−1 i=1 i=1 ˆ ti l L1 (ti − s)α −3 ds +∑ 2Γ( α − 2) ti−1 i=1 l

l−1

l−1

l l−1 l L4 L4 + ∑ L3 + ∑ L4 + ∑ ] i=1 i=1 i=1 2 i=1 i=1 i=1 2 ˆ η ˆ ti m L1 L1 α −2 (η − s) ds + ∑ (ti − s)α −2 ds +bd[ Γ(α − 1) tm Γ( α − 1) t i−1 i=1 ˆ ti ˆ ti m−1 l l l−1 l L1 L1 (ti − s)α −3 ds + ∑ (ti − s)α −3 ds + ∑ L3 + ∑ L4 + ∑ L4 ] +∑ i=1 Γ(α − 2) ti−1 i=1 Γ(α − 2) ti−1 i=1 i=1 i=1 ˆ ξ ˆ ti l L1 L1 (ξ − s)α −2 ds + ∑ (ti − s)α −2 ds +(c + d)b[ Γ(α − 1) tm Γ( α − 1) ti−1 i=1 ˆ ti ˆ ti l−1 l l l−1 l L1 L1 α −3 α −3 (ti − s) ds + ∑ (ti − s) ds + ∑ L3 + ∑ L4 + ∑ L4 ] +∑ i=1 Γ(α − 2) ti−1 i=1 Γ(α − 2) ti−1 i=1 i=1 i=1 1 1 1 [ (bc(1 + l)L1 ) + (bc(2l − 1)L1 + b(2d + c)(1 + l)L1 ) ≤ ac + ad + bc Γ(α + 1) Γ(α ) 3 1 (bc(2l − )L1 + b(2d + c)(2l − 1)L1 ) + bclL2 + Γ(α − 1) 2 3 +(bc(2l − 1) + b(2d + c)l)L3 + (bc(2l − ) + b(2d + c)(2l − 1))L4 ], 2

+ ∑ L2 + ∑ L3 + ∑


and

ˆ 1 ˆ s 1 1 α −1 ac[ (1 − s) φq ( | f (r, u(r))| dr)ds |m1 | ≤ ac + ad + bc Γ(α ) tl 0 ˆ s l ˆ ti 1 + ∑ t (ti − s)α −1φq ( 0 | f (r, u(r))| dr)ds Γ(α ) i=1 i−1 ˆ ti ˆ s l−1 (tl − ti ) α −2 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ ti s l−1 (tl − ti )2 (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ s l (1 − tl ) (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) 0 ti−1 i=1 ˆ ti ˆ s l−1 (1 − tl )(tl − ti ) α −3 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ˆ t s l i (1 − tl )2 (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 2Γ( α − 2) 0 ti−1 i=1 l

l−1

+ ∑ |Qi (u(ti ))| + ∑ (tl − ti) |Ii (u(ti ))| i=1 l−1

(tl − ti 2 i=1

+∑

l−1

i=1

)2

l

|Ji (u(ti ))| + ∑ (1 − tl ) |Ii (u(ti ))| i=1

l

(1 − tl )2 |Ji (u(ti ))|] 2 i=1 ˆ η ˆ s (η − s)α −2 φq ( | f (r, u(r))| dr)ds

+ ∑ (1 − tl )(tl − ti ) |Ji (u(ti ))| + ∑ i=1

1 Γ(α − 1) tm 0 ˆ ti ˆ s m 1 (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) ti−1 0 i=1 ˆ ti ˆ s m−1 (tm − ti ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) t 0 i−1 i=1 ˆ ti ˆ s m (η − tm ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) 0 t i−1 i=1

+ad[

m

m−1

m

i=1 ξ

i=1

m

m−1

m

i=1

i=1

i=1

+ ∑ |Ii (u(ti ))| + i=1

ˆ

∑ (tm − ti) |Ji(u(ti ))| + ∑ (η − tm ) |Ji (u(ti ))|]

ˆ s 1 α −2 (ξ − s) φq ( | f (r, u(r))| dr)ds +cb[ Γ(α − 1) tm 0 ˆ ti ˆ s m 1 (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) 0 t i−1 i=1 ˆ ˆ s m−1 (tm − ti ) ti (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 Γ(α − 2) ti−1 ˆ ti ˆ s m (ξ − tm ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) 0 ti−1 i=1 + ∑ |Ii (u(ti ))| +

∑ (tm − ti) |Ji(u(ti ))| + ∑ (ξ − tm) |Ji (u(ti ))|]}

479

480


≤

1 1 1 [ (ac(1 + l)L1 ) + (ac(2l − 1)L1 + (ad + bc)(1 + l)L1 ) ac + ad + bc Γ(α + 1) Γ(α ) 1 3 + (ac(2l − )L1 + (ad + bc)(2l − 1)L1 ) + aclL2 Γ(α − 1) 2 3 +(ac(2l − 1) + (ad + bc)l)L3 + (ac(2l − ) + (ad + bc)(2l − 1))L4 ]. 2

Therefore, 1 |Tu(t)| ≤ Γ(α ) +

ˆ

t

tk

α −1

(t − s)

k

1 ∑ Γ(α ) i=1

ˆ

ti

ti−1

k−1

(tk − ti) +∑ i=1 Γ(α − 1)

ˆ s φq ( | f (r, u(r))| dr)ds 0

ˆ s (ti − s)α −1 φq ( | f (r, u(r))| dr)ds ˆ

0

ti

ti−1

ˆ

α −2

(ti − s)

ˆ s φq ( | f (r, u(r))| dr)ds 0


ti (tk − ti)2 0 i=1 2Γ(α − 2) ti−1 ˆ ti ˆ s k (t − tk ) (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 1) 0 ti−1 i=1 ˆ ti ˆ s k−1 (t − tk )(tk − ti ) α −3 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ Γ(α − 2) ti−1 0 i=1 ˆ ˆ ti s k (t − tk )2 (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 k−1

+∑

k

k−1

+ ∑ |Qi (u(ti ))| + ∑ (tk − ti) |Ii (u(ti ))| i=1 k−1

i=1

(tk − ti 2 i=1

+∑

)2

k

|Ji (u(ti ))| + ∑ (t − tk ) |Ii (u(ti ))| i=1

k−1

k

(t − tk )2 |Ji (u(ti ))| + |m0 | + |m1 | 2 i=1 l ˆ ti ∑ (ti − s)α −1ds

+ ∑ (t − tk )(tk − ti ) |Ji (u(ti ))| + ∑ i=1

L1 ≤ Γ(α )

ˆ

t

tk

(t − s)α −1 ds +

ti−1

ˆ ti L1 (ti − s) ds + ∑ (ti − s)α −3 ds 2Γ( α − 2) ti−1 t i−1 i=1 ˆ ti ˆ ti l l−1 L1 L1 (ti − s)α −2 ds + ∑ (ti − s)α −3 ds +∑ Γ( α − 1) Γ( α − 2) ti−1 ti−1 i=1 i=1 ˆ ti l l l−1 l−1 l l−1 L1 L4 + ∑ L3 + ∑ L4 (ti − s)α −3 ds + ∑ L2 + ∑ L3 + ∑ +∑ i=1 2Γ(α − 2) ti−1 i=1 i=1 i=1 2 i=1 i=1 l−1

L1 +∑ i=1 Γ(α − 1)

ˆ

L1 Γ(α ) i=1

ti

α −2

l−1

k

L4 + |m0 | + |m1 | . i=1 2

+∑ Hence,

|Tu(t)| ≤

λ1 (1 + l)L1 (λ1 (2l − 1) + λ2 l)L1 + Γ(α + 1) Γ(α )

(8)


481

(λ1 (2l − 32 ) + λ2 (2l − 1))L1 + λ1 lL2 Γ(α − 1) 3 +(λ1 (2l − 1) + λ2 l)L3 + (λ1 (2l − ) + λ2 (2l − 1))L4 . 2 +

Since t ∈ [0, 1], therefore there exists a positive constant L, such that Tu ≤ L, which implies that the operator T is uniformly bounded. On the other hand, for any t ∈ Jk , 0 ≤ k ≤ l, we have ˆ t ˆ s 1 (Tu) (t) ≤ (t − s)α −2 φq ( | f (r, u(r))| dr)ds Γ(α − 1) tk 0 ˆ ti ˆ s k 1 α −2 (ti − s) φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ s k−1 (tk − ti ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) 0 ti−1 i=1 ˆ ti ˆ s k (t − tk ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ( α − 2) 0 t i−1 i=1 k

k−1

i=1 k

i=1

+ ∑ |Ii (u(ti ))| + ∑ (tk − ti ) |Ji (u(ti ))| + ∑ (t − tk ) |Ji (u(ti ))| + |m1 | i=1

≤

ac(1 + l)L1 (λ3 (1 + l) + ac(2l − 1))L1 1 [ + ac + ad + bc Γ(α + 1) Γ(α ) 3 (λ3 (2l − 1) + ac(2l − 2 ))L1 + aclL2 + Γ(α − 1) 3 ¯ +(λ3 l + ac(2l − 1))L3 + (λ3 (2l − 1) + ac(2l − ))L4 ] := L. 2

Hence, for t1 , t2 ∈ Jk , t1 < t2 , 0 ≤ k ≤ l, we have ˆ |(Tu)(t2 ) − (Tu)(t1 )| ≤

(Tu) (s) ds ≤ L(t ¯ 2 − t1 ),

t2

t1

which implies that T is equicontinuous on all Jk , k = 0, 1, 2, . . . , l. Thus, by the Arzela-Ascoli Theorem, the operator T : PC(J, R) → PC(J, R) is completely continuous. Theorem 7. Let lim

u→0

least one solution.

f (t, u) Qk (u) Ik (u) Jk (u) = 0, lim = 0, lim = 0 and lim = 0, then the problem (1) has at u→0 u→0 u u→0 u φ p (u) u

f (t, u) Qk (u) Ik (u) Jk (u) = 0, lim = 0, lim = 0 and lim = 0, therefore there exists a constant u→0 u→0 u u→0 u φ p (u) u r > 0 such that | f (t, u)| ≤ φ p (δ1 ) |φ p (u)| , |Qk (u)| ≤ δ2 |u| , |Ik (u)| ≤ δ3 |u| and |Jk (u)| ≤ δ4 |u| for 0 < |u| < r, where δi > 0 (i = 1, 2, 3, 4) satisfy the inequality Proof. Since lim

u→0

{

λ1 (1 + l)δ1 (λ1 (2l − 1) + λ2 l)δ1 (λ1 (2l − 32 ) + λ2 (2l − 1))δ1 + + Γ(α + 1) Γ(α ) Γ(α − 1) +λ1 l δ2 + (λ1 (2l − 1) + λ2 l)δ3 + (λ1 (2l − 32 ) + λ2 (2l − 1))δ4 } ≤ 1.

(9)

482


Let us set Ω = {u ∈ PC(J, R)| u < r} and take u ∈ PC(J, R) such that u = r, that is, u ∈ ∂ Ω. Then, by the process used to obtain (8), we have |Tu(t)| ≤ {

λ1 (1 + l)δ1 (λ1 (2l − 1) + λ2 l)δ1 (λ1 (2l − 32 ) + λ2 (2l − 1))δ1 + + + λ1 l δ2 Γ(α + 1) Γ(α ) Γ(α − 1) +(λ1 (2l − 1) + λ2 l)δ3 + (λ1 (2l − 32 ) + λ2 (2l − 1))δ4 } u .

(10)

Thus, it follows that Tu ≤ u , u ∈ ∂ Ω. Therefore, by Theorem 3, the operator T has at least one fixed point, ¯ which in turn implies that the problem (1) has at least one solution u ∈ Ω. Theorem 8. Assume that there exist positive constants Li (i = 1, 2, 3, 4) such that | f (t, u)| ≤ φ p (L1 ), |Qk (u)| ≤ L2 , |Ik (u)| ≤ L3 , |Jk (u)| ≤ L4 , for t ∈ J, u ∈ R and k = 1, 2, . . . , l.

(11)

Then the problem (1) has at least one solution. Proof. Let us show that the set V = {u ∈ PC(J, R)|u = μ Tu, 0 < μ < 1} is bounded. Let u ∈ V, then u = μ Tu, 0 < μ < 1. For any t ∈ J, we have ˆ t ˆ s μ α −1 (t − s) φq ( f (r, u(r))dr)ds u(t) = Γ(α ) tk 0 ˆ s ˆ μ k ti α −1 + ∑ t (ti − s) φq( 0 f (r, u(r))dr)ds Γ(α ) i=1 i−1 ˆ ˆ s k−1 μ (tk − ti) ti (ti − s)α −2 φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s k−1 μ (tk − ti )2 ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 ˆ ˆ s k μ (t − tk ) ti α −2 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ˆ s k−1 μ (t − tk )(tk − ti ) ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ˆ s k μ (t − tk )2 ti α −3 (ti − s) φq ( f (r, u(r))dr)ds +∑ 0 i=1 2Γ(α − 2) ti−1 k

k−1

i=1 k−1

i=1

+ ∑ μ Qi (u(ti )) + ∑ μ (tk − ti )Ii (u(ti )) k μ (tk − ti)2 Ji (u(ti )) + ∑ μ (t − tk )Ii (u(ti )) +∑ 2 i=1 i=1 k−1 k μ (t − tk )2 Ji (u(ti )) − μ m0 − μ m1t. + ∑ μ (t − tk )(tk − ti )Ji (u(ti )) + ∑ 2 i=1 i=1

Combining (11) and (12) and employing the procedure used to obtain (8), we obtain |u(t)| = μ |Tu(t)|

(12)


≤

1 Γ(α )

ˆ

t

tk

483

ˆ s (t − s)α −1 φq ( | f (r, u(r))| dr)ds

k

1 + ∑ Γ(α ) i=1

ˆ

0

ti

α −1

ti−1

k−1

(tk − ti ) i=1 Γ(α − 1)

+∑

k−1

)2

(ti − s) ˆ

ti

ti−1

ˆ

ˆ s φq ( | f (r, u(r))| dr)ds 0

ˆ s (ti − s)α −2 φq ( | f (r, u(r))| dr)ds

ti

0


(tk − ti 0 i=1 2Γ(α − 2) ti−1 ˆ ˆ ti s k (t − tk ) (ti − s)α −2 φq ( | f (r, u(r))| dr)ds +∑ 0 i=1 Γ(α − 1) ti−1 ˆ ti ˆ s k−1 (t − tk )(tk − ti ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ Γ(α − 2) 0 ti−1 i=1 ˆ ti ˆ s k 2 (t − tk ) (ti − s)α −3 φq ( | f (r, u(r))| dr)ds +∑ 2Γ( α − 2) t 0 i−1 i=1 +∑

k

k−1

+ ∑ |Qi (u(ti ))| + ∑ (tk − ti ) |Ii (u(ti ))| i=1 k−1

(tk − ti 2 i=1

+∑

i=1

)2

k−1

k

|Ji (u(ti ))| + ∑ (t − tk ) |Ii (u(ti ))| i=1

k

(t − tk )2 |Ji (u(ti ))| 2 i=1

+ ∑ (t − tk )(tk − ti ) |Ji (u(ti ))| + ∑ i=1

+ |m0 | + |m1 |

λ1 (1 + l)L1 (λ1 (2l − 1) + λ2 l)L1 (λ1 (2l − 32 ) + λ2 (2l − 1))L1 + + Γ(α + 1) Γ(α ) Γ(α − 1) 3 +λ1 lL2 + (λ1 (2l − 1) + λ2 l)L3 + (λ1 (2l − ) + λ2 (2l − 1))L4 2 := L, ≤

which implies that u ≤ L for any t ∈ J. So, the set V is bounded. Thus, by Theorem 4, the operator T has at least one fixed point. Hence the problem (1) has at least one solution. 4 Examples Example 1. For 2 < α ≤ 3, consider the following fractional order impulsive boundary value problem ⎧ ⎪ (φ4 (C Dα0+ u(t))) = t 3 u4 (t) + u5 (t) arctan u(t), ⎪ ⎪ ⎪ ⎪ 0 < t < 1, t = 14 , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ Δu( 14 ) = eu( 4 ) u2 ( 14 ), ⎪ ⎪ ⎪ ⎨ 2u5 ( 14 ) ( 1 ) = , Δu ⎪ 4 ⎪ 3 + 2u3 ( 14 ) ⎪ ⎪ ⎪ ⎪ Δu ( 14 ) = u2 ( 14 ) + sin(1 − cosu( 14 )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3u(0) − 2u ( 1 ) = 0, u(1) + 5u ( 45 ) = 0, ⎪ ⎪ ⎩ u (0) = 0. 3

(13)

484


where l = 1. Clearly, all the assumptions of Theorem 7 hold. Thus, by the conclusion of Theorem 7, we can get that the fractional order impulsive boundary value problem (13) has at least one solution. Example 2. Consider the following fractional order impulsive boundary value problem ⎧ 1 cos u(t) ⎪ ⎪ , 0 < t < 1, t = , (φ3 (C Dα0+ u(t))) = ⎪ 4 ⎪ 4 + u (t) 3 ⎪ ⎪ ⎪ 1 ⎪ −u2 ( 13 ) u( 13 ) ⎪ ⎪ + 5 sin(1 + e ), Δu( ) = e ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎨ 1 7 + 3u2 ( 13 ) , Δu ( ) = 3 2 + 5u2 ( 13 ) ⎪ ⎪ t ⎪ ⎪ ( 1 ) = e cos u(t) , ⎪ ⎪ Δu ⎪ ⎪ 3 5 + u3 (t) ⎪ ⎪ ⎪ ⎪ u(0) − 2u ( 15 ) = 0, 3u(1) + 5u ( 27 ) = 0, ⎪ ⎪ ⎪ ⎩C α D0+ u(0) = 0, u (0) = 0.

(14)

where l = 1. 1 3 e Obviously L1 = , L2 = 6, L3 = and L4 = , and the condition of Theorem 8 holds. Therefore, by 2 5 5 Theorem 8, the boundary value problem (14) has at least one solution. Acknowledgements We would like to thank the referees for their valuable comments and suggestions. References [1] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier, Amsterdam. [2] Lakshmikantham, V., Leela, S. and Vasundhara Devi, J. (2009), Theory of Fractional Dynamic Systems, Cambridge Academic Publisher, Cambridge. [3] Agarwal, R.P., Belmekki, M. and Benchohra, M. (2009), A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations, 2009, Art. ID 981728. [4] Ahmad, B. and Nieto, J.J. (2009), Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications. An International Journal, 58, 1838–1843. [5] Balachandran, K. and Trujillo, J.J. (2010), The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 72, 4587–4593. [6] Belmekki, M., Nieto, J.J. and Rodriguez-Lopez, R. (2009), Existence of periodic solution for a nonlinear fractional differential equation, Boundary Value Problems, Art. ID 324561, 18. [7] Benchohra, M., Hamani, S. and Ntouyas, S.K. (2009), Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 71, 2391–2396. [8] Darwish, M.A. and Ntouyas, S.K. (2010), On initial and boundary value problems for fractional order mixed type functional differential inclusions, Computers & Mathematics with Applications. An International Journal, 59, 1253– 1265. [9] Edelman, M. and Tarasov, V.E. (2009), Fractional standard map, Physics Letters A, 374(2), 279–285. [10] Machado,J.A.T. (2013), Fractional generalization of memristor and higher order elements, Communications in Nonlinear Science and Numerical Simulation, 18, 264–275. [11] Sabatier, J., Agrawal, O.P. and Machado, J.A.T. (2007), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht.


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About Utility of the Simplified Grünwald-Letnikov Formula Equivalent Horner Form Dariusz W. Brzeziński†, Piotr Ostalczyk Institute of Applied Computer Science, Lodz University of Technology, 18/22 Stefanowskiego St.,90-924 Łód´z, Poland Submission Info Communicated by Mark Edelman Received 7 November 2014 Accepted 2 March 2015 Available online 1 January 2016 Keywords Grünwald-Letnikov / Horner Form Fractional Order BackwardDifference/Sum Numerical Calculations Accuracy “Calculation Tail” concept Application inReal-Time Computations

Abstract First we discuss some crucial factors that determine numerical calculations accuracy of the Grünwald-Letnikov formula and its equivalent Horner form. Then we introduce simplified variants of both formulas and the concept of the calculation tail. We analyze the utility of its length for mitigation of a time and a memory shortages influence on the accuracy in realtime microprocessor calculations. Credibility of the conclusions is lent by the comparison of the results obtained on a PC and on a real-time DSP system.


1 Introduction There are several formulas [1–10] which can be applied to calculate fractional order integrals and derivatives numerically (FOD/I). They include the Riemann-Liouville/Caputo and the Grünwald-Letnikov formulas. The Riemann-Liouville/Caputo formula application obligates traditional approach to FOD/I numerical calculations, i.e. there can be applied well know methods of numerical integration as for example the Gauss Quadratures. However they must be heavily modified to obtain high accurate results [11]. The Grünwald-Letnikov formula enforces an approach in a form of the fractional order backward difference/sum (FOBD/S) approximation method. Still, the FOBD/S application effect is equivalent to integration approach to FOD/I numerical calculations [1, 12] but much simpler to code, calculate and apply practically. For this reason it is a usual choice for many technical applications, among others in control systems [13–22]. The Grünwald-Letnikov method’s core consists of a multiplication of some coefficients and a function’s values, of which difference/sum is to calculate. Previous works of ours [23, 24] and the analysis of the Grünwald-Letnikov formula confirm some crucial factors which influence the accuracy of numerical calculations with its application, i.e. an amount of applied coefficients in calculations and monotonicity of a function. † Corresponding


488

Dariusz W. Brzeziński, Piotr Ostalczyk / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 487–498

In more of our works [25, 26] we stated the fact that due to increasing number of necessary mathematical operations for a set accuracy, real-time microprocessor calculations applying the Grünwald-Letnikov formula can be flawed by a time and a memory shortages. To mitigate their influence on the accuracy of FOBD/S numerical calculations we introduce Horner’s form of the Grünwald-Letnikov formula as well as simplified variants of both formulas. After that we apply so called the calculation tail to simplified variants of the formulas and evaluate its length influence on the accuracy. The paper is constructed as follows: We begin, in Section 2, with mathematical introduction to the Grünwald-Letnikov formula and its equivalent Horner form. This section contains the description of both formulas’ simplified variants as well as the algorithm for the application of the calculation tail. Section 3 contains evaluation results for the Grünwald-Letnikov formula and its Horner equivalent form in context of an coefficients amount required for a set accuracy. We calculate some FOBD/S to evaluate functions’ characteristics which influence the accuracy of numerical calculations, i.e. their monotonicity, frequency or FOBD/S order. The results are required as reference values for an experiment described in next Section. In the main Section 4 we apply practically simplified variants of the Grünwald-Letnikov formula and its equivalent Horner form. We evaluate how the calculation tail of different lengths influence numerical calculations accuracy, i.e. how many of the coefficients can be removed during the calculations and how it decreases the final accuracy. This is the utility for the mitigation of the time and the memory shortages influence on the calculations accuracy of real-time microprocessor calculations. In section 5 we do some practical numerical accuracy comparison tests; we compare PC system simulated results with the results of similar as in the section 4 calculations using a real-time DSP system. It lends the credibility to the PC simulation results from section 3 and 4 and the final conclusions, which are presented in section 6. 2 Mathematical preliminaries The Grünwald-Letnikov formula of FOBD calculations is a pretty straightforward formula. Its construction and algorithmic simplicity make it predisposed for computer applications. The Horner form is the Grünwald-Letnikov formula to which the well known Horner’s scheme of polynomial’s value calculation is applied. The scheme possesses some significant computational advantages like lower computational complexity and natural method of reading input data. 2.1

¨ The Grunwald-Letnikov formula (GL)

For a given discrete-time, real bounded function f (k) = f0 , f1 , . . . , fk−1 , fk the Grünwald-Letnikov formula of the FOBD is defined as [1–5, 8, 10] GL (ν ) 0 Δk

k

(ν )

f (k) = ∑ ai

fk−i ,

(1)

i=0

where ν ∈ R is the FOBD order. FOBS is defined as the FOBD evaluated for negative order, fk is a discrete time (ν ) function and ai are the coefficients for i = 0, 1, 2, 3, . . . , k − 1, k. (ν ) The coefficients ai can be calculated applying the following formula ⎧ 0 for i = −1, −2, −3, . . . ⎨ (ν ) 1 for i = 0 (2) ai = ⎩ i ν (ν −1)···(ν −i+1) for i = 1, 2, 3, . . . . (−1) i! The relation between coefficients (2) leads to the recurrent formula (ν )

ai

(ν )

= ai−1 (1 −

1+ν ) for i = 1, 2, 3, . . . . i

(3)

Dariusz W. Brzeziński, Piotr Ostalczyk / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 487–498 Value

Value

Value

Value

1.0

1.0

1.0

1.0

0.5

0.5

0.5

0.5

0.0

N

0.0

N

0.0

N

0.0

0.5

0.5

0.5

0.5

1.0

1.0

1.0

1.0

(a)

489

(b)

(c)

N

(d)

Fig. 1 Ten first coefficients a(ν ) for FOBD (a), FOBDS (c) of order ν = 0.2 and FOBD (b), FOBS (d) of order 0.8. (ν )

Another calculations method of the coefficients ai involves application of Euler’s Gamma function. For ν ∈R Γ (i − ν ) (ν ) for i = 0, 1, 2, 3, . . . . ai = Γ (−ν ) Γ (i + 1) Formula (1) can be expressed in a matrix-vector form ⎡ GL (ν ) 0 Δk

2.2

f (k) =

(ν ) (ν ) a0 a1

···

(ν ) ak

fk

⎤

⎢ fk−1 ⎥ ⎢ ⎥ ⎢ .. ⎥ . ⎣ . ⎦ f0

(4)

¨ Horner form (H) of the Grunwald-Letnikov formula (ν )

Applying the same assumptions as in the Grünwald-Letnikov definition and utilizing the coefficients ci (ν ) ci

=

⎧ ⎨

0 1

⎩ i−1−ν i

for i = −1, −2, −3, . . . for i = 0 for i = 1, 2, 3, . . . ,

we can apply Horner’s scheme to (1) to calculate the FOBD/S [8, 9] (ν ) (ν ) (ν ) (ν ) (ν ) H (ν ) Δ f (k) = c + c + c + · · · + c . f f f + c f f k k−1 k−2 0 k 0 0 1 2 k−1 1 k

(5)

(6)

The formulas (1) and (6) are equivalent [9] GL (ν ) 0 Δk

(ν )

f (k) =H 0 Δk f (k) .

(7)

Analyzing formula (1) we can discover that the number of mathematical operations increases with the incrementing k. This arises the need of their reduction [23–25]. To reduce a calculation time and a computer memory as well as mathematical operations required, we can remove some of the coefficients (2), (5) from (1) and (6). According to [8, 9] we can apply the properties of the coefficients (2) and (5) for that purpose. The consideration is focused on the order interval (0, 1) because for ν 1 we can always split ν into an integer part n ∈ Z+ and a fractional part ν ∈ (0, 1) and perform the differentiation procedure as operators concatenation Dn+ν f (t) = Dν (Dn f (t)) , where D is differentiation operator [1].

490

2.3


¨ Simplified variants of the Grunwald-Letnikov formula (GLs) and its equivalent Horner form (Hs)

One can prove [9] that (ν ) lim a i→∞ i

= 0,

(8)

(ν )

= 1,

(9)

lim ci

i→∞

and for some i > k − L

(ν )

≈ 0,

(10)

(ν )

≈ 1,

(11)

ai ci

it is possible to define the simplified variants of the formulas (1) and (6): (ν ) for 0 < k ≤ L ∑ki=0 ai fk−i ( ν ) GLs 0 Δk,L f (k) = (ν ) k ∑i=k−L ai−k+L f2k−L−i for k > L, and Hs (ν ) 0 Δk,L

⎧ ⎨ c(ν ) fk + c(ν ) fk−1 + c(ν ) fk−2 + · · · + c(ν ) f + c(ν ) f for k ≤ L 1 0 0 1 2 k−1 k f (k) =

( ν ) ( ν ) (ν ) ⎩c for k > L. fk + c1 fk−1 + · · · + ck−L ∑Li=0 fi 0

(12)

(13)

3 Factors which influence GL and H accuracy of computations There are many factors which influence the accuracy of numerical calculations applying (1) and (6). They include an amount of the coefficients (2) and (5) applied during the calculations. How many of them are required to obtain results with a set accuracy depends in turn on the FOBD/S order, monotonicity of a function and frequency, in case of a periodic function. We may narrow our research to the coefficients (2) applied in the method (1) because [8, 9] (ν )

ai

(ν )

(ν )

= ai−1 ci .

The method presented in the algorithm of (1) suggests that there should be first analyzed some simple functions which differs from each other in monotonicity. We include monotonically increasing function f (k) = kh,

(14)

and monotonically decreasing function f (k) = 1 − kh.

(15) t−t0 k ,

Functions (14)–(15) as well as the others presented in the paper (18)–(23) are evaluated for: t = kh, h = t0 = 0, t = 1, k = 0, 1, 2, 3, . . . , k − 1, k. They equal implicitly 0 for t < 0 (e.g. they are multiplied by unit step). First we calculate FOBD of order ν = 0.45. FOBS are evaluated for negative order ν = −0.45. Before we present the results, here are some important explanations to the numerical implementation. All programs for the numerical experiment are written in C++ with the use of double precision. Required values of the FOD/I assumed as exact for the accuracy assessment are calculated applying high accuracy algorithms [11]. The examined accuracy is expressed as relative error in context of an amount of the coefficients (2) v c (16) er (N) = 1 − , ve


491

where vc is a calculated value, ve a value assumed as exact one and N is the number of applied coefficients (2) in FOBD/S calculations. Bearing in mind that the average analogue-digital converter has a resolution of 12 bits, the set accuracy is assumed as er (N) ≤ 1.0e − 04

(17)

in context of an amount of the applied coefficients (2). The experiments were conducted using high-accuracy PC computer armed with Intel i7 2600K processor and 64-bit Linux OS with G++ compiler.

Fig. 2 An amount of the coefficients (2) required for the accuracy (17). FOBD/S of order ν = 0.45 for the functions (14) and (15).

Fig.2 contains the comparison chart of an amount of the coefficients (2) required for FOBD/S computations with the accuracy (17) in case of functions (14)-(15). It shows that the most influencing factor in this part of the evaluation is monotonicity of a function. The first dozen of the coefficients (2) (presented in Fig.1) multiplied by the last values of the function contribute the most in the final value of the FOBD/S. The rest of them merely decreases an excess or insufficiency. For this reason if the function is monotonically increasing (14) there is required fewer coefficients to complete the calculations with the set accuracy than in the case of a function which is monotonically decreasing (15). A case of a constant function f (k) = 1 (kh)

(18)

is discussed in the next section. Now we analyze in which way the increasing/decreasing the order of FOBD/S influences an amount of coefficients (2) required for the accuracy (17). For this purpose we calculate FOBD/S of orders 0.1, 0.5 and 0.9 of the functions (14), (15) and (18). Fig.3 contains comparison charts of an amount of the coefficients (2) required for the accuracy (17) for the selected FOBD/S orders for the functions (14),(15) and (18). It shows that the increasing/decreasing order decreases/increases an amount of the required coefficients respectively. In case of (18) the FOBD/S calculations consist of the coefficients (2) summation. An amount of required coefficients for the accuracy (17) depends solely on the calculated order wherein the fractional integral of the order ν = −0.5 requires highest amount of them. In case of the fractional derivatives for this function, required amount increases with an calculated order.

492


(a)

(b)

Fig. 3 An amount of the coefficients (2) required for the accuracy (17). FOBS (a) and FOBD (b) of order ν = 0.1, 0.5 and 0.9 for the functions (14), (15) and (18).

Next we examine some other functions. They include exponentially decaying function f (k) = e−2(kh) ,

(19)

f (k) = e−2(kh) sin (2π kh) ,

(20)

exponentially decaying sine function

exponentially decaying high frequency sine function f (k) = e−2(kh) sin (8π kh) ,

(21)

f (k) = sin (4π kh) ,

(22)

high frequency sine function

and π /2 shifted sine function i.e. cosine function f (k) = cos (4π kh) .

(23)

We calculate FOBD of order ν = 0.45. FOBS are evaluated for negative order ν = −0.45. Fig.4 contains the comparison chart of the coefficients (2) amount required for FOBD/S computations with the accuracy (17) for the functions (19)-(23). As it is to see, frequency of a periodic function has enormous influence on an coefficients amount required for the set accuracy (17). An amount of required coefficients increases intensively with a frequency of a function. If additionally a bounding box of a function is of increasing or constant character and there is low order to compute, an amount of required coefficients for the accuracy (17) could reach many millions, even billions (see charts for the functions (21) and (23) in Fig.4).


493

Fig. 4 An amount of the coefficients (2) required for the accuracy (17). FOBD/S of order ν = 0.45 for the functions (19)-(23).

4 Determining The calculation tail length influence on the FOBD/S numerical calculations accuracy. experiment details Expanding the knowledge obtained in the experiment described in Section 3, the following experiment involves evaluation of the so called calculation tail length influence on the accuracy of numerical calculations of FOBD/S for selected functions. The length is denoted from now on as L. Evaluated percentage length of the tail L is: 5, 10, 20 and 30% of the former amount of coefficients (2) and (5) required for a set accuracy (see Figs 2 and 3). Decreased accuracy criterion presented in Tabs 1-4 after cut off of L coefficients applying (13) and (14) is relative error expressed in % casted to 3 decimal digits (to obtain coherent results with Section 5): vc (24) er (L) = 1 − · 100%, ve where vc is a calculated value by application of the formulas (13) or (14) with the tail of the length L, ve is a reference value obtained by application high accuracy algorithms [11], which have been mentioned already in Section 3. The results presented in Tabs 1-4 shows that the equivalent Horner’s form of the Grünwald-letnikov formula (14) application enables to reduce significantly computational complexity of FOBD/S. The calculation tail of 5% and 10% length does not decrease overall accuracy over allowed level (17). It even increases accuracy slightly in some cases. Higher coefficients percentage removal decreases the accuracy, especially for coefficients demanding functions. However, if there is accepted certain accuracy loss (for example 5% from the reference (17)), there can be removed even 30% of them. Table 1 Error (24) for L = 5%

ν 0.1 0.5 0.9 −0.1 −0.5 −0.9

GLs

Hs

GLs

Hs

Function 18 GLs Hs

0.5 2.6 4.7 0.5 2.5 4.4

-0.01 0.08 0.22 -0.01 0.01 -0.01

4.5 2.5 0.5 5.5 7.4 9.3

0 0 0 0 0 0

49.7 7.7 1.1 49.2 7.2 0.8

14

15

0.11 0.08 0.01 0.13 0.1 0.02

20

23

GLs

Hs

GLs

Hs

793.3 166.9 75.1 761.1 219.6 14.8

0.19 0.04 0 0.35 5.37 0.15

9.8 39.6 351.8 28.5 78.6 405.2

-0.01 0 -0.01 0 0.16 0.74

494


Table 2 Error (24) for L = 10%

ν

GLs

Hs

GLs

Hs

Function 18 GLs Hs

0.1 0.5 0.9 −0.1 −0.5 −0.9

1.06 5.41 9.95 1.05 5.13 9.05

0.05 0.43 1.03 0.03 0.12 0.03

9.05 5.13 1.05 10.94 14.62 18.14

0 0.01 0 -0.01 0 0

100 15.67 2.27 97.9 13.84 1.06

14

15

0.53 0.41 0.09 0.55 0.4 0.1

20

23

GLs

Hs

GLs

Hs

717.9 271.6 205.6 309.9 1075.9 49.4

1.25 0.29 0.02 2.81 34.53 0.95

54 26 531 84.3 165.8 695.3

0.02 0.07 0.04 0.04 0.07 2.84


ν 0.1 0.5 0.9 −0.1 −0.5 −0.9

GLs

Hs

GLs

Hs

Function 18 GLs Hs

2.25 11.8 22.2 2.2 10.55 18.15

0.28 2.15 5.25 0.22 0.6 0.19

18.2 10.56 2.21 21.76 28.44 34.55

0.02 0.08 0.04 0 0.04 0.01

202.29 32.92 4.93 193.38 25.22 0.01

14

15

2.49 2 0.53 2.44 1.74 0.42

20

23

GLs

Hs

GLs

Hs

903.5 78 40.4 1230.1 1538.1 19.2

4.39 1.05 0.09 8.84 117.32 3.16

171.5 121.9 190.1 190.42 242.13 573.8

0.1 0.25 0.17 0.17 1.79 7.96


ν 0.1 0.5 0.9 −0.1 −0.5 −0.9

GLs

Hs

GLs

Hs

Function 18 GLs Hs

3.62 19.52 37.89 3.49 16.32 27.4

0.79 6.07 15.3 0.61 1.58 0.5

27.46 16.33 3.5 32.45 41.43 41.29

0.08 0.33 0.17 0.05 0.23 0.08

307.3 52.2 8.1 285.98 33.86 3.26

14

15

6.5 5.42 1.5 6.21 4.28 0.99

20

23

GLs

Hs

GLs

Hs

1239.5 210 59 1332 731.9 11.7

5.89 1.44 0.12 11.79 153.89 4.1

190.1 239.3 551.5 171.7 123.8 195.7

0.1 0.24 0.16 0.16 1.19 7.85

The Grünwald-letnikov formula (1) is unsuitable for the procedure of the the calculation tail length application due to the fact that GLs (13) is actually GL(1) with reduced by k − L number of the coefficients (2). 5 Credibility of the experiments results: comparison of a PC system and a real-time DSP system Considering the problems of a real-time microprocessor calculations there is a necessity to lend the credibility to PC system simulation results which were presented in the past sections. The most obvious method is to compare them to a real-time calculations system results in the similar disciplines. The following experiment is conducted using high-accuracy PC computer and real-time DSP system. The real-time system presented in Fig.5 consisted of DSP processor, RS232 port, I2C interface, digital to analogue


495

Fig. 5 The scheme of the real-time calculation system.

converter (DAC) and the oscilloscope. A PC system has been already presented in Section 3. Communication between the PC and the DSP was maintained by serial port using RS232 and the digital signal was sent from DSP to DAC through I2C interface. The DAC used in the evaluation had 12 bits resolution. It was possible to change voltage with smallest step equals 1.4mV applying maximum voltage possible to enforce (5.74V). The calculations were carried out using the DSP processor. Data were sent to it via RS232 then the calculations of FOBD/S were carried out applying formulas (13) and (14). Obtained values were casted on 0 − 4096 range due to the limitation of the used DAC. Data were sent from the DSP to the DAC using I2C interface and the calculated values as voltage values were measured by 0.001V exact oscilloscope. Finally the error (24) was calculated. On high-accuracy PC system the values of calculated FOBD/S were obtained by C++ programs. Results were casted to three decimal digits to ensure measurement compatibility with described real-time system before there was error (24) calculated. There were used 600 of the coefficients (2) and (5) to compute reference values, because such an amount is used in practical, real-time applications (in this case the number of possible coefficients was determined by the amount of available memory in tested DSP-system). There were calculated FOBD/S of the functions (18) and (19) wherein the parameters 4 and 10 added to them ensured higher value of voltage displayed by the oscilloscope. It minimized the influence of the noise on the results readout. The parameters do not change coefficients requirements of the functions for a set accuracy (17) in comparison to the same functions without the parameters added: f (k) = 4 (kh) ,

(25)

f (k) = 10 e−2(kh) .

(26)

The core part of the experiment consisted of three stages • computing reference values of FOBD/S applying 600 of the coefficients (2) and (5) and calculating relative errors to the assumed exact values obtained by [11], • comparing calculated values provided by the high-accuracy PC system and the real-time DSP system, • assuming that decreasing the calculations accuracy by 0.1% will not affect the proper operation of a reallife device, the influence of the calculations tail length L was investigated applying formulas (13) and (14). The results presented in Tabs 5-7 confirm the conclusions already presented in Section 4:

496


1. The simplified variant of the Grünwald-Letnikov formula is unsuitable for the coefficients removal procedure 2. Horner’s simplified variant application enables to remove 5% of the coefficient without loss of accuracy over assumed 0.1% 3. An coefficients amount possible to truncate from 600 depends on the factors presented in Section 3

Table 5 Percentage error applying 600 coefficients: PC vs RTS. −0.1

Order System 25 26 Function

−0.5

−0.9

0.1

0.5

0.9

PC

RTS

PC

RTS

PC

RTS

PC

RTS

PC

RTS

PC

RTS

0 0.002

0.004 0.001

0 0.002

0.001 0.001

0 0.002

0.003 0.001

0 0.013

0.002 0.044

0 0.002

0.002 0.008

0.001 0.004

0.003 0.005

Table 6 Percentage error applying 600 coefficients: PC vs DSP. −0.1

Order System 25 26 Function

−0.5

−0.9

0.1

0.5

0.9

PC

DSP

PC

DSP

PC

DSP

PC

DSP

PC

DSP

PC

DSP

0 0.001

0 0.002

0 0.002

0 0.002

0 0.002

0 0.002

0 0

0 0

0 0

0 0

0.001 0

0.001 0.001

Table 7 An amount of coefficients possible to remove (L) increasing the error by 0.1% over refernce values applying 600 coefficients: GLs vs Hs. −0.1

Order

−0.5

−0.9

0.1

0.5

0.9

Formula

GLs

Hs

GLs

Hs

GLs

Hs

GLs

Hs

GLs

Hs

GLs

Hs

25 26 Function

1 1

25 7

1 1

25 18

1 1

21 15

1 1

39 15

1 1

27 13

1 1

22 14

The results provided by the real-time system verify the results of the PC system. Some differences between oscilloscope and DSP readings are caused by interferences in the electrical system used in the experiment. The results displayed by the oscilloscope are charged with the well known voltage measurement errors. Additionally there is an error caused by the DAC. Above causes were confirmed by the obtained values comparison between the high-accuracy PC system and the DSP system. Additional experiment involving measurement of oscilloscope reading of constant voltage input confirmed above conclusion. Applying voltage 1.380V the noises presented by the oscilloscope were ±0.012V . 6 Final conclusions Fractional order derivatives/integrals calculations applying fractional order backward differences/sums are a very computation intense process if there is to meet a certain level of accuracy. We can quickly obtain FOD/I


497

of simple monotonic functions with accuracy below 10−4 mark with a handful of coefficients by application of the Grünwald-Letnikov formula. However there are required many hundreds of thousands of them for FOD/I of high-frequency periodic functions. We are forced to apply over 2.5 billions of the coefficients to cross 10−8 accuracy mark for low orders of FOBD/S. In Section 3 we presented crucial factors which influence the accuracy of the Grünwald-Letnikov method and its Horner equivalent form: 1. Function’s monotonicity 2. Fractional order which is to calculate 3. Frequency of a periodic function Combination of the above factors increases their overall influence. The most demanding case is when there is to calculate a low order of FOD/I for a high frequency periodic function of which bounding box is either constant or increasing. Therefore a certain computational accuracy becomes difficult task for the real-time calculations. Horner’s scheme application to the Grünwald-Letnikov formula as well as the calculation tail of a different length enables not only for more computational friendly application of the method but also for removal of some significant amount of the coefficients while maintaining a set accuracy at the same time in case of the most used functions. However it does not help much in case of the most coefficients demanding functions. How many of the coefficients can be removed depends on type of a function and its monotonicity, frequency in case of periodic functions. However in case of some functions it is possible to remove even 30% of the coefficients without substantial loss of accuracy. Section 5 includes the calculation tail length application analysis and comparison results between a PC and a real-time DSP system. They confirm credibility of the PC simulation: it is possible to remove 5% of the coefficients with no accuracy loss over the set 0.1% in comparison to the reference values. The Grünwald-Letnikov method is not suitable at for the coefficients removal procedure at all. Relative errors of 100% or higher magnitude for almost all of the tested functions are the evidence. We stated in the introduction that Horner’s equivalent form of the Grünwald-Letnikov formula and the application of the calculation tail length can be helpful for a time and a memory shortages influence on accuracy in real-time microprocessor calculations. The procedure and its results included in the following paper confirms it. Additionally the presented procedure has been successfully applied in a robot arm control described in [26]. References [1] Podlubny, I. (1999), Fractional Differential Equations: Academic Press, San Diego, CA. [2] Miller, K. S. and Ross, B. (1993), An Introduction To The Fractional Calculus and Fractional Differential Equations: John Willey and Sons INC., New York,NY. [3] Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012), Fractional Calculus Models and Numerical Methods: World Scientific Publishing Co.Pte. Ltd., Singapore. [4] Ostalczyk, P. (2008), Zarys rachunku róz˙ niczkowego i całkowego ułamkowych rze¸dów: Wydawnictwo Politechniki Łódzkiej, Łód´z, Poland. (in Polish). [5] Kilbas, A. A., Srivastava, H.M. and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations: Elsevier Science. [6] Ortigueira, M.D., Machado, J.A.T. and Sa da Costa, J. (2005), Which Differintegration?, IEE Proceedings - Vision, Image and Signal Processing, 152(6). [7] Ortigueira, M.D. and Machado, J.A.T. (2014), What is a Fractional Derivative, J.Comput. Phys. [8] Ostalczyk, P. (2009), A note on the Grünwald-Letnikov fractional-order backward difference. Physica Scripta, 136, 1-5 [9] Ostalczyk, P. (1995), Fractional-order Backward Difference Equivalent Forms. Fractional Differentiation and Its Applications. Systems Analysis, Implementation and Simulation, System Identification and Control. Ubooks Verlag, Neusäss,Germany, 545-556

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[10] Valerio, D., Trujillo, J. J., Rivero, M., Machado, J.A.T. and Baleanu, D. (2013), Fractional Calculus: A Survey of Useful Formulas, The European Physical Journal Special Topics, 222, 1827-1846. [11] Brzeziński, D.W. and Ostalczyk, P. (2014), High-accuracy Numerical Integration Methods for Fractional Order Derivatives and Integrals Computations: Bulletin of the Polish Academy of Sciences Technical Sciences, Vol.62, no.4. [12] Diethelm, K. (2004), The Analysis of Fractional Differential Equations: Springer-Verlag. [13] Petras, I. and Vinagre, B.M. (2002), Practical Application of Digital Fractional-order Controller to Temperature Control. Acta Montanistica Slovaca, 7, 131-137 (13) [14] A¨strom ¨ K. J. and Hägglund T. (1995), PID controllers: Theory, design and tuning, Instrument Society of America. [15] Chen Y.Q., Petras I. and Xue D. (2009), Fractional Order Control - A Tutorial, 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA. [16] Gene F. (2000), Digital Signal Processor Trends, Texas Instruments, 52-55 [17] Franklin G.F., Powell J.D. and Ememi-Naeini A. (2002), Feedback Control of Dynamic Systems. Prentice Hall, Upper Saddle River, New Yersey. [18] Jacquot R.G. (1994), Modern Digital Control Systems. Marcel Dekker, Inc, New York [19] Monje C.A., Chen YQ., Vinagre B., Xue D. and Feliu V. (2010), Fractional-order Systems and Controls Fundamentals and Applications. Advances in Industrial Contro. Springer London Limited. [20] Ogata K. (1987), Discrete-Time Control Systems, Prentice Hall International Editions, Englewood Cliffs [21] Silva, M.F. and Tenreiro Machado J.A. (2006), Fractional Order PD-Joint Control of Legged Robots, Journal of Vibration and Control, 12(12), 1483-1501 [22] Shenga, H., Sunb, H., Coopmansc, C., Chenc Y.Q. and Bohannanc, G. (2010), Physical Experimental Study of Variable-order Fractional Integrator and Differentiator. Proceedings of FDA 2010. The 4th IFAC Workshop Fractional Differentiation and its Applications. Badajoz, Spain. [23] Ostalczyk, P., Duch, P. and Sankowski, D. (2011), Fractional Order Backward Difference Grunwald-Letnikov and Horner Simplified Forms Evaluation Accuracy Analysis. [24] Brzeziński, D.W. and Ostalczyk, P. (2012), The Grünwald-Letnikov Formula and Its Horner’s Equivalent Form Accuracy Comparison and Evaluation For Application To Fractional Order PID Controler: IEEE Explore Digital Library: IEEE Conference Publications-17th International Conference On Methods and Models In Automation and Robotics (MMAR). [25] Ostalczyk, P., Duch, P., Brzeziński, D.W. and Sankowski, D.(2014), Order Functions Selection in the Variable, Fractional-Order PID Controller. Advances in Modeling and Control of Non-integer Order Systems, Springer Lecture Notes in Electrical Engineering. 320, 159-170 [26] Ostalczyk P., Brzeziński D.W., Duch P., Łaski M. and Sankowski D. (2013), The Variable, Fractional-order Discretetime PD Controller in The IISv1.3 Robot Arm Control, Central European Journal of Physics, Vol. 11 (6), 750-759



The Double Exponential Formula as a Gauss Quadratures Replacement for Numerical Integration Dariusz W. Brzeziński†, Piotr Ostalczyk Institute of Applied Computer Science, Lodz University of Technology, 18/22 Stefanowskiego St., 90-924 Łód´z, Poland Submission Info Communicated by Mark Edelman Received 12 November 2014 Accepted 14 January 2015 Available online 1 January 2016 Keywords Double Exponential Formula Gauss Quadratures Arbitrary Precision Accuracy of Numerical Integration

Abstract We propose to replace the Gauss Quadratures with a numerical integration method known as the Double Exponential (DE) Formula. The numerical quadrature built upon it is at least equivalently accurate and much simpler to customize and apply in situations when tabulated values of the Gauss Quadratures’ nodes and weights can not be applied. The DE Formula was developed for integrals with endpoint singularities. However, we confirm that it can be successfully applied to any integral and interval, for which the Gauss Quadratures have been usually selected. To remain compact, the following presentation focuses only on the most difficult integrals, e.g. the improper integrals and the integrals with endpoint singularities. The main part of the paper consists of the calculations accuracy comparison between numerical quadrature based upon the DE Formula and the Gauss-Laguerre, the Gauss-Hermite or Gauss-Chebyshev Quadratures. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Singularities and infinite limits issues are the most difficult to deal with in numerical integration. There are no general solutions. Ordinarily, a selected method of numerical integration must be applied to individual problem cases [1–19]. The application of numerical integration methods developed for general use often gives solutions with errors on the unsatisfactory level. However, there is a possibility of increasing the accuracy by adjusting the integrand or restricting the integration interval. The literature on the subject [1,3,4,8,9] gives some advice for preparations of the integrand. Recommended preparations include manual adjustment by transforming the integrand prior the application of numerical method of integration, by exclusion of the singularity or singular section from the integration interval or by stretching of the problematic section and application of the adaptive algorithm to it. Integration of the reciprocal function can also be applied in case of a simple, single-valued functions. The application of the above proceedings can be associated with the loss of calculations accuracy. Some of the operations can not be programmed as an automatic integrator [20] which hinders successful computer application. † Corresponding


500

D.W. Brzeziński,P.Ostalczyk / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 499–509

A wide range of numerical methods still exists there, which involve the application of polynomials in integration. The polynomials’ properties are selected for the specific integration problem, and thereby they assure high accuracy results. The ordinarily sufficient method of the Gauss Quadratures application is reduced to the use of tabulated values of nodes and weights. They are freely available [14] and easy to incorporate into a computer program. However, the integration process can become difficult, if conditions enforce high precision nodes and weights computations or the integrand does not include weight function required by a quadrature [10–13, 15–19]. The first case requires the application of mathematical formulas for nodes and weights and the second manual extraction of the weight function from the integrand prior the integration. Generally, the Gauss Quadratures can be applied to a specific types of functions only. It is conditioned by the linkage to their weight functions (with exclusion of the Gauss-Legendre Quadrature, in which the weight function equals 1). The quadratures are usually applied to compute the most difficult integrals, e.g. the improper integrals and the integrals with end point singularities which was the reason of selection for the following accuracy comparison. Some important details of the Gauss Quadratures’ numerical application requirements as well as the formulas for the nodes and the weights computations for example quadratures are presented below there. The presentation is only intended as a background to underline the numerical construction simplicity of the quadrature built upon the DE Formula, which is presented in the next chapter. Comprehensive details about the Gauss Quadratures can be found in the literature on the subject [3, 4, 9, 13, 14]. The Gauss-Laguerre Quadrature is applied to compute the integrals in the half-infinite interval though, only if the integrand includes the exponential function f (x) = e−x . The Laguerre polynomials are defined by the following formula Ln (x) =

d n n −x (x e ). e−x n! dxn 1

(1)

The weights of the quadrature can be calculated in the following way Ak =

n!

xk [Ln (xk )]2

,

(2)

where xk are the zeros of the Laguerre polynomials (1) obtained by any numerical method. In case of the Gauss-Hermite Quadrature for infinite interval, the integrand expression must include the 2 exponential function f (x) = e−x . The Hermite polynomials are defined by the Rodrigues formula 2

Hn (x) = (−1)n ex

d n −x2 (e ). dxn

The weights of the quadrature can be calculated applying the formula √ 2n+1 n! π , Ak = [H (xk )]2

(3)

(4)

where xk are the zeros of the Hermite polynomials (3) obtained by any numerical method. In order to successfully apply the Gauss-Chebyshev Quadrature in the range (−1, 1), the integrand expres√ sion must include the function f (x) = 1/ 1 − x2 . The Chebyshev polynomials can be calculated applying the following explicit formula Tn (x) =

n2

n (−1)m (n − m − 1)!(2x)n−2m . m!(n − 2m)! m=0 2

∑

(5)


501

The zeros of the Chebyshev polynomials are expressed by the formula xk = cos

2k − 1 π, 2n

(6)

where the weights are fixed. Their values are

π . (7) n Apart from the strict application conditions the quadratures’ formulas for the nodes and the weights require a knowledge of higher mathematics and the numerical methods. All of the above suggests, that although the Gauss Quadratures are indeed the ultimate tools for a highaccuracy numerical integration of difficult integrands, they have the strict application conditions and are complex to construct. In a nutshell, they cannot be applied as a tool of general application [20]. In this context, as a replacement, we want to demonstrate an exponential type transformation formula known as the Double Exponential Formula (in short the DE Formula). It joins an independent variable substitution in integrand and the Trapezoidal Rule. It can be applied to integrate a wide range of functions which include functions with end point singularities, in the infinite and the half-infinite intervals. It can also be used to compute derivatives and integrals of fractional orders, applying difficult to integrate numerically, the Riemann-Liouville and Caputo formulas [21]. The DE Formula has been known for many years, but never achieved a wider popularity due to poor publicity and some drawbacks. The drawbacks are mainly associated with the limitations of the single and double precision variables applied in the initial programming at the time of development [22], e.g. under- and over-flow often occurring when calculating the weights. There were two developed methods of solution of this particular problem in the seventies. They are described in [22]. Another idea was presented in 2007 [19]. All the solutions unfortunately decreases the overall accuracy abilities of the method. We applied the DE Formula during the research on the high accuracy numerical integration method for fractional order derivatives and integrals applying the Riemann-Liouville and Caputo formulas. The method turned up so versatile that it was chosen as replacement for the Gauss Quadratures. We programmed it applying modern programming languages (C++ and Python) and enhanced its accuracy abilities with arbitrary precision application (GNU GMP/MPFR for C++ and mpmath for Python). The application of arbitrary precision also enabled to eliminate the drawbacks in form of under- and over-flows in the weights’ computations. We also attempted, although unsuccessful due to the memory shortages, to apply the method for fractional order derivatives and integrals real time microprocessor calculations (as a replacement for the Grünwald-Letnikov method). The main goal of this article is to present the computations accuracy abilities of the DE Formula, enhanced by the modern programming technology, with emphasis on its constructions and applications simplicity in comparison to the requirements and the formulas, on which the Gauss Quadratures are built upon. Ak =

2 The double exponential transformation and the trapezoidal rule The DE Formula joins the exponential type independent variable substitution in the integrand and the Trapezoidal Rule. The Trapezoidal Rule (and especially its composite variant) is one of the simplest algorithms of numerical quadrature, often underestimated due to its low accuracy for integrating generic integrands over finite interval. However, for integrating of analytic functions over R the Trapezoidal Rule is particularly effective, especially for fast converging integrands (at least at single-exponential rate), for which the truncation of the finite sum has minimal impact on the integration result [23, 24]. The method of transforming any analytic function into one over R is the independent variable transformation. The general idea of this kind of variable transformation was proposed for the first time by Korbov [25]. It also

502


became a subject of works of Schwartz, Stenger, Haber, Takahasi, Sidi and the others, who gave it different names [26–31]. In the following paper the idea of this kind of analytic function transformation into fast converging one over R is based on the transformation, which was proposed by Schwartz [26]. It become known as the Tanh rule (since x = tanht). It can be presented as follows: Let us consider the integral ˆ b f (x) dx, (8) I= a

where f (x) is integrable on interval (a, b). The function f (x) may have singularity at x = a or x = b or at both ends. First we apply the following variable transformation x = φ (t) , φ (−∞) = a, φ (∞) = b. We obtain

ˆ I=

∞ −∞

f (φ (t)) φ (t) dt.

(9)

(10)

It is important that φ (t) possesses the property such as φ (t) decreases its values to 0 at least at single exponential rate as t → ±∞ (11) φ (t) → exp (−c exp (|t|)) , where c is some constant. After that, it is best to apply the Trapezoidal Rule to the transformed integrand expression [22, 23] I=h

∞

∑

n=−∞

f (φ (nh)) φ (nh) ,

where h is sampling step. Due to the property (11) truncation of the summation process can be done at some arbitrarily chosen n = −N− and n = +N+ , I=h

+N+

∑

n=−N−

f (φ (nh)) φ (nh) ,

(12)

N = −N− + N+ + 1, where N states the amount of sampling points of the integrated function. Since φ (nh) as well as the whole expression f (φ (nh)) φ (nh) converges at least at exponential rate at large |n|, the quadrature formula (12) is called the Double Exponential [22, 28, 29, 31]. Due to truncation of the summation process (12) at some arbitrary chosen n = −N− , n = +N+, function f (x) can have singularities at x = a and/or x = b as long as it is integrable over the integration interval. Two kinds of errors should be taken into consideration when implementing the DE Formula: discretization error, due to the use of the Trapezoidal Rule to approximate an integral and truncation error, because of truncation of infinite sum at some N. The optimal strategy is to get both errors equal [19, 22]. The subinterval width h, that defines the evaluation step and the number of sampling points are key values in such strategy. The source [19] suggests the following value of h for the DE Formula h∼

log (2π N ω /c) , N

where c is some constant to be taken, usually 1 or π /2 and ω is the distance to the nearest singularity of the integrand defined in [28, 32].


503

Equations (13)-(15) present three different transformations with varied convergence rate [18, 24]. Transformation (13) has the slowest convergence and transformation (15), the fastest respectively:

x = φ (t) = tanh t p , φ (t) = x = φ (t) = tanh x = φ (t) = tanh

π 2

π 2

pt p−1 , p = 1, 3, 5, . . . , cosh2 t p

sinh t , φ (t) =

sinh t 3 , φ (t) =

π 2 cosh t cosh2 ( π2 sinh

t)

(13)

,

(14)

3 2 3 2 π t cosh t . cosh2 ( π2 sinh t 3 )

fx

(15)

ft t

1

1

x

x

ta

(a)

(b)

t

(c)

Fig. 1 Transformation (14) applied to the function f (x) = √ 1

1−x2

(c) transformed integrand.

ta

: (a) Initial integrand, (b) transforming expression and

Applying the transformation (14) to a function according to the formula (12), we obtain the following trapezoidal form N b−a b+a b−a xi + )wi ( ), (16) S = h∑ f( 2 2 2 i=1 where xi = f (tanh(

π sinh ti )), 2

ti = −ta + (i − 1)h, i = 1, 2, 3, . . . , N, h = wi =

2ta , N

π 2 cosh ti cosh2 ( π2 sinh ti )

are the nodes, the sampling points and the weights of the DE Quadrature respectively. 3 Remarks on the numerical implementation 3.1

Appropriate transforming expression selection

The DE Transformation changes the shape of the integrand to convex-concave. The application of the Trapezoidal Rule to the transformed integrand causes the errors to compensate. Therefore, the calculations accuracy depends on the appropriate selection of the transforming expression. For integrating the functions which do not include the exponential part, the transforming expression (14) can be considered as optimal (see Fig.1).

504


For integrating the functions with the exponential function part and for the infinite intervals, the DE Formula must be applied with slightly modified transforming expressions. They still must have similar properties, e.g. their derivatives must decrease more or less exponentially [19, 30]: ˆ I=

∞

0

exp(−x) f (x)dx → x = exp(t − exp(−t)), ˆ

π f (x)dx → x = exp( sinh t), 2

(18)

exp(−x2 ) f (x)dx → x = exp(t − exp(−t)),

(19)

I= ˆ I=

∞

−∞

∞

(17)

0

ˆ I=

∞

−∞

π f (x)dx → x = sin( sinh t). 2

(20)

The differences in decaying rate of the transforming expressions (13)-(15) and (17)-(20) can be utilized according to the properties of the integrand, e.g. the transforming expression (17) is especially dedicated to 2 functions which fall off rapidly (e−x or e−x ) at ∞. The formula (17) is double exponential at x = 0 but only single exponential at x = ∞. Transforming expression converges double-exponentially only in case of combination with the exponential fall-off of the integrated function [19]. fx

ft

0

x

(a)

ta

ta

(b)

t

(c)

Fig. 2 Transformation (17) applied to the integrand with the exponential function, e.g. f (x) = integrand, (b) transforming expression and (c) transformed integrand.

fx

−x e√ x

x ∈ 0, +∞): (a) Initial

ft

a

b

x

(a)

Fig. 3 Transformation (18) applied to the function f (x) = expression and (c) transformed integrand.

(b) 1 , (x+1)2

a

b

(c)

x ∈ 0, ∞ : (a) Initial integrand, (b) transforming

t

D.W. Brzeziński,P.Ostalczyk / Discontinuity, Nonlinearity, and Complexity 4(4) (2015) 499–509 fx

505

ft

x

(a)

ta

ta

(b)

t

(c)

2

Fig. 4 Transformation (19) applied to integrand with e−x function, e.g. f (x) = e (b) transforming expression and (c) transformed integrand.

−x2 2

, x ∈ (−∞, +∞): (a) Initial integrand,

The DE Formula which uses the Trapezoidal Rule assumes the following form ˆ S=

t

−t

f (x (t)) x (t) dt = h

N 2

∑

k=− N2

f (x (kh)) x (kh) , h =

2ta , k = 1, 2, 3, . . . , N. N

(21)

The appropriate selection of the integration interval −ta , ta have an impact on the accuracy. The source [19] suggests the interval (−4, 4) for typical applications. It should be widened so much, as the endpoints of the integration interval should contribute to the summation. 3.2

Application of arbitrary precision

The calculation of the DE Formula weights from formula (16) can cross the double precision arithmetic limits. First, the denominator in the weight function formula sec2 (c sinh t) can overflow. Second, the selection of the parameter ta > 3.7 can cause underflow and the occurrence of uncontrolled errors (the computed integral values replaced by random numbers). Both problems can be solved applying some customization of the weights computations [19, 22] which unfortunately causes the accuracy to decrease. Because of the fact that the higher the value of the parameter ta , the nearer the singularity we integrate and the higher the accuracy we usually obtain, the above mentioned drawbacks can be eliminated applying arbitrary precision programming libraries GNU MPFR [32] and GNU GMP [33]. The application of the libraries increases the overall computations accuracy [34–36] as well. The GNU Multiple Precision Floating-Point Reliable Library (MPFR) is an arbitrary precision package for C language and is based on GNU Multiple-Precision Library (GMP). MPFR supports arbitrary precision floating point variables. It also provides proper rounding of all implemented operations and mathematical functions [37–40]. The library enables a user to set the precision of the arbitrary precision variables by specifying the number of bits to use in the mantissa of the floating point number. Due to the design of the library it is possible to work with any precision between 2 bits and maximum bits allowed for a computer. The ability of MPFR to set the precision exactly to the desired precision in bits is the major difference of this library compared to the competitors. It is also the reason for choosing it for the following investigation, because setting the parameter ta > 4.7 causes the values of the weights to have values much smaller than the double precision variables can hold. The most common errors in numerical calculations are caused by wrong rounding. The MPFR library supports proper rounding in compliance with the IEEE 754-2008 standard. It implements four of the rounding modes specified by the standard, as well as an additional one not included in it, e.g. round away from 0.

506


4 Accuracy comparison details We selected a few difficult functions to integrate: ˆ

1

cos x √ dx, 1 − x2 −1 ˆ 1 1 √ I= dx, 1 − x4 −1 ˆ ∞ 5 e−x x 6 dx, I= I=

0 ∞

ˆ I=

0

ˆ

I=

ˆ I= ˆ I=

∞

x2

−∞

ˆ I=

1 dx, (x + 1)2

∞ −∞

0 1

−1

e− 2 dx, 2

(22) (23) (24) (25) (26)

e−x−x dx,

(27)

e5 sin x dx,

(28)

1−x 6 x dx. 1+x

(29)

2π

The Gauss-Chebyshev Quadrature (denoted as Che) is applied to calculate the integrals: (22)-(23), the Gauss-Laguerre Quadrature (Lag): (24)-(25), the Gauss-Hermite Quadrature (Her): (26)-(27), the Gauss-Legendre (Leg) and the Gauss-Kronrod (Kro) Quadratures (28) and the Gauss-Jacobi Quadrature (Jac)(29). The DE Quadrature (DE) (21) is constructed according to integrals’ properties applying (14) and (17)-(20) transforming expressions to calculate the integrals (22)-(29). The accuracy criterion of numerical calculations in the following comparison is expressed as a measurement of relative error vc er (n) = |1 − | ve where vc is a calculated value, ve a value assumed as exact one and n the number of subintervals the interval is divided into. 5 Results and conclusions The Double Exponential Formula and the quadrature built upon it is a very capable numerical integration method. It can be applied as a replacement for the Gauss Quadratures due to its positive acceptance for the integration of a wide range of function types. This includes improper integrals, integrals with endpoint singularities, integrals of periodic functions as well as integrals in the Riemann-Liouville / Caputo fractional derivatives and integrals formulas [20, 21]. In the following paper there were only presented the computational accuracy of some difficult integrals, for which an usual selection always have been the Gauss-Laguerre, the Gauss-Hermite and the Gauss-Chebyshev Quadratures. However, during the research, we also tested with equal success the DE Formula replacement for computations of other integrals types, for which the Gauss-Legendre, the Gauss-Jacobi and the Gauss-Kronrod Quadratures have been usually applied. The Gauss-Kronrod Quadrature is a modification of the Gauss-Legendre


Fig. 5 Computational error er (n) for (22) and (23).




507

Quadrature modification enhanced for accuracy. It is considered as general purpose numerical integration method in industrial applications [10]. The comparison results for Leg, Jac, Kro and DE are presented in Fig.8. The DE Quadrature provides an equivalent, and in many cases better accuracy than the respective Gauss Quadratures, although 10, 20 or even 30 times more sampling points are necessary. This amount does not exceed even the common requirements of the Trapezoidal Rule. The comparison results for Che, Lag, Her and DE are presented in Figs 5-7. A great advantage of the DE Quadrature is its construction’s simplicity [21]. It is sufficient to compare any Gauss Quadrature’s nodes and weights calculations formulas (1)-(7) with the one of DE quadrature (16) to notice the difference in favour of the latter. There are required values of the transformed expression and its derivative in a given interval only. However, to optimal usage of accuracy abilities of the DE Formula, the transforming expression (13)-(15), (17)-(20) must be selected appropriately. This requires the knowledge of how fast the integrand convergent is. Moreover, to exclude the possibility of the under- and overflow occurrence during the method construction and application as well as an increase of the overall calculations accuracy, it is recommended to apply arbitrary

508


precision mathematical libraries [32, 33]. References [1] Demidowicz, B.P., Maron, I.A., and Szuwałowa (1965), Metody Numeryczne: Państwowe Wydawnictwo Naukowe, Warszawa. (in Polish). [2] Mysovskih, I.P. (1969), Lectures On Numerical Methods, Noordhoff. [3] Kythe, P.K. and Schaferkotter, M.R. (2005), Handbook of Computational Methods for Integration, Chapman & Hall/CRC. [4] Burden, R.L. and Faires, J.D. (2003), Numerical Analysis, 5th ed.: Brooks/Cole Cengage Learning, Boston. [5] Chapra, S.C. and Canale, R.P. (2010), Numerical Methods for Engineers, Sixth Edition, McGraw-Hill Companies, Inc. [6] Hjorton-Jensen, M. (2009), Computational Physics, University of Oslo. [7] Shampine, L.F., Allen, R.C. and Pruess, S. (1997), Fundamentals Of Numerical Computing, Wiley. [8] Mathews, J.H. and Fink, K.D. (1999), Numerical Methods Using Matlab, 3rd Edition, Prentice Hall. [9] Krylov, V.I. (1967), Priblizhennoe vychislenie integralov, 2e izd. Nauka. (in Russian). [10] Kahaner, D., Moler, C. and Nash, S. (1989), Numerical Methods And Software, Prentice Hall. [11] Conte, S.D. and de Boor, C. (1980), Elementary Numerical Analysis. An Algorithmic Approach, 3rd Edition, McGrawHill Book Company. [12] Quarteroni, A., Sacco, R. and Salieri, F. (2000), Numerical Mathematics, Springer. [13] Kiusalaas, J. (2005), Numerical Methods in Engineering With Python, Cambridge University Press. [14] Abramowitz, M. and Stegun, I.A. (1968), Handbook of Mathematical Functions, Applied Mathematics Series Dover Publications, NY. [15] Stroud, A.H. and Secrest, D. (1966), Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ. [16] Desmarais, R.N. (1975), Programs For Computing Abscissas and Weights For Classical and Nonclassical Gaussian Quadrature Formulas, NASA Technical Note, NASA TN D-7924, NASA, Washington D.C. [17] Gautschi, W. (2004), Numerical Mathematics And Scientific Computation. Orthogonal Polynomials: Computation and Approximation, Oxford Science Publication. [18] Lau, H.T. (1995), A Numerical Library In C For Scientists And Engineers, CRC Press. [19] Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (2007), Numerical Recipes. The Art of Scientific Computing, Third Edition, Cambridge University Press, 172-179. [20] Brzeziński, D. and Ostalczyk, P. (2012), Numerical Evaluation of Fractional Differ-integrals of Some Periodical Functions via the IMT Transformation: Bulletin of the Polish Academy of Sciences Technical Sciences, 60(2), 285–292. [21] Brzeziński, D.W. and Ostalczyk, P. (2014), High-accuracy Numerical Integration Methods for Fractional Order Derivatives and Integrals Computations: Bulletin of the Polish Academy of Sciences Technical Sciences, 62(4). [22] Mori, M. (1990), Developments in The Double Exponential Formulas for Numerical Integration, Proceedings of the International Congress of Mathematicians, Kyoto, Japan. [23] Stenger, F. (1973), Integration Formulae Based On The Trapezoidal Formula, J. Inst. Math. Appl., 103–114. [24] Waldvogel, J. (2011), Towards A General Error Theory of the Trapezoidal Rule, Approximation and Computation, Springer Verlag. [25] Korobov, N.M. (1963), Number-Theoretic Methods of Approximate Analysis, GIFL, Moscow. (in Russian). [26] Schwartz, C. (1969), Numerical Integration of Analytic Functions, Journal of Computational Physics, 4, 19–29. [27] Haber, S. (1999), The Tanh Rule For Numerical Integration, SIAM J. Numer. Anal., 14(4). [28] Takahasi, H. and Mori, M. (1974), Double Exponential Formulas for Numerical Integration, Publ. RIMS Kyoto Univ., 9, 721–741. [29] Takahasi, H. (1973), Quadrature Formulas Obtained by Variable Transformation, Numer. Math., 21, Springer Verlag, 206-219. [30] Sidi, A. (1993), A New Variable Transformation for Numerical Integration, in H. Brass and G.Hammerlin (editors) Numerical Integration IV , Birkhauser, Berlin ISNM 112 , pp. 359-373. [31] Mori, M. and Sugihara, M. (2001), The Double-exponential Transformation in Numerical Analysis, Journal of Computational and Applied Mathematics, 127, 287-296. [32] The GNU Multiple Precision Floating-Point Reliable Library, https://mpfr.org/. [33] The GNU Multiple Precision Arithmetic Library, https://gmplib.org/. [34] Evans, G.A., Forbes, R.C., and Hyslop, J.(1984), The Tanh Transformation for Singular Integrals, Intern. J. Computer Math 15, 339–358. [35] Overton, M. (2001), Numerical Computing with IEEE Floating Point Arithmetic, SIAM. [36] Wilkinson, J.H. (1994), Rounding Errors in Algebraic Processes, Dover, New York. [37] Brisebarre, N. and Muller, J.M. (2007), Correct Rounding of Algebraic Functions, Theoretical Informatics and Applications, Jan-March 2007, 47, 71–83.


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Igor Belykh Department of Mathematics & Statistics Georgia State University 30 Pryor Street, Atlanta, GA 30303-3083 USA Email: [email protected]

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Andrey Shilnikov Department of Mathematics and Statistics Georgia State University, 100 Piedmont Ave SE Atlanta GA 30303, USA Fax: +1 404 413 6403 Email: [email protected]

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N.H. Ibragimov Department of Mathematics, IHN Blekinge Institute of Technology S-371 79 Karlskrona, Sweden Fax: +46 455 385 407 Email: [email protected]

Raul Rechtman Centro de Investigacion en Energia Universidad Nacional Autonoma de Mexico Priv. Xochicalco S/N, Temixco Morelos 62580, Mexico Fax: +52 555 622 9791 Email: [email protected]

Todd Young Department of Mathematics Ohio University Athens, OH 45701, USA Email: [email protected]

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 4

December 2015

Contents Fractional Dynamics and Systems with Power-Law Memory M. Edelman, J.A. Tenreiro Machado........................................................................

381－382

Fractional Calculus: Models, Algorithms, Technology J.A. Tenreiro Machado………......……………………………..…………………..

383－389

Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo αFamilies of Maps M. Edelman………...……..…………………………………..………...…………..

391－402

Analysis of Terrorism Data-series by means of Power Law and Pseudo Phase Plane Antonio M. Lopes, J.A. Tenreiro Machado…………………………....…………...

403－411

Adaptive Memory Identification of Fractional Order Systems Yang Zhao, Yan Li, Fengyu Zhou.………………………………….…..…………..

413－428

The Optimal Control Problem for Linear Systems of Non-integer Order with Lumped and Distributed Parameters V.A. Kubyshkin, S.S. Postnov..…..…………………………….………..…….……

429－443

Sliding Mode Control of Fractional Lorenz-Stenflo Hyperchaotic System Jian Yuan, Bao Shi……………………………………….....……..…………….....

445－455

Hybrid Projective Synchronization in Mixed Fractional-order Complex Networks with Different Structure Li-xin Yang, Jun Jiang, Xiao-jun Liu……...……….…………………...……….....

457－465

Nonlinear Four-point Impulsive Fractional Differential Equations with p-Laplacian Operator Fatma Tokmak Fen, Ilkay Yaslan Karaca……...……………..………...……….....

467－486

About Utility of the Simplified Grünwald-Letnikov Formula Equivalent Horner Form Dariusz W. Brzezinski, Piotr Ostalczyk………....………………...…………….....

487－498

The Double Exponential Formula as a Gauss Quadratures Replacement for Numerical Integration Dariusz W. Brzezinski, Piotr Ostalczyk……...…...…………...…...…………….....

499－509

Available online at https://lhscientificpublishing.com/Journals/DNC-Download.aspx

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