Journal of Applied Nonlinear Dynamics

Volume 4 Issue 2 June 2015

ISSN 2164‐6457 (print) ISSN 2164‐6473 (online)

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics Editors J. A. Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrical Engineering Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal Fax:+ 351 22 8321159 Email: [email protected]

Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA Fax: +1 618 650 2555 Email: [email protected]

Associate Editors J. Awrejcewicz Department of Automatics and Biomechanics K-16, The Technical University of Lodz, 1/15 Stefanowski St., 90-924 Lodz, Poland Fax: +48 42 631 2225, Email: [email protected]

Stefano Lenci Dipartimento di Ingegneria Civile Edile e Architettura, Universita' Politecnica delle Marche via Brecce Bianche, 60131 ANCONA, Italy Fax: +39 071 2204576 Email: [email protected]

Miguel A. F. Sanjuan Department of Physics Universidad Rey Juan Carlos Tulipán s/n 28933 Mostoles, Madrid, Spain Fax: +34 916647455 Email : [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University Balgat, 06530, Ankara, Turkey Fax: +90 312 2868962 Email: [email protected]

Shaofan Li Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, CA 94720-1710, USA Fax : +1 510 643 8928 Email: [email protected]

C. Steve Suh Department of Mechanical Engineering Texas A&M University College Station, Texas 77843-3123 USA Fax:+1 979 845 3081 Email: [email protected]

Nikolay V. Kuznetsov Mathematics and Mechanics Faculty Saint-Petersburg State University Saint-Petersburg, 198504, Russia Fax:+ 7 812 4286998 Email: [email protected]

C. Nataraj Department of Mechanical Engineering Villanova University, Villanova PA 19085, USA Fax: +1 610 519 7312 Email: [email protected]

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119991 Moscow, Russia Fax: +7 495 939 0397 Email: [email protected]

Editorial Board Ahmed Al-Jumaily Institute of Biomedical Technologies Auckland University of Technology Private Bag 92006 Wellesley Campus WD301B Auckland, New Zealand Fax: +64 9 921 9973 Email:[email protected]

Giuseppe Catania Department of Mechanics University of Bologna viale Risorgimento, 2, I-40136 Bologna, Italy Tel: +39 051 2093447 Email: [email protected]

Mark Edelman Yeshiva University 245 Lexington Avenue New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Alexey V. Borisov Department of Computational Mechanics Udmurt State University, 1 Universitetskaya str., Izhevsk 426034 Russia Fax: +7 3412 500 295 Email: [email protected]

Liming Dai Industrial Systems Engineering University of Regina Regina, Saskatchewan Canada, S4S 0A2 Fax: +1 306 585 4855 Email: [email protected]

Xilin Fu School of Mathematical Science Shandong Normal University Jinan 250014, China Email: [email protected]

Continued on back materials

Journal of Applied Nonlinear Dynamics Volume 4, Issue 2, June 2015

Editors J. A. Tenreiro Machado Albert Chao-Jun Luo

L&H Scientific Publishing, LLC, USA

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Journal of Applied Nonlinear Dynamics 4(2) (2015) 101–116

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Study of the Effect of the Coupling in a Dispersion-managed Dual Core Optical Fiber Using the Collective Variables Approach Roger Bertin Djob1†, Aurélien Kenfack-Jiotsa2 , and Timoléon Crépin Kofané3 1 Laboratory

of Mechanics, Department of Physics, Faculty of Sciences, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon 2 Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher Teachers’ Training College, University of Yaounde I, P.O. Box 47 Yaoundé, Cameroon 3 Laboratory of Mechanics, Department of Physics, Faculty of Sciences, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon Submission Info

Abstract

Keywords

This paper highlights the interaction between two gaussian pulses propagating inside a dual core optical fiber by the mean of collective variables (CVs) approach. The main result is that energies of the signals being propagated in such fiber with linear coupling always end up being compensated whatever their amplitudes at the entry. It also appears convergence or divergence of the temporal positions of the neighboring solitary waves.

Coupled QCGLE Dual core optical fiber DM soliton CVs

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

Communicated by Albert C.J. Luo Received 4 June 2014 Accepted 30 January 2015 Available online 1 July 2015

1 Introduction During last decades, telecommunications by fiber optics took a great extend. Fiber optics replaced electric cables and huge device. Optical fibers are widely used in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths (data rates) than wire cables. An optical fiber is a flexible, transparent fiber made of high quality extruded glass (silica) or plastic, slightly thicker than a human hair. It can function as a waveguide, or light to transmit light between its two ends. Power over optical fiber cables can also work to deliver an electric current for low-power electric devices. The propagation of intense light pulses in a standard telecommunication fiber induce a host of nonlinear phenomena [1, 2]. The combined effects of nonlinear phenomena and the fiber chromatic dispersion lead to many interesting dynamical processes which are difficult to understand in terms of the original pulse field, but can be easily understood by applying a collective variables (CVs) approach. † Corresponding author. Email address: [email protected]

ISSN 2164 − 6457, eISSN 2164 − 6473/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/JAND.2015.06.001

102

R.B. Djob et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 101–116

The pulse dynamics become even more complicated in the case of systems that are not integrable. For instance, in optical communications, it was shown that pulses propagating in dispersion-managed (DM) fiber links, called DM solitons possess a richer temporal and spectral structure of modes than a simple collective entity [3]. That is, a DM soliton is not only able to translate like a whole entity, but can also vibrate like a diatomic molecule. Hence, it is useful, especially in such non integrable systems, to associate new variables, called collective variables, with localized collective phenomena, in order to simplify somehow the description of the pulse dynamics. For example, such CVs may represent the amplitude of a pulse, its temporal position, the pulse width, and so on. The number of CVs that can be introduced into the system is usually determined by the physics under consideration. Then one must derive a transformation which allows us to express the original field equation in terms of CVs. The generation of a train of soliton pulses from continuous wave light in optical fibers was first suggested by Hasegawa [4], and first realized experimentally by Mollenauer et al. [5]. Optical solitons may soon be the primary carriers for long- and short- distance information transmission because, unlike pulses in a linear dispersive fiber, solitons are self-confined, propagating for a long-distance without changing shape [5]. A well-known example of an equation which admits pulse-like soliton solutions is the nonlinear Schr¨ odinger equation (NLSE) [6]. The losses can be compensated by the erbium-doped amplifiers [2]. A well-known model for the study of pulse propagation in doped fiber amplifiers is the one-dimensional Quintic Complex Ginzburg-Landau Equation (QCGLE). Optical solitons propagating in optical fibers may induce a host of nonlinear phenomena such as parametric wave mixing, stimulated Raman scattering, or self-steepening [7–11]. Also, among other related to, the modulation instability arising in the cubic quintic NLSE has been investigated recently for pulse propagation. Dispersion-managed (DM) solitons are attracting considerable interest in optical communication systems because of their superb characteristics which are not observed with conventional solitons. Furthermore, transmission performance are sometimes degraded by perturbations (as linear waves). Actually, the propagation of pulse in fiber links is always destabilized. The use of transmission control methods such as guiding filters [12–14] were studied in order to stabilize DM solitons propagation. In addition, it is shown that nonlinear gain is expected to be more beneficial to DM solitons than to conventional solitons [15–22], in order to stabilize DM soliton transmission. Recently, [23–27] studied, using CVs, propagation of pulse soliton in optical fiber with a single core obeying quintic and third order dispersion CGLE. They investigated equations of motion numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyze effects of initial amplitude and width on the propagating pulse. They also illustrated influence of the third order dispersion and the cubic-quintic non linear coefficients on the evolution of these parameters. In optics, linearly coupled CGL equations serve as models of ring lasers based on dual-core fibers [28, 29]. Systems of this type are also interesting by themselves, as models in which various dynamical regimes can be studied, such as soliton lattices [30]. Such a symmetric model was introduced in recent work [31], where various symmetric and asymmetric stable solitary pulses solutions were found, and transitions between them through supercritical and subcritical bifurcations, which account for the symmetry breaking in the system, were studied in detail. Besides its general interest, the model is also of direct relevance to optics, as it describes a symmetric dual-core (i.e., twin-core) fiber. It is based on the following equations: iψ1,x + pψ1,tt + q|ψ1 |2 ψ1 = iγψ1 + c|ψ1 |4 ψ1 + kψ2 , iψ2,x + pψ2,tt + q|ψ2 |2 ψ2 = iγψ2 + c|ψ2 |4 ψ2 + kψ1 ,

(1)

with ψ being the normalized envelopes of the field (ψ1 the field inside the left core and ψ2 the field inside the right core); x and t the spatial and temporal coordinates. p, q,γ , c are complex coefficients: p = pr + ipi , q = qr + iqi , γ = γr + i γi and c = cr + ici .


103

The parameters pr , pi , qr , qi , cr , ci , γr , are real constants. They are commonly expressed as functions of z (without loosing their constant character), i.e., pr = pr (z), pi = pi (z), qr = qr (z), qi = qi (z), cr = cr (z), ci = ci (z), γr = γr (z), and γi = γi (z), respectively. pr measures the wave dispersion, pi the spectral filtering, qr and qi represent the nonlinear coefficient and the nonlinear gain-absorption coefficient, respectively. We noted that nonlinear gain helps to suppress the growth of radiative background (linear mode) which always accompanies the propagation of nonlinear stationary pulses in real fiber links. cr and ci stand for the higher-order correction terms to the nonlinear refractive index and the nonlinear amplification absorption, respectively. γr and γi represent the linear gain and the frequency shift, respectively. k is a real coefficient called coupling coefficient. Because of the proximity of the cores, waves propagating inside the two fibers may interact. The coupling coefficient k traduces this interaction. The paper is organized as follows. In Sect. 2, we present the CVs theory for the coupled QCGLE equations and establish the equations of motion of the pulse parameters. In Sect. 3, we describe the evolution of solitary pulses inside both cores and influence of the linear coupling. Two cases are specified: initial symmetric pulses and initial asymmetric pulses. Sect. 4 concludes the paper.

2 Derivation of the CVs equations of motion 2.1

CVs method for the coupled QCGLE

We consider a decomposition of the original field in such manner that solutions of equations 1 are respectively the sum of approximation functions (ansatze) and residual fields g(x,t) [13, 14]:

ψ1 (x,t) = f1 (u1 , . . . , uN ,t) + g1 (x,t), ψ2 (x,t) = f2 (v1 , . . . , vN ,t) + g2 (x,t),

(2)

ui and vi are space-dependant CVs describing behaviors of some parameters of these fields. The ansatz functions f1 and f2 are chosen to be the best representations of the fields. Determination of CVs is based on the minimization of the residual field energy. The expressions of the residual field energies are respectively: ˆ +∞ ˆ +∞ 2 ε1 = |g1 (x,t)| dt = |ψ1 (x,t) − f1 (ui ,t)|2 dt, −∞ −∞ (3) ˆ +∞ ˆ +∞ 2 2 ε2 = |g2 (x,t)| dt = |ψ2 (x,t) − f2 (vi ,t)| dt, i = 1, . . . , N. −∞

−∞

Expressions of total energies in each core yield: ˆ +∞ |ψ1 (x,t)|2 dt, E1 = −∞ ˆ +∞ |ψ2 (x,t)|2 dt. E2 =

(4)

−∞

The energies of the residual fields become: ˆ +∞ ˆ +∞ 2 ε1 = | f1 (ui ,t)| dt − 2Re ( ψ1 (x,t) f1 (ui ,t) dt), −∞ −∞ ˆ +∞ ˆ +∞ 2 ε2 = | f2 (vi ,t)| dt − 2Re ( ψ2 (x,t) f2 (vi ,t) dt), −∞

−∞

(5)

104


indicates the conjugated complex. The ansatz must satisfy the minimization of the residual fields energies. This condition establishes the first constraint equations:

∂ ε1 , ∂ ui ∂ ε2 . Ci2 = ∂ vi Ci1 =

(6)

In fact they read: Ci1 Ci2

ˆ = 2Re (− = 2Re (−

+∞

−∞ ˆ +∞ −∞

g1 f1,ui dt), (7) g2 f2,vi dt),

f1,ui and f2,vi represent first derivatives of f1 and f2 with respect to ui and vi respectively. The second constraint equations yield: C˙i1 = 0, C˙2 = 0;

(8)

i

C˙i1 and C˙i2 are the first derivatives with respect to x of Ci1 and Ci2 respectively. The second constraint equations are: C˙i1 = 2Re [− C˙i2 = 2Re [−

ˆ

+∞

−∞ ˆ +∞ −∞

ˆ g1 f1,ui u j dt) − g2 f2,vi v j dt) −

+∞

−∞ ˆ +∞ −∞

g1,x f1,ui dt] = 0, (9) g2,x f2,vi dt] = 0;

fxi is the first derivative with respect to xi of f . Using bare approximation i.e. g ≈ 0 and neglecting variations of g i.e gtt ≈ 0, the constraint equations become: C˙i1 =2Re [∑(< f1,ui · f1,u j > − < f1,ui · i(pr + ipi ) · f1,tt >

− < f1,ui · ( γr + i γi ) · f1 > − < f1,ui · i(c3r + ic3i ) · | f1 |2 f1 >

+ < f1,ui · i(c5r + ic5i ) · | f1 |4 f1 > + < f1,vi · k · f2 >] = 0 C˙i2 =2Re [∑(< f2,vi · f2,v j > − < f2,vi · i(pr + ipi ) · f2,tt >

(10)

− < f2,vi · ( γr + i γi ) · f2 > − < f2,vi · i(c3r + ic3i ) · | f2 |2 f2 >

+ < fvi · i(c5r + ic5i ) · | f2 |4 f2 > + < f2,vi · k · f1 >] = 0, ´ +∞ where: < (· · · ) >= −∞ (· · · ) dt and Re (· · · ) indicates the real part of the complex (· · · ). We finally obtain the constraint equations in the form:

∂ C1 ˙ ][U ] + [R1 ] = 0, ∂U ∂ C2 ˙ ][V ] + [R2 ] = 0, [C˙2 ] = [ ∂V [C˙1 ] = [

(11)


105

which leads to

∂ C1 −1 1 ] [R ], ∂U ∂ C2 −1 2 ] [R ], [V˙ ] = −[ ∂V [U˙ ] = −[

1

(12)

2

[ ∂∂CU ] and [ ∂∂CV ] are square matrices whose elements are: Ci1j = −2Re (< f1,ui · fu j >),

(13)

Ci2j = −2Re (< f2,vi · fv j >). 1

2

1

2

[ ∂∂CU ]−1 and [ ∂∂CV ]−1 are inverse matrices of [ ∂∂CU ] and [ ∂∂CV ], [R] are column vectors whose elements are: R1i =2Re (− < f1,ui · i(pr + ipi ) · f1,tt > − < f1,ui · ( γr + i γi ) · f1 > − < f1,ui · i(qr + iqi ) · | f1 |2 f1 > + < f1,ui · i(cr + ici ) · | f1 |4 f1 >) + < f1,ui · k · f2 >), R2i

(14)

=2Re (− < f2,vi · i(pr + ipi ) · f2,tt > − < f2,vi · ( γr + i γi ) · f2 > − < f2,vi · i(qr + iqi ) · | f2 |2 f2 > + < f2,vi · i(cr + ici ) · | f2 |4 f2 >) + < f2,vi · k · f1 >).

2.2

Application of the CVs method to determine evolution of gaussian pulse solitons parameters inside the two cores

We consider gaussian pulse solitons propagating inside a symmetric dual core optical fiber. The ansatze are written as: −

(t−u2 )2 u2 3

f1 (u1 , u2 , u3 , u4 , u5 , u6 ,t) = u1 e

−

f2 (v1 , v2 , v3 , v4 , v5 , v6 ,t) = v1 e

(t−v2 )

u4

ei 2

(t−u2 )2 +iu5 (t−u2 )+iu6

,

(15)

2

v2 3

e

v i 24

2

(t−v2 ) +iv5 (t−v2 )+iv6

.

function of the pulse propagating in the left core. Its parameters are defined f1 is the approximation u4 as: u1 , u2 , 2 ln(2)u3 , 2π , 2uπ5 and u6 the pulse amplitude, temporal position, pulse width (FWHM), chirp, frequency and phase, respectively. v1 , v2 , 2 ln(2)v3 , 2vπ4 , 2vπ5 and v6 are, respectively, the pulse amplitude, temporal position, pulse width (FWHM), chirp, frequency and phase of the approximation function of the pulse propagating in the right core. Hence, from development of section 2, the following inverse matrices occur: ⎡ ⎤ √ √ 3/4 u √2π 0 −1/2 u √2π 0 0 0 3 1 ⎢ ⎥ √ √ √ ⎢ ⎥ 2u u u 2 u u 2 3 4 3 3 5 ⎢ 0 0 1/2 u 2 √π 1/2 u 2 √π ⎥ 0 1/2 u 12 √π ⎢ ⎥ 1 1 ⎢ ⎥ √ √ ⎢ ⎥ 2u 3 ⎢ ⎥ √2 √ −1/2 0 0 0 0 2 π 1 u π u ⎢ ⎥ 1 1 ∂ C −1 ⎢ ⎥, √ √ ] =⎢ (16) [ ⎥ ∂U 0 −2 u 3 u 22 √π ⎥ 0 0 0 16 √π u 122 u 5 ⎢ 3 1 3 ⎢ ⎥ √ ⎢ √ ⎥ √ 4 u 2 +4 u 2 ⎢ ( ) 3 4 u5 2 ⎥ ⎢ ⎥ 0 1/2 uu4 u23√π2 0 0 1/2 u 3 u 2 √π 1/2 u 3uu24√ π 1 3 1 1 ⎢ ⎥ √ ⎣ ⎦ √ √ √ 2 2 u3 u5√ 2 u3 u4 u√5 2 2√ 1 2(2 u3 u√5 +3) 0 1/2 u 2 π 0 −2 u 3 u 2 π 1/2 u 2 π 4 u 2u π 1

3

1

1

1

3


106

⎡

√ √2 3 π

3/4 v

0

⎢ √ ⎢ 2v ⎢ √3 0 1/2 ⎢ v 12 π ⎢ √ ⎢ ⎢ √2 −1/2 0 v1 π ∂ C2 −1 ⎢ ⎢ ] =⎢ [ ∂V 0 0 ⎢ ⎢ ⎢ √ ⎢ ⎢ 0 1/2 vv4 v23√π2 1 ⎢ ⎣ √ 0 1/2 vv3 v25√π2

√ √2 1 π

−1/2 v

1

0

0

0

0

0

√ 2v √3 v1 2 π

√ v4 v3√ 2 v1 2 π

√ v3 v5√ 2 v1 2 π

0

0

16 √π v 122 v

3

√ 2√ π

3 2 3 v1

0

5

0 −2 v

1/2

0

√

0 0

1/2

1/2

√ 2√ 3 2 π 3 v1 √ v 3 v4 √ v5 2 2 v1 π

−2 v

√ 2 1/2 π √ √ 2 2 v3 v4 v√ 1 2(2 v3 v√5 +3) 5 2 2 2 4 v1 π v1 v3 π

(v3 4 v4 2 +4 ) √

1/2

0

v3 3 v1 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(17)

Elements of column vector [R1 ] are: √ √ √ √ 2π (u43 u24 + 4u23 u25 + 4)pi 1 1 = − u1 − u1 u3 2πγr + u31 u3 π qi + u51 u3 6π qi 4 u3 3 2u2 v2 −u22 −v22 π 2 2 − 2ke u3 +v3 v1 v3 u3 , 2 u3 + v23 √ √ √ √ 1 u21 u5 2π (3u43 u24 + 4u23 u25 + 12)pr 1 1 + u3 u5 u21 2πγi + u3 u5 u41 π qr + u3 u5 u61 6π qr R2 = − 4 u3 3 2u2 v2 −u22 −v22 √ π v3 u3 u1 v1 (2v2 − 2u2 ) 2 2 − 2ke u3 +v3 , 3 (u23 + v23 ) 2 1 √ 3pi u43 u24 + 4pi u23 u25 + 8pr u23 u4 − 4pi 1 2 √ 1 4√ 1 6√ u u u 6π qi − 2 πγ + π q + R13 = − u21 2π r i 8 2 1 4 1 18 1 u23 2 √ 2u2 v2 −u2 2 −v2 2v3 u23 (u2 − v2 )2 u1 v1 π π 3 u2 +v2 3 3 − 2k( + u1 v1 v3 )e , 5 (u23 + v23 )3 (u23 + v23 ) 2 √ √ √ √ √ 1 3 1 1 1 √ √ R14 = − pr u21 u3 2π + pr u21 u53 u24 2π + pr u21 u33 u25 2π − pi u33 u21 2 π u4 − u21 2u33 πγi 8 32 8 4 8 1 4 3√ 1 6√ 3√ 1 √ u1 v1 v3 u53 (u2 − v2 )2 − u1 u3 π qr − u1 6u3 π qr − 2k( π , 5 16 72 2 (u23 + v23 ) 2 2u2 v2 −u22 −v2 2 π 1 3 3 u23 +v23 )e + u1 v1 v3 u3 4 (u23 + v23 )3 √ √ √ 1 R15 = pr u21 u33 u4 u5 2π − pi u3 u5 u21 2 π , 2 √ √ √ 1 √ (u4 u2 + 4u23 u25 + 4)pr 1 √ √ − u21 2u3 πγi − u41 u3 π qr − u61 6u3 π qr , R16 = u21 2π 3 4 4 u3 3 R11

and elements of column vector [R2 ] are: √ √ √ 1 √ (v4 v2 + 4v23 v25 + 4)pi 1 √ √ − v1 2v3 πγr + v31 v3 π qi + v51 6v3 π qi R21 = − v1 2π 3 4 4 v3 3 2u2 v2 −u22 −v22 π 2 2 − 2ke u3 +v3 u1 v3 u3 , 2 u3 + v23

(18)


107

√ √ √ √ (3v43 v24 + 4v23 v25 + 12)pr √ 1 1 R22 = − v21 v5 2π + v3 v5 v21 2 πγi + v3 v5 v41 π qr + v3 v5 v61 6π qr , 4 v3 3 4 2 2 2 2 √ √ 3pi v3 v4 + 4pi v3 v5 + 8pr v3 v4 − 4pi 1 2 1 1 4√ 1 6√ − 2 πγ + π q + R23 = − v21 2π v v v 6π qi r i 8 2 1 4 1 18 1 v23 − 2k(2e

√ u1 v23 u3 (u2 −v2 )2 v1 π (2u2 v2 −u22 −v22 ) 5 (u23 +v23 )(u23 +v23 ) 2

+e

2 2u2 v2 −u2 2 −v2 2 u2 3 +v3

u1 v1 u33

π (u23 + v23 )3

),

√ √ √ √ √ 1 1 3 1 1 R24 = − pr v21 v3 2π + pr v21 v33 v25 2π + pr v21 v53 v24 2π − pi v33 v21 2π v4 − v21 v33 2πγi 8 8 32 4 8 √ 5 2 √ √ √ 1 1 1 v u3 (u2 − v2 ) u1 v1 π − v41 v33 π qr − v61 6v33 π qr − 2k( 3 5 16 72 2 (u23 + v23 ) 2 2 2u2 v2 −u2 2 −v2 1 π 3 3 u2 +v2 3 3 + u1 v1 v3 u3 )e , 4 (u23 + v23 )3 √ √ √ 1 R25 = pr v21 v33 v4 v5 2π − pi v3 v5 v21 2 π , 2 √ √ 1 v2 2π (v43 v24 + 4v23 v25 + 4)pr √ 1 √ √ − 2π v21 v3 γi − v41 v3 π qr − v61 6v3 π qr . R26 = 1 4 v3 3

(19)

We finally obtain the following explicit analytical expressions for the CVs equations of motion: 6 4 2 √ 2u2 u4 v1 v3 − 4u2 u43 v1 v2 v3 − 108 72 u3 v1 v3 + 2u3 v1 v2 v3 u˙1 = − 2k( 2 3 u23 (u23 + v23 )( 5/2) √ √ (u −v )2 −2u43 v1 v33 − 36u23 v1 v53 − u22 +v22 1 √ −36 2pi u1 u63 u25 − 72 2pi u1 u43 u25 v23 )e 3 3 − 2( + 108 72 u23 (u23 + v23 )2 u23 (u23 + v23 ) 72 √ √ √ √ √ √ −72 2u1 u43 v23 γr − 36 2pi u1 u23 u25 v43 − 72 2pi u1 u43 − 144 2pi u1 u23 v23 − 72 2pi u1 v43 − 36 2u1 u63 γr u23 (u23 + v23 )2 √ √ −36 2u1 u23 v43 γr + 45qi u31 u63 + 90qi u31 u43 v23 + 45qi u31 u23 v43 + 16 6qi u51 u63 + u23 (u23 + v23 )2 √ √ √ √ √ 32 6qi u51 u43 v23 + 16 6qi u51 u23 v43 + 36 2pr u1 u63 u4 + 72 2pr u1 u43 u4 v23 + 36 2pr u1 u23 u4 v43 + ), u23 (u23 + v23 )2 (u −v )2

1 √ −4u2 u23 v1 v3 + 4u23 v1 v2 v3 − u22 +v22 2k e 3 3 u˙2 = 3 2 u1 (u23 + v23 ) 2 √ √ √ √ 1 √ 2pi u1 u43 u4 u5 + 2pi u1 u23 u4 u5 v23 + 2 2pr u1 u23 u5 + 2 2pr u1 u5 v23 + 2 , 1 2 u1 (u2 + v2 ) 2 3

3

2

(u −v ) 1 √ 144u22 u43 v1 v3 − 288u2 u43 v1 v2 v3 − 36u63 v1 v3 + 144u43 v1 v22 v3 + 36u23 v1 v53 − u22 +v22 3 3 2k e u˙3 = 5 36 u3 u1 (u23 + v23 ) 2 √ √ √ √ √ 1 √ 9 2pi u1 u83 u24 + 18 2pi u1 u63 u24 v23 + 9 2pi u1 u43 u24 v43 − 36 2pi u1 u43 − 72 2pi u1 u23 v23 + 2( 36 u3 u1 (u23 + v23 )2 √ √ −36 2pi u1 v43 + 9qi u31 u63 + 18qi u31 u43 v23 + 9qi u31 u23 v43 + 36 2pr u1 u23 u4 v43 + u3 u1 (u23 + v23 )2 √ √ √ √ √ 4 6qi u51 u63 + 8 6qi u51 u43 v23 + 4 6qi u51 u23 v43 + 36 2pr u1 u63 u4 + 72 2pr u1 u43 u4 v23 + ), u3 u1 (u23 + v23 )2

(20)


108

(u −v )2

1 √ −144u22 u43 v1 v3 + 288u2 u43 v1 v2 v3 − 144u43 v1 v22 v3 − 72u43 v1 v33 − 72u23 v1 v53 − u22 +v22 u˙4 = − 2k e 3 3 5 9 u43 u1 (u23 + v23 ) 2 √ √ √ √ √ 1 √ −36 2pr u1 u43 − 72 2pr u1 u23 v23 − 36 2pr u1 v43 + 9 2pr u1 u83 u24 + 18 2pr u1 u63 u24 v23 − 2( 9 u43 u1 (u23 + v23 )2 √ √ √ √ 9 2pr u1 u43 u24 v43 − 36 2pi u1 u63 u4 − 72 2pi u1 u43 u4 v23 − 36 2pi u1 u23 u4 v43 + 9qr u31 u63 + 18qr u31 u43 v23 + u43 u1 (u23 + v23 )2 √ √ √ 9qr u31 u23 v43 + 4 6qr u51 u63 + 8 6qr u51 u43 v23 + 4 6qr u51 u23 v43 + ), u43 u1 (u23 + v23 )2 √ (u −v )2 2pi u1 u63 u24 u5 1 √ −4u2 u43 u4 v1 v3 + 4u43 u4 v1 v2 v3 − u22 +v22 1√ 3 3 + 2k e 2( u˙5 = 3 2 2 u23 u1 (u23 + v23 ) u23 u1 (u23 + v23 ) 2 √ √ √ 2pi u1 u43 u24 u5 v23 + 4 2pi u1 u23 u5 + 4 2pi u1 u5 v23 ), + u23 u1 (u23 + v23 ) √ −2u2 u63 u5 v1 v3 − 2u2 u43 u5 v1 v33 + 2u63 u5 v1 v2 v3 + 2u43 u5 v1 v2 v33 − 2u22 u43 v1 v3 + 4u2 u43 v1 v2 v3 u˙6 = 2k( 5 u23 u1 (u23 + v23 ) 2 √ (u −v )2 −144u43 v1 v22 v3 − 72u43 v1 v33 − 72u23 v1 v53 − u22 +v22 1 √ 72 2pi u1 u63 u4 u25 v23 3 3 + )e + 2( 5 72 u23 u1 (u23 + v23 )2 u23 u1 (u23 + v23 ) 2 √ √ √ √ √ 36 2pi u1 u43 u4 u25 v43 + 36 2pr u1 u63 u25 + 72 2u1 u43 v23 γi + 36 2u1 u23 v43 γi − 144 2pr u1 u23 v23 + u23 u1 (u23 + v23 )2 √ √ √ √ √ −36 2pi u1 u63 u4 + 16 6qr u51 u63 − 72 2pi u1 u43 u4 v23 − 36 2pi u1 u23 u4 v43 + 32 6qr u51 u43 v23 u23 u1 (u23 + v23 )2 √ √ √ √ 16 6qr u51 u23 v43 + 36 2pi u1 u83 u4 u25 + 72 2pr u1 u43 u25 v23 + 36 2pr u1 u23 u25 v43 + 45qr u31 u63 + u23 u1 (u23 + v23 )2 √ √ √ −72 2pr u1 u43 − 72 2pr u1 v43 + 90qr u31 u43 v23 + 45qr u31 u23 v43 + 36 2u1 u63 γi + ), u23 u1 (u23 + v23 )2 and: √ 2u1 u22 u3 v43 − 4u1 u2 u3 v2 v43 − 12 u1 u53 v23 − 2u1 u33 v43 + 2u1 u3 v22 v43 v˙1 = − 2k( 5 v23 (u23 + v23 ) 2 √ (u −v )2 6 − 22 22 − 108 1 √ −36 2pi u43 v1 v23 v25 72 u1 u3 v3 u3 +v3 + − 2( 5 )e 72 u23 (u23 + v23 )2 v23 (u23 + v23 ) 2 √ √ √ √ −72 2pi u23 v1 v43 v25 − 36 2pi v1 v63 v25 − 72 2pi u43 v1 − 144 2pi u23 v1 v23 + u23 (u23 + v23 )2 √ √ √ √ −72 2pi v1 v43 − 36 2u43 v1 v23 γr − 72 2u23 v1 v43 γr − 36 2v1 v63 γr + 45qi u43 v31 v23 u23 (u23 + v23 )2 √ √ √ 45qi v31 v63 + 16 6qi u43 v51 v23 + 32 6qi u23 v51 v43 + 16 6qi v51 v63 + u23 (u23 + v23 )2 √ √ √ 72 2pr u23 v1 v43 v4 + 36 2pr v1 v63 v4 + 36 2pr u43 v1 v23 v4 + 90qi u23 v31 v43 + ), u23 (u23 + v23 )2

(21)


109

(u −v )2

1 √ 4u1 u2 u3 v23 − 4u1 u3 v2 v23 − u22 +v22 v˙2 = 2k e 3 3 3 2 v1 (u23 + v23 ) 2 √ √ √ √ 1 √ 2pi u23 v1 v23 v4 v5 + 2pi v1 v43 v4 v5 + 2 2pr u23 v1 v5 + 2 2pr v1 v23 v5 + 2 , 1 2 u1 (u2 + v2 ) 2 3

3

2

(u −v ) 1 √ 144u1 u22 u3 v43 − 288u1 u2 u3 v2 v43 + 36u1 u53 v23 + 144u1 u3 v22 v43 − 36u1 u3 v63 − u22 +v22 3 3 2k e v˙3 = 5 36 v3 v1 (u23 + v23 ) 2 √ √ √ √ √ 1 √ 9 2pi u43 v1 v43 v24 + 18 2pi u23 v1 v63 v24 + 9 2pi v1 v83 v24 − 36 2pi u43 v1 − 72 2pi u23 v1 v23 + 2( 36 v3 v1 (u23 + v23 )2 √ √ −36 2pi v1 v43 + 9qi u43 v31 v23 + 18qi u23 v31 v43 + 9qi v31 v63 + 4 6qi u43 v51 v23 + v3 v1 (u23 + v23 )2 √ √ √ √ √ 8 6qi u23 v51 v43 + 4 6qi v51 v63 + 36 2pr u43 v1 v23 v4 + 72 2pr u23 v1 v43 v4 + 36 2pr v1 v63 v4 + ) v3 v1 (u23 + v23 )2 (u −v )2

1 √ −144u1 u22 u3 v43 + 288u1 u2 u3 v2 v43 − 72u1 u53 v23 − 72u1 u33 v43 − 144u1 u3 v22 v43 − u22 +v22 2k e 3 3 , v˙4 = − 5 9 v43 v1 (u23 + v23 ) 2 √ √ √ √ √ 1 √ −36 2pr u43 v1 − 72 2pr u23 v1 v23 − 36 2pr v1 v43 + 9 2pr u43 v1 v43 v24 + 18 2pr u23 v1 v63 v24 2( − 9 v43 v1 (u23 + v23 )2 √ √ √ √ 9 2pr v1 v83 v24 − 36 2pi u43 v1 v23 v4 − 72 2pi u23 v1 v43 v4 − 36 2pi v1 v63 v4 + 9qr u43 v31 v23 + 18qr u23 v31 v43 + v43 v1 (u23 + v23 )2 √ √ √ 4 6qr u43 v51 v23 + 8 6qr u23 v51 v43 + 4 6qr v51 v63 + 9qr v31 v63 + ), v43 v1 (u23 + v23 )2 (u −v )2

1 √ 4u1 u2 u3 v43 v4 − 4u1 u3 v2 v43 v4 − u22 +v22 2k e 3 3 v˙5 = 3 2 v23 v1 (u23 + v23 ) 2 √ √ √ √ 1 √ 2pi u23 v1 v43 v24 v5 + 2pi v1 v63 v24 v5 + 4 2pi u23 v1 v5 + 4 2pi v1 v23 v5 , + 2 2 v23 v1 (u23 + v23 ) √ u1 u2 u33 v43 v5 + u1 u2 u3 v63 v5 − u1 u33 v2 v43 v5 − u1 u3 v2 v63 v5 − u1 u22 u3 v43 + 2u1 u2 u3 v2 v43 v˙6 =2 2k( 5 v23 v1 (u23 + v23 ) 2 1 1 5 2 3 4 2 4 − (u2 −v2 )2 2 2 2 u1 u3 v3 − 2 u1 u3 v3 − 144u1 u3 v2 v3 − )e u3 +v3 5 2 2 2 v3 v1 (u3 + v3 ) 2 √ √ √ √ 1 √ 72 2pi v1 v63 v4 v25 u23 + 36 2pi v1 v43 v4 v25 u43 + 36 2pi v1 v83 v4 v25 + 36 2pr u43 v1 v23 v25 + 2( 72 v23 v1 (u23 + v23 )2 √ √ √ √ √ −144 2pr u23 v1 v23 − 36 2pi v1 v63 v4 + 16 6qr v51 v63 − 36 2pi u43 v1 v23 v4 − 72 2pi u23 v1 v43 v4 v23 v1 (u23 + v23 )2 √ √ √ √ 32 6qr u23 v51 v43 + 36 2v1 v63 γi − 72 2pr u43 v1 − 72 2pr v1 v43 + 45qr u43 v31 v23 + 90qr u23 v31 v43 + v23 v1 (u23 + v23 )2 √ √ √ √ √ +45qr v31 v63 + 36 2pr v1 v63 v25 + 36 2u43 v1 v23 γi + 72 2u23 v1 v43 γi + 72 2pr u23 v1 v43 v25 + 16 6qr u43 v51 v23 ). v23 v1 (u23 + v23 )2

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3 Numerical results 3.1

Case of initial symmetric pulses

Initial symmetric pulses are pulses having the same parameters and symmetric temporal positions. In this subsection, we fixed initial conditions as follows: (u1 , u2 , u3 , u4 , u5 , u6 ) = (1, −1, 1, 0, 0, 0) for the left core and (v1 , v2 , v3 , v4 , v5 , v6 ) = (1, 1, 1, 0, 0, 0) for the right core. We considered quintic QCGLE as perturbation to the NLSE. Thus, we took pi , qi , ci coefficients very small and γi = 0. The space dependence of the dispersion coefficient is explained by the fact that in DM optical fiber there is zero dispersion which consists of anomalous-dispersion (pr = d1 > 0) fiber with a length z1 = 0.1, followed by a normal-dispersion (pr = d2 < 0) fiber with a length z2 = 0.02. Hence a portion of DM fiber has a length za = z1 + z2 . The numerical study of the evolution of neighboring pulses parameters along the propagation distance is carried out using standard fourth order Runge-Kutta method with the spatial step 2 · 10−3 . Numerical computations will be done for zero-average dispersion setting d1 = 1 and d2 remains, in all the text, −4d1 . QCGLE coefficients are set as: p = 1 − i0.6, q = 1 − i0.046, c = 0.1 + i0.0016. right core 2

1.5

1.5 amplitude

amplitude

left core 2

1 0.5 0 0

1 0.5

20

40 space

0 0

60

20

40 space

60

(a) Amplitude variation in the left core

right core 3.5

3

3

2.5

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width

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2

2

1.5

1.5

1

1

0.5 0

20

40 space

60

0.5 0

20

40 space

60

(b) Width variation in the right core

Fig. 1 Amplitude and width variation of pulses in the two cores for different values of the linear coupling coefficient: k = −0.1 (red color), k = 0.0 (black color) and k = 0.1 (green color).


111

temporal position

4

2

0

−2

−4 0

10

20

30

space

40

50

60

70

Fig. 2 Variations of the temporal position of pulses propagating inside the two cores for different values of the linear coupling coefficient: k = −0.1 (red color) and k = 0.1 (green color) and and k = 0.0 (black color).

k=0.1

amplitudes ratio

2

1.5

1

0.5

0 0

10

20

30

space

40

50

60

70

Fig. 3 Evolution of amplitudes ratio when the coupling coefficient is equal to 0.1.

k=0.1

Widths ratio

2 1.5 1 0.5 0 0

10

20

30

space

40

50

60

70

Fig. 4 Evolution of widths ratio when the coupling coefficient is equal to 0.1.

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power in the left core power in the right core

power

8

6

4

2

0 0

20

40 space

60

80

Fig. 5 Variations of light power inside the two cores when the coupling coefficient is equal to 0.1.

Note that by setting the initial frequencies u50 = 0 and v50 = 0, the temporal positions and frequencies remain zero if besides k the coupling coefficient is null (Equations 20 and 21). Figure 1 shows the evolution of the pulse amplitude and width during the propagation distance for different values of the linear coupling coefficient: (a): k = −0.1 (red color), (b): k = 0.0 (black color), (c): k = 0.1 (green color). One observes that the evolution of these parameters in the two cores is identical. Besides, one observes convergence of the temporal positions of two cores when k > 0, which is traduced by the fact that their temporal position shift in the same direction and end by joining. However their temporal positions move away if k < 0 (see Figure 2). Figures 3 and 4 representing the amplitudes and widths ratios evolution show that they are equal to 1 for each. That observation confirm the identical evolution of the amplitude and width showed on Figure 1. We also studied the evolution of light power propagating inside each core. Recall that its expressions are respectively obtained by the relations: ˆ ∞ √ √ (| f1 |)2 dt = 1/2 2 π u1 2 u3 , P1 = ˆ−∞ (22) ∞ √ √ 2 2 (| f2 |) dt = 1/2 2 π v1 v3 . P2 = −∞

It appears on Figure 5 that the energies in both cores are equal all along the fiber because their power diagrams are superposed. 3.2

Case of initial asymmetric pulses

In this subsection, we fixed initial amplitude of the right core at 75% of the amplitude of the left core. The other parameters and coefficient remain the same. On Figure 6, quite observations are made with the above subsection. The amplitudes ratio and widths ratio of light inside the two cores primarily oscillate before stabilizing to 1 (see Figure 8 and Figure 9). So after a certain distance lights in both core propagate with the same amplitude and width. These observation are confirmed on Figure 10 which shows the variations of light power inside each core. We remark on that figure that the cores exchange their power at the beginning, the one with the greater energy transfer it to his neighbor, before reaching an equilibrium where energies are equal. We also note as in the case of initial symmetric waves, convergence and divergence of the temporal positions (see Figure 7) depending on the sign of the coupling coefficient. For positive value of k, the temporal positions converge on the left core side and join while for negative value, the temporal positions diverge. The divergence in the left core is more amplified than the one in the right core.

R.B. Djob et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 101–116 right core

2

2

1.5

1.5 amplitude

amplitude

left core

1 0.5 0 0

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20

40 space

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60

20

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(a) Amplitude variation in the left core left core

right core

3.5

4

3

3.5 3 width

width

2.5 2 1.5

2 1.5

1 0.5 0

2.5

1 20

40 space

0.5 0

60

20

40 space

60

(b) Width variation in the right core

Fig. 6 Amplitude and width variation of pulses in the two cores for different values of the linear coupling coefficient: k = −0.1 (red color), k = 0.0 (black color) and k = 0.1 (green color). 5

temporal position

4 3 2 1 0 −1 −2 0

10

20

30

space

40

50

60

70

Fig. 7 Variations of the temporal position of pulses propagating inside the two cores for different values of the linear coupling coefficient: k = −0.1 (red color) and k = 0.1 (green color) and k = 0.0 (black color).

4 Conclusions In this work, we successfully derived the CVs equations of motion for the coupled QCGLE with the help of the CVs treatment of DM dual core optical fiber. The dynamics of pulses parameters deeply modified due to the effects of linear coupling.

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R.B. Djob et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 101–116 k=0.1

amplitudes ratio

2

1.5

1

0.5 0

10

20

30

space

40

50

60

70

Fig. 8 Evolution of amplitudes ratio when the coupling coefficient is equal to 0.1. k=0.1 3

Widths ratio

2.5 2 1.5 1 0.5 0

10

20

30

space

40

50

60

70

Fig. 9 Evolution of widths ratio when the coupling coefficient is equal to 0.1. k=0.1 15

power in the left core power in the right core

power

10

5

0 0

10

20

30

space

40

50

60

70

Fig. 10 Variations of light power inside each core when the coupling coefficient is equal to 0.1.

We noticed that symmetric initial pulses propagate with the same amplitude, width and energy while for asymmetric initial pulses, there is a compensation phenomenon. In both cases, we noticed that when the coupling is taken into account, temporal positions may converge or diverge depending on the sign of the coupling coefficient.


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Thought the necessity of manufacturing fibers with multi-cores is benefit in industry because it reduces the number of devices, we must avoid and reduce the effects of coupling if we need originate emergent signals.

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Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems Y. Boukal1,2 , M. Darouach1 , M. Zasadzinski1 †, and N.E. Radhy2 1

Université de Lorraine, Centre de Recherche en Automatique de Nancy (CRAN UMR-7039, CNRS), IUT de Longwy, 186 rue de Lorraine 54400, Cosnes et Romain, France 2 Universit´ e Hassan II, Faculté des Sciences Ain-Chock, Laboratoire Physique et Matériaux Microélectronique Automatique et Thermique BP: 5366 Maarif, Casablanca 20100, Morocco Submission Info Communicated by Albert C.J. Luo Received 10 September 2014 Accepted 12 January 2015 Available online 1 July 2015 Keywords

Abstract This paper considers unknown input functional fractional order observer design for fractional-order linear time-invariant (LTI) systems with a constant time delay. After given the existence conditions of such observer, based on the fractional order Lyapunov stability approach, a sufficient condition for the asymptotic stability of the estimation error is given in a linear matrix inequality (LMI) formulation. The obtained results are illustrated by a numerical example.

Fractional-order time-delay system Functional fractional-order observer Unknown inputs Asymptotic stability Linear matrix inequality (LMI) ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The recent years, there has been a considerable interest and research activities to the applications of fractional calculus in many branches of science and engineering. Many processes in the nature have been successfully modelled by fractional order differential equations and their dynamics are represented in continuous-time domain by differential equations of non integer-order α , in physics, chemistry and biology, such as electrical engineering, viscoelasticity, modeling of neurons, diffusion processes etc (see [1–7]). The main reason for describing these systems by fractional model is the much progress has also been made in a set of definitions, theoretical methods and numerical analysis of fractional calculus. Delay systems (also called hereditary, with memory, dead-time or with after-effects) met in many practical systems, interconnected real systems and appear naturally in many engineering process applications. In fact, in any situation in which transmission delays cannot be ignored. Moreover, time-delay † Corresponding author. Email address: [email protected]


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Y. Boukal et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 117–130

phenomena in the inner dynamics of those processes are due to transportation of material, energy, or information, such as chemical processes, thermal, hydraulic, economics, mechanics systems (many examples are given in [8, 9]), and is well known as a source of instability. This explains the attention that has been devoted to the problems of stability, stabilization, control and observer design for this class of systems. The question of stability is of main interest in physical, chemical and biological systems. Recently, there has been some advances in control theory of fractional-order systems, but there is a few stability results for this class of systems with a pure time-delay, such as the sufficient conditions of a finite-time stability in state-space form for homogeneous and non-homogeneous systems which are given in [10] and with multi-state time delay in [11]. New frequency domain methods for the analysis of this class with delays have been proposed in [12]. Some results related to the bounded-input bounded-output stability of fractional-order control systems with delays are available in [13–15]. In [16] stability analysis of linear fractional order neutral system with multiple delays by algebraic approach has been proposed. We can conclude that there is not a general stability theory for fractional-order systems with delay. The lack of a general theorem of asymptotic stability, and hence makes the stability analysis of fractional order time-delay systems a challenging task. The control of dynamical systems is often realized with the assumption that the entire state vector is available trough output measurement. Since this is not generally true in the practice, it is necessary to design observers which produce an estimate of this state vector. However, there are only a few results regarding the state estimation for the fractional-order system [17–20]. In some applications it is necessary to estimate the state of a system in the presence of completely unknown input. Control of unknown input fractional order systems with delay is a subject of both practical and theoretical importance, so it is important to design a unknown input fractional order observers which are able to estimate the state of a system in the presence of unknown input disturbances. Despite the importance of system state estimation in many practical engineering applications, to the best of the authors knowledge, there is no work in which the problem of designing unknown input fractional order observer for unknown input fractional order systems with delay is investigated. In this work, a method is proposed to design a functional observer for fractional-order systems with delay and unknown inputs. This approach is divided into two parts. In the first one, the conditions for the existence of the functional observer are given. In the second one, to prove the convergence to zero of the estimation error, we use the fractional order Lyapunov stability approach, and a sufficient condition for the asymptotic stability of the estimation error is given in a linear matrix inequality (LMI) formulation. Finally, a numerical example is presented to illustrate the observer design method applied to the state estimation problem for fractional order systems with delay.

2 Preliminaries 2.1

Fractional-Order differential equation

Formulations of non-integer order derivatives fall into two main classes: the Riemann-Liouville derivative and the Grˆ unward-Letnikov derivative, on one hand, defined as [21, 22] ˆ dn t 1 f (τ ) RL α dτ , (n − 1) ≤ α < n (1) a Dt f (t) = n Γ(n − α ) dt a (t − τ )α −n+1 or the Caputo derivative on the other, defined as [22, 23] C α a Dt

f (t) =

1 Γ(n − α )

ˆ a

t

d n f (τ ) dt n dτ , (t − τ )α −n+1

(n − 1) ≤ α < n

(2)


119

with n ∈ N and α ∈ R+ , where Γ(.) is the Gamma function. The physical interpretation of the fractional derivatives and the solution of fractional differential equations are given in [22,23]. Here and throughout the paper, only the Caputo definition is used since its Laplace transform allows the utilization of initial values of classical integer-order derivatives with clear physical interpretations. Proposition 1. [24] Between the two above definitions, the main difference concerns the initial condition as can be seen in the following relationships C α 0 Dt RL α 0 Dt

α f (t) =RL 0 Dt ( f (t) −

f (t) =C0 Dtα f (t) +

m−1

tk ), k!

(3a)

t k−α f k (0+ ). Γ(k − α + 1)

(3b)

∑

f k (0+ )

k=0 m−1

∑

k=0

Proposition 2. [24] Let us consider the Riemann-Liouville fractional derivative of an order α . Then, we have at −α RL α . (4) D (a) = t 0 Γ(1 − α ) Remark 1. [21] The fractional derivative of any real order q of fractional derivative of order p < 0 is given by q p a Dt (a It f (t))

q+p

=a Dt

f (t).

(5)

Remark 2. [21] The most important property of Riemann-Liouville fractional derivative is that for p > 0 and t > a we have p −p a Dt (a It f (t))

= f (t).

(6)

Remark 3. [25] The q-th order fractional derivative of function g(x(t)) = x(t)2 with respect to t is given by [25] q α (7) 0 Dt g(x(t)) = x(t)0 Dt x(t) + px , where px =

∞

Γ(1 + α )

(α −k)

∑ Γ(1 + k)Γ(1 − K + α ) (0 Dtk x(t))(0 Dt

x(t)).

(8)

k=1

Remark 4. [21] The formula for the evaluation of the fractional derivative of a composite function is given by p a Dt F(h(t)))

=

(t − a)−p F(h(t)) Γ(1 − p) ∞ K 1 hr (t) ar p k!(t − a)k−p k m , +∑ ∑ F (h(t)) ∑ ∏ ar ! r! r=1 k=1 k Γ(k − p + 1) m=1

where the ∑ extends over all combinations of non-negative integer values of a1 , a2 , . . . , ak such that k

∑ rar = k,

r=1

k

∑ ar = m. r

(9)


120

Remark 5. The binomial coefficient

n k

, which is defined as n n! = k k!(n − k)!

(10)

can be generalized, using the Gamma function. We note that for any natural number n, n! = Γ(n + 1), thus, we have, for any integer n and for any integer k less than or equal to n n Γ(n + 1) . (11) = Γ(k + 1)Γ(n − k + 1) k One might extend the relation beyond the above boundaries to include that if k is strictly negative or strictly greater than n, it was agreed that the binomial coefficient is zero. 2.2

Stability of fractional-order systems

Definition 1. Definition of the Mittag-Leffler Stability [26]. The solutions of t0 Dtα x(t) = f (t, x) is said to be Mittag-Leffler stable if (12) || x(t) ||≤ {m[x(t0 )]Eα (−λ (t − t0 )α }b , where t0 is the initial time, α ∈ (0, 1), λ > 0, b > 0, m(0) = 0, m(x) ≥ 0, and m(x) is locally Lipschitz on x ∈ B ∈ Rn with Lipschitz constant m0 . β

Lemma 3. [27], [26] Let x = 0 be an equilibrium point for the non-autonomous fractional order Dt x(t) = f (x,t) where β ∈ [0, 1]. Assume that there exists a Lyapunov function V (t, x(t)) and class-K functions μi (i = 1, 2, 3) satisfying μ1 ||x(t)|| ≤ V (t, x(t) ≤ μ2 ||x(t)|| (13) and

β

Dt V (t, x(t) ≤ −μ3 ||x(t)||. Then the equilibrium point of system

q (Dt x(t)

(14)

= f (x,t)) is Mittag-Leffler stable.

Remark 6. [26] For both Riemann-Liouville and Caputo fractional derivative definition, if the conβ ditions of Lemme 3 are satisfied, then it can be concluded system Dt x(t) = f (x,t) is Mittag-Leffler stable. Remark 7. [27] Mittag-Leffler stability implies asymptotic stability. With remark 7, the stability concept used in the sequel is the asymptotic stability since this stability is achieved if conditions (13) and (14) in lemma 3 hold.

3 Main results In real applications, generally, the systems studied are subject to unknown disturbances, also called unknown inputs. The unknown inputs can be a combination of unmeasured disturbances, unknown control actions, or unmodelled system dynamics. A linear fractional-order time-delay system with unknown inputs can be represented by the following form : ⎧ α D x(t) = A0 x(t) + A1 x(t − τ ) + Buu(t) + Bd d(t), ⎪ ⎪ ⎨ y(t) = Cx(t) + Dd d(t), (15) z(t) = T x(t), 0 < α < 1, ⎪ ⎪ ⎩ x(t) = ψ (t),t ∈ [−τ , 0],


121

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is the input vector, y(t) ∈ R p the measurement output vector, d(t) ∈ Rk is the unknown input vector and z(t) ∈ Rr is the functional state. A0 ∈ Rn×n , A1 ∈ Rn×n , Bu ∈ Rn×m , Bd ∈ Rn×k , C ∈ R p×n , Dd ∈ R p×k and T ∈ Rr×n are known constant matrices. τ is a pure time delay assumed to be constant and known with τ > 0. Dα is used to denote the fractional derivative of order α . The associated function ψ (t) of the initial states is taken as Hartley and Lorenzo [29]. A, B, C and L are known constant system matrices of appropriate dimensions. Remark 8. The notation Dα is used if the knowledge of the choice of the derivative in not necessary for understanding the mathematical development. In the sequel, if the Caputo derivative is used, then Dα is replaced by Ca Dtα , while if the Riemann-Liouville derivative is considered, then Dα is replaced by RL Dα . a t In order to reconstruct the state variable of the system (15), as in [30] we consider the functional observer of the form ⎧ α ⎨ D η (t) = N0 η (t) + N1η (t − τ ) + Hu(t) + J0y(t) + J1 y(t − τ ), (16) 0 < α < 1, zˆ(t) = η (t) + Ey(t), ⎩ η (t) = η0 ,t ∈ [−τ , 0], where η (t) ∈ Rr is the state vector of observer and zˆ(t) ∈ Rr is the estimate of the functional z(t). The matrices N0 , N1 , J0 , J1 , H and E are unknown and of appropriate dimensions which must be determined such that zˆ(t) converges asymptotically to z(t). Define the error e(t) = z(t) − zˆ(t), the dynamic of the fractional-order error is given by Dα e(t) = Dα z(t) − Dα zˆ(t) = N0 e(t) + N1 e(t − τ ) + (PA0 − N0 P − J0C)x(t) + (PA1 − N1 P − J1C)x(t − τ ) + (PB − H)u(t) (17) +(PBd + N0 EDd − J0 Dd )d(t) + (N1 EDd − J1 Dd )d(t − τ ) − EDd Dα , d(t) where P = T − EC. The following theorem gives the conditions for the existence and stability of the functional observer (16). Theorem 4. For 0 < α < 1, system (16) is an asymptotic functional observer guaranteed lim zˆ(t) − t→+∞

z(t) = 0, for any x(0), zˆ(0) and u(t) if there exists matrices N0 , N1 , J0 , J1 , H and E such that Dα e(t) = N0 e(t) + N1 e(t − τ )

is

stable.

(18a)

PA0 − N0 P − J0C =0,

(18b)

PA1 − N1 P − J1C =0,

(18c)

H =PB,

(18d)

PBd + N0 EDd − J0 Dd =0,

(18e)

N1 EDd − J1 Dd =0,

(18f)

EDd =0,

(18g)

Proof. Under conditions (18b– 18g) and from (17), one can see that the fractional-order dynamic of this observer error is given by Dα e(t) = N0 e(t) + N1 e(t − τ ) with 0 < α < 1. In this case, for any z(0), zˆ(0) and u(t), lim e(t) = 0 if (19) is stable. This proves the theorem. t→+∞

(19)


122

Now the design of the functional observer is reduced to find matrices N0 , N1 , J0 , J1 , H and E such that (18b–18g) are satisfied. By using the definition of P, equations (18b) and (18c) can be written as N0 T + ECA0 + K0C = TA0 ,

(20a)

N1 T + ECA1 + K1C = TA1 ,

(20b)

K0 Dd + ECBd = T Bd ,

(20c)

K1 Dd = 0,

(20d)

EDd = 0,

(20e)

where K0 = J0 − N0 E and K1 = J1 − N1 E. Notice that once matrix E is determined, we can easily compute the value of matrix H from (18d) of Theorem 4. Equations (20a–20e) can be written as

(21) N0 N1 K0 K1 E M1 = M2 , where

⎡

⎤ T 0 0 0 0 ⎢ 0 T 0 0 0⎥ ⎢ ⎥ ⎥ M1 = ⎢ ⎢ C 0 Dd 0 0 ⎥ , ⎣ 0 C 0 Dd 0 ⎦ CA0 CA1 CBd 0 Dd

M2 = TA0 TA1 T Bd 0 0 .

The necessary and sufficient condition for the existence of the solution of (21) can then be given by the following lemma. Lemma 5. [31] The equation (21) is solvable if and only if the following condition holds

M1 = rank M1 . rank M2

(22)

In this case, the general solution of (21) is given by

N0 N1 K0 K1 E = M2 M1+ − Z(I − M1 M1+ ), where Z is an arbitrary matrix of appropriate dimension, and satisfying M1 M1+ M1 = M1 . From (23), we obtain

where

M1+

(23)

is the generalized inverse matrix

N0 = A0 − ZB0,

(24a)

N1 = A1 − ZB1,

(24b)

K0 = A2 − ZB2,

(24c)

K1 = A3 − ZB3,

(24d)

T A0 = (M2 M1+ ) I 0 0 0 0 ,

T B0 = (I − M1 M1+ ) I 0 0 0 0 ,

T A1 = (M2 M1+ ) 0 I 0 0 0 ,

T B1 = (I − M1 M1+ ) 0 I 0 0 0 .


Matrices J0 , J1 and H are obtained from

⎧ ⎨ J0 = K0 + N0 E, J1 = K1 + N1 E, ⎩ H = (T − EC)B.

123

(25)

By using this results we can compute all the parameters of the functional fractional-order observer (16) if matrix parameter Z is known. Under condition (22) and by using (24a), (24b), the observer error dynamics can be written as Dα e(t) = (A0 − ZB0)e(t) + (A1 − ZB1)e(t − τ ),

0 < α < 1.

(26)

Then, the design of the functional observer (16) is reduced to the determination of the free matrix parameter Z such that condition (18a) of Theorem 4 is satisfied, i.e, the asymptotic stability of the estimation error dynamics is guaranteed. Now, the determination of the observer gain Z is solved such that the asymptotic stability of the observation error (26) with 0 < α < 1 holds. Firstly, we start by given the asymptotic stability condition of the autonomous fractional-order system given by the following plant : Dα e(t) = [(A0 + A1 ) − Z(B0 + B1 )]e(t),

0 < α < 1,

(27)

because this is a necessary condition for ensuring the asymptotic convergence of e(t) in the case where τ = 0. Lemma 6. [32], [28] The fractional-order system Dα x(t) = Ax(t)

(28)

is asymptotically stable where 0 < α < 1, if and only if there exist two real symmetric matrices Pk1 ∈ Rn×n , k = 1,2 and two skew-symmetric matrices Pk2 ∈ Rn×n , k = 1,2 such that 2

2

∑ ∑ Sym(Γi j ⊗ (APi j )) < 0

(29)

i=1 j=1

where

P11 −P12

P12 > 0, P11

P21 −P22

P22 > 0, P21

π π ⎤ sin(α ) − cos(α ) 2π Γ11 = ⎣ π2 ⎦ , sin(α ) cos(α ) 2 2 ⎡ π π ⎤ cos(α ) sin(α ) 2π 2π ⎦ , Γ12 = ⎣ − sin(α ) cos(α ) 2 2 ⎤ ⎡ π π sin(α ) cos(α ) 2 π Γ21 = ⎣ π2 ⎦ , − cos(α ) sin(α ) 2 2 ⎡ π π ⎤ − cos(α ) sin(α ) 2 ⎦. Γ22 = ⎣ π2 π − sin(α ) − cos(α ) 2 2 ⎡

(30a)

(30b)

(30c)

(30d)

124


Now, on the basis of the Lyapunov approach, we develop a sufficient asymptotic stability condition of the delayed estimation error dynamic in terms of linear matrix inequality (LMI). It is given in the following theorem. Theorem 7. Suppose that conditions (18b–18g) are satisfied. Then there exists a parameter matrix Z such that system (26) is asymptotically stable ∀t > τ +t0 , if there exist matrices P = PT > 0, Q = QT > 0 and Y satisfying the following LMI: AT0 P + PA0 − BT0 Y T −Y B0 PA1 −Y B1 0 and Q = QT > 0, the fractional derivative of V (t) along the trajectory of (26) by using the proposition 1 is given by : C α 0 Dt V (t)

= Ψ1 + Ψ2 ,

where Ψ1 and Ψ2 are given by Ψ1 = 2C0 Dtα (eT (t)Pe(t)) α T T RL α = (RL 0 Dt e (t))Pe(t) + e (t)P(0 Dt e(t)) − ϒ1 + ϒ2 ,

Ψ2 =C0 Dtα eT (t − τ )Qe(t − τ ) (t − τ )−α T e (t − τ )Qe(t − τ ) − ϒ3 + ϒ4 + ϒ5 Γ(1 − α ) (t0 )−α T e (t − τ )Qe(t − τ ) − ϒ3 + ϒ4 + ϒ5 . < Γ(1 − α ) =

(34)


125

Now, by replacing (32a, 32b, 33a, 33b, 33c) into (34), we obtain C α 0 Dt V (t)

α T T RL α ≤ (RL 0 Dt e (t))Pe(t) + e (t)P(0 Dt e(t)) +

(t0 )−α T e (t − τ )Qe(t − τ ) + ϒ Γ(1 − α )

(35)

with ϒ = −ϒ1 + ϒ2 − ϒ3 + ϒ4 + ϒ5 . Considering (26), inequality (35) can be rewritten as follows C α T T T T T 0 Dt V (t) ≤ e (t)(N0 P + PN0)e(t) + e (t − τ )N1 Pe(t) + e (t)PN1 e(t − τ ) +

(t0 )−α T e (t − τ )Qe(t) + ϒ. Γ(1 − α ) (36)

Under remark 5, we have ϒ2 = ϒ4 = ϒ5 = 0,

(37)

and, without loss of generality, one can easily conclude that t −α ||e(0+ )||2 Γ(1 − α ) − 2t α ||e(0+ )||2 ϒ3 = Q Γ(1 − α ) ϒ1 = 2P

≥

0,

(38a)

≥

0.

(38b)

Then by using inequalities (38a) and (38b), we obtain C α 0 Dt V (t)

≤ eT (t)(N0T P + PN0 )e(t) + eT (t − τ )N1T Pe(t) + eT (t)PN1 e(t − τ ) +

Consequently, we have

C α T 0 Dt V (t) ≤ X

with X =

e(t) . Then if e(t − τ )

N0T P + PN0 N1T P

PN1

(t0 )−α T e (t − τ )Qe(t). (39) Γ(1 − α )

(t0 )−α Γ(1−α ) Q

AT0 P + PA0 − BT0 Y T −Y B0 PA1 −Y B1 (t0 )−α AT1 P − BT1 Y T Γ(1−α ) Q

X

(40)

< 0,

(41)

we have C0 Dtα V (t) < 0 (see lemma 3 and remark 7), where the parameter Z is given by Z = P−1Y . So, it is shown that the fractional delayed estimation error is asymptotically stable if the LMI (31) holds. This completes the proof. 4 Numerical example Consider the LTI fractional-order system with delay described by: ⎧ −4 2 1 4 1 0.9 0 ⎪ 0.5 ⎪ x(t) + x(t − 0.35) + u(t) + d(t), D x(t) = ⎪ ⎪ 1 −1 −2 0.1 0 0.2 −0.6 ⎪ ⎪ ⎪ ⎨ 1 −2 0.5 0.1 y(t) = x(t) + d(t), 0.5 1 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ 10 ⎪ ⎪ ⎩ z(t) = x(t). 01

(42)

The results of Section 3 are applied to system (42), then we can see that condition (22) is satisfied. A feasible solution of the LMI in Theorem 7 is given by


126

P=

10+3

1.1229 1.3815 , 1.3815 3.1952

0.5698 1.1365 0.3207 0.6402 −0.0006 −1.1371 −0.0002 −0.6402 0.0003 −0.0002 . and Y 0.6873 1.3748 −0.1603 −0.3198 0.0003 −1.3740 0.0001 0.3204 −0.0002 0.0001 the asymptotic stability of (26) is obtained. Then the parameter matrix Z is obtained as 0.5188 1.0313 0.7420 1.4811 −0.0014 −1.0332 −0.0006 −1.4816 0.0007 −0.0005 −1 Z =P Y = . −0.0092 −0.0156 −0.3710 −0.7405 0.0007 0.0167 0.0003 0.7409 −0.0004 0.0002 = 10+3

Using the algorithm in Section 3, the functional observer parameters are given by −6.2699 2.8849 −0.2970 0.1485 , N1 = , N0 = 2.8849 −1.9425 0.1485 −0.0742 1.4795 3.0572 0 0.2466 , E= , J0 = −0.7397 −1.5286 0 0.8767 0 3.3907 0.8767 , H= . J1 = 0 −1.6953 −0.4384 In the simulation studies, we consider the case where the dynamics of states and outputs signals of the system are affected by unknown input d(t). Figures 1 and 2 display the unknown inputs signals d1 (t) and d2 (t) comportment. We consider the case where, after three seconds, the first component of unknown inputs vector is activated for a duration lower than forty seconds. Then in the interval between five and sixty-five seconds, the disturbance who affect the system is coming from the second component of the unknown inputs vector. Figures 3 and 4 illustrate the simulation of functional states system z1 (t), z2 (t) and their estimates zˆ1 (t), zˆ2 (t) for the case where the system is affected by unknown inputs. We can observe that one still has zˆ1 (t) and zˆ2 (t) converge to z1 (t) and z2 (t) respectively despite the perturbations d1 (t) and d2 (t). Next, Figures 5 and 7 show that estimation error converges to zero. Figures 6 and 8 show the estimation error behaviour near the origin. It is clear that the Unknown Input Observer ensures the asymptotic tracking of the system states despite the presence of unknown inputs. We can see clearly that the estimation error is not influenced by unknown inputs variation, knowing that these unknown inputs act on the states of the system. All figures 1-8 show the performances of our observer design approach.

5 Conclusion In this paper, the design problem of functional fractional-order observers for fractional-order linear systems with delayed states and unknown inputs is studied. Our approach involves a procedure which decouples the estimation error from the values of the state and unknown inputs. We give the existence conditions of such observers, and by using the fractional-order Lyapunov stability approach, a sufficient condition for the asymptotic stability of the estimation error has been given in a LMI formulation. The fact that our observer uses delays allows it to be applied to a much larger class of systems such as delayed diffusion process modelled by fractional order derivative with unknown inputs, fault detection, diagnosis and design observer-based controller for this class of systems. Simulation results demonstrate the effectiveness of the proposed design method.

Y. Boukal et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 117–130 8

Unkown Input d1(t)

7

Amplitude

6 5 4 3 2 1 0 −1 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 1 Evolution of Unknown Input d1 (t). 8

Unkown Input d2(t)

6

Amplitude

4 2 0 −2 −4 −6 −8 −10 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 2 Evolution of Unknown Input d2 (t).

Functional State Z1 Estimation of the functional state Z1

3 2.5

Amplitude

2 1.5 1 0.5 0 −0.5 −1 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 3 Evolution of the vector z1 (t) and its estimate zˆ1 (t).

127


Functional State Z2 Estimation of the functional state Z2

1

Amplitude

0.5 0 −0.5 −1 −1.5 −2 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 4 Evolution of the vector z2 (t) and its estimate zˆ2 (t).

0.8

Estimation Error Z1

0.7

Amplitude

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 5 Evolution of the estimation error e1 (t).

Estimation Error Z1

0.9 0.8 0.7

Amplitude

128

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 6 The behaviour of the estimation error e1 (t) at the origin.

Y. Boukal et al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 117–130 0.05

129

Estimation Error Z2

0

Amplitude

−0.05 −0.1 −0.15 −0.2 −0.25 −0.3 −0.35 −0.4 0

10

20

30

40

50

60

70

80

90

100

Time (s)

Fig. 7 Evolution of the estimation error e2 (t). 0.1

Estimation Error Z2

Amplitude

0 −0.1 −0.2 −0.3 −0.4 −0.5 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Fig. 8 The behaviour of the estimation error e2 (t) at the origin.

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A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth Wesson1 and Richard Rand2† 1 Center

for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA of Mathematics and Mech. & Aero. Eng., Cornell University, Ithaca, NY 14853, USA

2 Professor

Submission Info Communicated by J.A. T. Machado Received 30 October 2014 Accepted 13 January 2015 Available online 1 July 2015 Keywords Replicator equation Quasiperiodic forcing Floquet theory Harmonic balance Numerical integration

Abstract Evolutionary dynamics combines game theory and nonlinear dynamics to model competition in biological and social situations. The replicator equation is a standard paradigm in evolutionary dynamics. The growth rate of each strategy is its excess fitness: the deviation of its fitness from the average. The game-theoretic aspect of the model lies in the choice of fitness function, which is determined by a payoff matrix. Previous work by Ruelas and Rand investigated the RockPaper-Scissors replicator dynamics problem with periodic forcing of the payoff coefficients. This work extends the previous to consider the case of quasiperiodic forcing. This model may find applications in biological or social systems where competition is affected by cyclical processes on different scales, such as days/years or weeks/years. We study the quasiperiodically forced Rock-Paper-Scissors problem using numerical simulation, and Floquet theory and harmonic balance. We investigate the linear stability of the interior equilibrium point; we find that the region of stability in frequency space has fractal boundary. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The field of evolutionary dynamics combines game theory and nonlinear dynamics to model population shifts due to competition in biological and social situations. One standard paradigm [1, 2] uses the replicator equation, (1) x˙i = xi ( fi (x) − φ ), i = 1, . . . , n where xi is the frequency, or relative abundance, of strategy i; the unit vector x is the vector of frequencies; fi (x) is the fitness of strategy i; and φ is the average fitness, defined by

φ = ∑ xi fi (x).

(2)

i

The replicator equation can be derived [3] from an exponential model of population growth,

ξ˙i = ξi fi ,

i = 1, . . . , n.

† Corresponding

author. Email address: [email protected]


(3)

132

Elizabeth Wesson, Richard Rand /Journal of Applied Nonlinear Dynamics 4(2) (2015) 131–140

where ξi is the population of strategy i, assuming that fi depends only on the frequencies: fi = fi (x). The derivation consists of a simple change of variables: xi ≡ ξi /p where p = ∑i ξi is the total population. The game-theoretic component of the replicator model lies in the choice of fitness functions. Define the payoff matrix A = (ai j ) where ai j is the expected reward for a strategy i individual vs. a strategy j individual. We assume the population is well-mixed, so that any individual competes against each strategy at a rate proportional to that strategy’s frequency in the population. Then the fitness fi is the total expected payoff for strategy i vs. all strategies: fi (x) = (Ax)i = ∑ ai j x j .

(4)

j

In this work, we generalize the replicator model to systems in which the payoff coefficients are quasiperiodic functions of time. Previous work by Ruelas and Rand [4,5] investigated the Rock-PaperScissors replicator dynamics problem with periodic forcing of the payoff coefficients. We also consider a forced Rock-Paper-Scissors system. The quasiperiodically forced replicator model may find applications in biological or social systems where competition is affected by cyclical processes on different scales, such as days/years or weeks/years.

2 The model 2.1

Rock-Paper-Scissors games with quasiperiodic forcing

Rock-Paper-Scissors (RPS) games are a class of three-strategy evolutionary games in which each strategy is neutral vs. itself, and has a positive expected payoff vs. one of the other strategies and a negative expected payoff vs. the remaining strategy. The payoff matrix is thus ⎞ ⎛ 0 −b2 a1 (5) A = ⎝ a2 0 −b3 ⎠ . −b1 a3 0 We perturb off of the canonical case, a1 = · · · = b3 = 1, by taking ⎛ ⎞ 0 −1 − F(t) 1 + F(t) A=⎝ 1 0 −1 ⎠ , −1 1 0

(6)

where the forcing function F is given by F(t) = ε ((1 − δ ) cos ω1 t + δ cos ω2t).

(7)

For ease of notation, write (x1 , x2 , x3 ) = (x, y, z). The dynamics occur in the simplex S ≡ {(x, y, z) ∈ R | x, y, z ∈ [0, 1], x + y + z = 1},

(8)

but since x, y, z are the frequencies of the three strategies, and hence x + y + z = 1, we can eliminate z using z = 1 − x − y. Therefore, the region of interest is T , the projection of S into the x − y plane: T ≡ {(x, y) ∈ R | x, y, x + y ∈ [0, 1]}.

(9)

See Figure 1. Thus the replicator equation (1) becomes x˙ = −x(x + 2y − 1)(1 + (x − 1)F(t)),

(10)

y˙ = y(2x + y − 1 − x(x + 2y − 1)F(t)).

(11)


133

1.0

0.5

z

0.0

x

0.5

0.0 1.0 0.5

1.0 0.0

y

Fig. 1 A curve in S and its projection in T .

Note that x˙ = 0 when x = 0, y˙ = 0 when y = 0, and x˙ + y˙ = (x + y − 1)(xF(t)(x + 2y − 1) − x + y)

(12)

so that x˙ + y˙ = 0 when x + y = 1, which means that x + y = 1 is an invariant manifold. This shows that the boundary of T is invariant, so trajectories cannot escape the region of interest. It is known [6] that in the unperturbed case (ε = 0) there is an equilibrium point at (x, y) = ( 13 , 13 ), and the interior of T is filled with periodic orbits. We see from Equations (10) and (11) that this interior equilibrium point persists when ε = 0. Numerical integration suggests that the Lyapunov stability of motions around the equilibrium point depends sensitively on the values of ω1 and ω2 . See Figure 2. We investigate the stability of the interior equilibrium using Floquet theory and harmonic balance, as well as by numerical methods. 2.2

Linearization

To study the linear stability of the equilibrium point, we set x = u + 13 , y = v + 13 , substitute these into (10) and (11) and linearize, to obtain 1 u˙ = − (u + 2v)(3 + 2F(t)), 9 1 v˙ = (F(t)(u + 2v) + 3(2u + v)). 9

(13) (14)

The linearized system (13)-(14) can also be written [7] as a single second-order equation on u, by differentiating (13) and substituting in expressions for v˙ from (14) and v from (13). This gives us 1 g(t)u¨ − g(t) ˙ u˙ − g2 (t)u = 0, 9

(15)

g(t) = −3 − 2F(t) = −3 − 2ε ((1 − δ ) cos ω1 t + δ cos ω2 t).

(16)

where

Now that we have a linear system with coefficients that are functions of time, we use Floquet theory to determine the stability of the origin.


134 x

x

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

100

200

300

400

500

600

700

t

0

100

200

(a) ω1 = ω2 = 1.2

300

400

500

600

700

t

(b) ω1 = ω2 = 0.9

Fig. 2 Numerical solutions for x(t) with identical initial conditions x(0) = y(0) = 0.33 and parameters ε = 0.9, δ = 0.6, but with different ω1 , ω2 .

3 Floquet theory Floquet theory is concerned with systems of differential equations of the form dx = M(t)x, dt

M(t + T ) = M(t).

We have the system (13) and (14), which can be written as 1 g(t) 2g(t) u u u˙ ≡ B(t) , = 1 v v˙ 9 2 (9 − g(t)) −g(t) v

(17)

(18)

where g(t) is as in (16). In general, B(t) is not periodic, since ω1 and ω2 are rationally independent. However, the set of points for which ω1 and ω2 are rationally dependent is dense in the ω1 − ω2 plane, and solutions of (18) must vary continuously with ω1 and ω2 , so it is reasonable to consider only the case that F(t), and hence g(t) and B(t), are in fact periodic. Assume that ω2 = ab ω1 in lowest terms, where a and b are relatively prime integers. Then we can make the change of variables τ = ω1 t, so ω2t = ab τ . Since a and b are relatively prime, we see that F, and hence g and B, have period T = 2π b in τ . Thus (18) becomes 1 u u B( τ ) , B(τ + 2π b) = B(τ ), (19) = v v ω1 where u indicates du/d τ . This has the same form as (17), so we can apply the results of Floquet theory. Suppose that there is a fundamental solution matrix of (19), u1 (τ ) u2 (τ ) , (20) X (τ ) = v1 (τ ) v2 (τ ) where

1 u1 (0) = , 0 v1 (0)

u2 (0) 0 = . 1 v2 (0)

(21)


135

Then the Floquet matrix is C = X (T ) = X (2π b), and stability is determined by the eigenvalues of C:

λ 2 − (trC)λ + detC = 0.

(22)

We can show [8] that detC = 1, as follows. Define the Wronskian W (τ ) = det X (τ ) = u1 (τ )v2 (τ ) − u2 (τ )v1 (τ ).

(23)

Notice that W (0) = det X (0) = 1. Then taking the time derivative of W and using (19) gives dW = u1 (τ )v2 (τ ) + u1 (τ )v2 (τ ) − u2 (τ )v1 (τ ) − u2 (τ )v1 (τ ) dτ 1 1 (g(τ )(u1 + 2v1 )v2 + u1 (9u2 − (u2 + 2v2 )g(τ )) = 9ω1 2 1 − g(τ )(u2 + 2v2 )v1 − u2 (9u1 − (u1 + 2v1 )g(τ ))) = 0. 2 This shows that W (τ ) = 1 for all τ , and in particular W (T ) = detC = 1. Therefore, √ trC ± trC2 − 4 , λ= 2

(24)

(25)

which means [8] that the transition between stable and unstable solutions occurs when |trC| = 2, and this corresponds to periodic solutions of period T = 2π b or 2T = 4π b. Given the period of the solutions on the transition curves in the ω1 − ω2 plane, we use harmonic balance to approximate those transition curves.

4 Harmonic balance We seek solutions to (15) of period 4π b in τ : u=

∞

kτ

kτ

∑ αk cos( 2b ) + βk sin( 2b ).

(26)

k=0

Since ω2 = ab ω1 where a and b are relatively prime, any integer k can be written as na + mb for some integers n and m [9, 10]. That is, there is a one-to-one correspondence between integers k and ordered pairs (m, n). We can therefore write the solution as u=

∞

∞

∑ ∑

αmn cos(

ma + nb ma + nb τ ) + βmn sin( τ) 2b 2b

(27)

∑ ∑

αmn cos(

mω2 + nω1 mω2 + nω1 t) + βmn sin( t). 2 2

(28)

m=0 n=−∞ ∞ ∞

=

m=0 n=−∞

We substitute a truncated version of (28) into (15), expand the trigonometric functions and collect like terms. This results in cosine terms whose coefficients are functions of the αmn , and sine terms whose coefficients are functions of the βmn . Let the coefficient matrices of these two sets of terms be Q and R, respectively. In order for a nontrivial solution to exist, the determinants of both coefficient matrices must vanish [8]. We solve the equations det Q = 0 and det R = 0 for relations between ω1 and ω2 . This gives the approximate transition curves seen in Figure 3. It has been shown [4,7] that in a periodically √ forced RPS system (i.e. δ = 0 in our model) there are tongues of instability emerging from ω1 = 2/n 3 in the ω1 − ε plane. Our harmonic balance analysis


136

Ω2 2.0

Ω2 2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.5

1.0

1.5

Ω1 2.0

(a) = 0.5, δ = 0.6

0.0

0.5

1.0

1.5

Ω1 2.0

(b) = 0.9, δ = 0.6

Ω2 2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

Ω1 2.0

(c) = 1.3, δ = 0.6

Fig. 3 Transition curves predicted by harmonic balance with −5 ≤ m ≤ 5, 0 ≤ n ≤ 5 for various values of ε .

√ √ is consistent with this: we observe bands of instability around ω1 = 2/ 3 and ω2 = 2/ 3, which √ get broader as ε increases. We also see narrower regions of instability along the lines nω1 + mω2 = 2/ 3, for each n, m used in the truncated solution (28). Thus the boundary of the region of instability exhibits self-similarity when we consider ω1 , ω2 ∈ [0, 21−k ] for k = 0, 1, . . . .


137

5 Numerical integration In order to check the results of the harmonic balance method, we generate an approximate stability diagram by numerical integration of the linearized system (15). For randomly chosen parameters (ω1 , ω2 ) ∈ [0, 2], we choose random initial conditions (u(0), u(0)) ˙ on the unit circle - since the system is linear, the amplitude of the initial condition needs only to be consistent between trials. We then integrate the system for 1000 time steps using ode45 in Matlab. This is an explicit Runge-Kutta (4,5) method that is recommended in the Matlab documentation for most non-stiff problems. We considered a motion to be unstable if max |u(t)| > 10. The set of points (ω1 , ω2 ) corresponding to unstable motions were plotted using matplotlib.pyplot in Python. See Figure 4. Each plot in Figure 4 contains approximately 5 × 104 points. We note that the unstable regions given by numerical integration appear to be consistent with the transition curves predicted by harmonic balance (Figure 3). The regions of instability around √ √ ω1 = 2/ 3 and ω2 = 2/ 3 are visible for all tested values of ε and δ , and as ε increases, more tongues √ of the form nω1 + mω2 = 2/ 3 become visible. 6 Lyapunov exponents A second, and more informative, numerical approach for determining stability is the computation of approximate Lyapunov exponents. This is a measure of a solution’s rate of divergence from the equilibrium point [11], and is defined as 1 λ = lim sup ln |u(t)|. t→∞ t

(29)

If the limit is finite, then u(t) ∼ eλ t or smaller as t → ∞. A positive Lyapunov exponent indicates that the solution is unstable. We do not find any negative Lyapunov exponents, but note [8] that the system (15) can be converted to a Hill’s equation ˙ 2 − 18g(t)g(t) ¨ 4g(t)3 + 27g(t) )=0 (30) z¨ − z( 2 36g(t) by making the change of variables u = g(t)z. Since g(t) is bounded, u is bounded if and only if z is bounded. And since there is no dissipation in (30), stable solutions correspond to λ = 0. We approximate the Lyapunov exponents numerically by integrating as above, and taking

λ≈

1 ln |u(t)|. 900 1.

(23)

Using Taylor’s series, X(η ; γ ) can be expanded in terms of γ as follows. ¯ 1 ∂ ` X(η ; γ ) ¯¯ X(η ; γ ) = X0 (η ) + ∑ X` (η )γ , where X` (η ) = . `! ∂ γ ` ¯γ =0 `=1 ∞

`

(24)

The auxiliary parameters are selected as γ = 0 and γ = 1 from Eq. (15), one may write X(η ; 0) = X0 (η ), X(η ; 1) = X(η ).

(25)

Thus as γ increases from 0 to 1 and X(η ; γ ) varies from the initial guess X0 (η ) to the solution X(η ) of the governing equations respectively. The auxiliary parameters are selected so that the series solutions converge for γ = 1 and the particular solution is ∞

X(η ) = X0 (η ) + ∑ X` (η ),

(26)

`=1

where X = f ∗ and g∗ . Therefore, we get the general approximate analytical solutions ( f` , g` ) in terms of special solutions ( f`∗ , g∗` ) are given by f` (η ) = f`∗ (η ) +C1 +C2 eη +C3 e−η , g` (η ) =

g∗` (η ) +C4 eη

−η

+C5 e

.

(27) (28)

We solve the Eqs. (27) and (28) one after the other in the order ` = 1, 2, 3, . . . by means of the symbolic computation software Mathematica. It is shown that the solution for the velocity profile can expressed as an infinite series of any desired order. 3.2

Convergence of the homotopy solutions

In this section, we prove that the base functions (i.e, Eqs. (10) and (11)) converges to the solutions of the governing Eqs. (6) and (7). That is, we show that lim fN (η ) = 0, lim gN (η ) = 0.

N→∞

N→∞

(29)

146

R. Seshadri, S.R. Munjam /Journal of Applied Nonlinear Dynamics 4(2) (2015) 141–152

From Eqs. (19)–(23) we have N

lim [ ∑ cX ℜ`X (η )] = lim

N→∞

N→∞

`=1

N

∑ LX [X` (η ) − Ω` X`−1 (η )]

`=1

N

N

`=1

`=1

= lim LX [ ∑ X` (η ) − ∑ Ω` X`−1 (η )] N→∞

= lim LX fN (η ) N→∞

∵ zN =

N

N

`=1

`=1

∑ X` (η ) − ∑ Ω` X`−1 (η )

(30)

= LX lim zN (η ), N→∞

N

lim [ ∑ cX ℜ`X (η )] = 0; η ∈ [0, ∞),

N→∞

`=1

where X = f and g. Eq. (30) gives that infinite sequences S1 , S2 , S3 , S4 , S5 , S6 ......, where (SN = [∑N`=1 ℜ`X (η )]) converges to zero. Now, from Eq. (21) N

∑ ℜ`f (η ) =

`=1 N

lim [ ∑ ℜ`f (η )] =

N→∞

`=1

=

N

`−1

000 0 0 0 2 (η ) + ∑ [ f`−1− j f j00 − f`−1− ∑ { f`−1 j f j ] + M[1 −V0 − f `−1 (η )] + (1 − 2V0 +V0 )},

`=1

j=0

∞

`−1

000 0 0 0 2 (η ) + ∑ [ f`−1− j f j00 − f`−1− ∑ { f`−1 j f j ] + M[1 −V0 − f `−1 (η )] + (1 − 2V0 +V0 )} j=0

`=1

d3 dη

∞

∞ `−1

`=1

`=1 j=0 ∞

( f (η )) + ∑ 3 ∑ `−1

+M[1 −V0 ] − M =

d ( ∑ f`−1 (η )) + (1 − 2V0 +V02 ) d η `=1

∞ ∞ d3 ∞ 0 0 ( f ( η )) + [ fm−1− j f j00 − fm−1− j ∑ ∑ ∑ j f j] d η 3 j=0 j=0 m= j+1

+M[1 −V0 ] − M( =

0 0 ∑ [ f`−1− j f j00 − f`−1− j f j]

d ∞ ( ∑ f j (η ))t) + (1 − 2V0 +V02 ) d η j=0

∞ d3 ∞ d2 ∞ ( f ( ( ∑ f j (η ))) η )) + ( f ( η ))( j j ∑ ∑ d η 3 j=0 d η 2 j=0 j=0

−(

(31)

d ∞ d ∞ ( ∑ f j (η )))( ( ∑ f j (η ))) d η j=0 d η j=0

+M[1 −V0 ] − M(

d ∞ ( ∑ f j (η ))) + (1 − 2V0 +V02 ). d η j=0

Eqs. (29) and (31) we have ∞ d3 ∞ d2 ∞ d ∞ d ∞ ( f ( η )) + ( f ( η ))( ( f ( η ))) − ( ( f ( η )))( ( ∑ f j (η ))) ∑ j ∑ j ∑ j ∑ j d η 3 j=0 d η 2 j=0 d η j=0 d η j=0 j=0

+M[1 −V0 ] − M(

d ∞ ( ∑ f j (η ))) + (1 − 2V0 +V02 ) = 0, d η j=0

(32)

and from Eq. (22) N

N

`−1

`=1

`=1

j=0

∑ ℜ`g (η ) = ∑ {g00`−1 (η ) + Pr ∑ [ f`−1− j (η )g0`−1 (η )]},

(33)

R. Seshadri, S.R. Munjam /Journal of Applied Nonlinear Dynamics 4(2) (2015) 141–152 N

lim [ ∑ ℜ`g (η )] =

N→∞

`=1

= = = =

∞

147

`−1

∑ {g00`−1 (η ) + Pr ∑ [ f`−1− j (η )g0`−1 (η )]} j=0

`=1

d2 dη

∞ `−1

`=1 ∞

`=1 j=0 ∞ ∞

( g (η )) + Pr ∑ 2 ∑ `−1

d2 dη 2 d2 dη 2 d2 dη

∞

∑ [ f`−1− j (η )g0j (η )]

( ∑ g`−1 (η )) + Pr ∑ `=1 ∞

∞

[ fm−1− j (η )g0j (η )]

j=0 m= j+1 ∞

( ∑ g j (η )) + Pr ∑ j=0 ∞

∑

∑

j=0 m= j+1 ∞

[ fm−1− j (η )g0j (η )]

( g (η )) + Pr( ∑ f j (η ))( 2 ∑ j j=0

j=0

d ∞ [ ∑ g j (η )]). d η j=0

(34)

Eqs. (29) and (34) we have ∞ d2 ∞ d ∞ ( g ( η )) + Pr( f ( η ))( [ ∑ g j (η )]) = 0. j j ∑ ∑ d η 2 j=0 d η j=0 j=0

(35)

From Eq. (20) we have ∞

∑

j=0

∞

f j (0) = 0,

∑

f j0 (0) = V0 ,

∞

∑ f j0 (∞) = 1 −V0 ,

(36)

j=0

j=0 ∞

∑ g j (0) = 0,

j=0

∞

∑ g j (∞) = 1.

(37)

j=0

Therefore, Eqs. (32) and (35) with Eqs. (36) and (37) shows that the Eqs. (10) and (11) converges. It is customary in HAM analysis to draw c0 -curves to identify the interval of optimal convergence control parameters(i.e, c f and cg ) inside which any suitable values can be chosen. From Fig. 1. it is shown that choosing c f in the interval [−1.4, −0.1] does not affect the values of shear stress rates and it is [−1.2, −0.2] for heat transfer rate. The c f and cg curves have been drawn for 35th order of HAM solution. 4 Optimal values of convergence parameters In order to choose the optimal value of convergence control parameter c0 in the interval [−1.4, −0.1], We compute the averaged residual errors suggested by [17]. EX '

1 `

i

`

j=0

i=0

∑ (N[ ∑ Xi ( j∆x)])2 ,

(38)

where X = f and g, ∆x = 10 ` and ` = 20. The HAM-based Mathematica package BVPh 2.0* has been utilized to compute the averaged residual errors of Eqs. (6) and (7). Hence, we use Eq. (38) to find the convergence control parameters for entire calculations. The averaged squared residual errors at different orders of approximation for f (η ) and g(η ) are shown in Fig. 2. The averaged squared residual error decreases rapidly with convergence control parameters c f = −0.96021, and cg = −1.07759. Thus the presence of convergence control parameter in HAM can greatly accelerate the convergence of series solution. *Refer. http://numericaltank.sjtu.edu.cn/BVPh.htm

148

R. Seshadri, S.R. Munjam /Journal of Applied Nonlinear Dynamics 4(2) (2015) 141–152 V0=0.2, Pr=0.7, M=2.0

0.1

g'H0L

4

Residual Error

f ''H0L, g'H0L

1

f ''H0L

6

2 0

0.01 0.001 10-4

-2 -1.5

-1.0

-0.5

10-5

0.0

5

c f , cg

10

15

20

25

30

Order of approximation

Fig. 2 Averaged squared residual error at different order of approximations.

Fig. 1 c f and cg -curves. M=2.0

Pr=0.7, M=2.0

1.2

1.0 æ

1.0

æ æ æ

æ

0.8

æ

æ

gHΗL

æ

æ

æ

æ æ æ

1.0

1.5

2.0

V0=0.0 V0=0.25 V0=0.5 V0=0.75 V0=1.0

0.4 0.2

æ

æ

0.5

æ

æ

0.0

0.6

æ

æ

æ

0.0

æ

0.4

æ

V0=0.0 V0=0.25 V0=0.5 V0=0.75 V0=1.0

æ

æ

0.6

0.2

æ

f 'HΗL

Kumari et al.@4D

æ

0.8

0.0

2.5

0

2

Η

4

6

8

10

Η

Fig. 3 Effect of ratio of stretching velocity on velocity profile.

Fig. 4 Effect of ratio of stretching velocity on temperature profile.

Table 1 Convergence of Homotopy solution for different orders of approximation for − f 00 (0) and −g0 (0) when ( V0 = 0.2 and Pr = 0.72.) `

− f 00 (0)

−g0 (0)

1

1.109728

0.577622

5

1.142037

0.562039

10

1.189254

0.552891

15

1.189254

0.552891

20

1.189254

0.552891

25

1.189254

0.552891

30

1.189254

0.552891

35

1.189254

0.552891

5 Results and discussion Here, we have studied the effect of various physical parameters such as magnetic parameter M, ratio of the stretching velocity to composite V0 on velocity and temperature profiles as well as on skin friction and heat transfer rates. The approximate analytical solutions are obtained in the form of general series solving with computational software such as Mathematica. The expressions for f and g are evaluated up to tenth order to keep the results up to sixth decimal places accuracy. Computations have been carried out for velocity and temperature profiles and for skin friction coefficients and heat transfer rates for several combinations of parameters V0 and M, Pr and only few important results are presented in the form of tables and figures. Figures 3 and 4 show that the effect of ratio of stretching velocity to free stream velocity V0 on the


149

Pr=0.7, V0=1.0 1.0 1.0

0.0 M=0.0 1.0 M=1.0 2.0 M=2.0 4.0 M=4.0

0.0 V0 = 0.0

0.6 0.6

0.8

gHΗL

f 'HΗL H L

0.8

0.5 V0 = 0.5

0.4

M=0.0 M=1.0 M=2.0 M=4.0

0.4 0.2

1.0 V0 = 1.0

0.2

0.6

0.0

0.0 0

1

2

3

4

0

5

2

4

6

Η Η

Fig. 5 Effect of Magnetic parameter on velocity profile. æ æ æ æ æ æ æ

M=2.0

0.55

g'H0L

æ

æ

æ æ æ

Pr=0.7

æ æ

æ

-2

M=0.0 M=1.0 M=2.0 M=4.0

Kumari et al.@4D

0.50

æ

-1

æ æ

æ æ æ

f ''H0L

0.60

M=4.0

æ æ æ

0

0.45

æ

-3

M=0.0 M=1.0

æ

1

10

Fig. 6 Effect of Magnetic parameter on temperature profile.

3 2

8

Η

0.0

0.2

0.4

0.6

0.8

0.40 0.0

1.0

0.2

V0

0.4

0.6

0.8

1.0

V0

Fig. 7 Effect of Magnetic parameter on Skin friction coefficient.

Fig. 8 Effect of Magnetic parameter on Heat transfer rate.

Table 2 Variation of Heat transfer rate g0 (0) for various Prandtl number and magnetic parameter

V0

Pr

g0 (0) M = 0.0

M = 2.0

M = 4.0

0.80552 9

0.81252 9

0.81952 9

0.0

1.4 7.0

1.10489 5

1.13989 5

1.17489 5

0.5

1.4

0.83225 3

0.83225 3

0.83225 3

7.0

1.25382 8

1.25382 8

1.25382 8

1.4

0.86220 1

0.85520 1

0.84820 1

7.0

1.42500 4

1.39000 6

1.35499 9

1.0

velocity profile f 0 (η ) and temperature profile g(η ) for fixed magnetic parameter M = 2.0. The velocity profiles change significantly as V0 increase from 0 to 1. The velocity profile for V0 = 0.5 does not have any effect along the wall normal direction whereas the effects of the profiles are reversed from V0 < 0.5 to V0 > 0.5 and all the profiles cross each other at η = 0.5. The flow velocity depends strongly on the given boundary conditions i.e, Equation (9) on velocity only. It is seen that there is a slight overshoot in temperature profile for V0 = 0.0, but the increasing values of V0 temperature profiles smoothly march to its free stream values. Figure 5 shows that the effect of magnetic parameter M on the velocity profiles f 0 (η ). Here again, due to the strong dependence of boundary conditions on V0 there is an increasing effect for V0 > 0.0 and decreasing effect for V0 < 0.0. To show these effects the profiles are plotted for three different values of V0 . It observed that when V0 < 0.5, the velocity profiles increases for increasing the magnetic parameter. It means that the flow has a boundary layer structure, and thickness of the boundary layer decreases

150


with effect of magnetic parameter M. When V0 = 0.5, the velocity profiles are constant for various values of M. By closed form solution from [4] given that, f (η ) = η2 and f 0 (η ) = 12 for any values of η , which means for this case the flow field is not effected by magnetic field. When V0 = 1.0, the velocity profiles for various values of M decrease, in this case also, the thickness of the boundary layer decrease with effect of magnetic parameter. The temperature variation for different magnetic parameters are presented in Fig. 6 for V0 = 1.0, Pr = 0.7. It is observed that temperature profile decreases as increasing magnetic parameter. The effect of magnetic parameter M on the skin friction coefficient f 00 (0) versus V0 is shown in Fig. 7. It is observed that, for any fixed M, the f 00 (0) decreases with increasing V0 and is positive for V0 < 0.5, zero for V0 = 0.5 and negative for V0 > 0.5. The zero skin friction of this case is not surprising since the surface and the fluid moves with the same velocity. Also f 00 (0) increases with M at the region V0 ∈ [0.0, 0.5) and decreases with increasing M at the region V0 ∈ (0.5, 1.0]. For this reason the magnetic field induces a force along the surface direction which supports the motion at the region V0 ∈ [0.0, 0.5). So that dimensionless velocity increased and hence the skin friction increases with the magnetics field. The magnetic field induces a force along the surface which opposes the motion at the region V0 ∈ (0.5, 1.0]. Clearly the dimensionless velocity is reduced and hence surface skin friction is reduced with increasing magnetic parameter. The heat transfer rate g0 (0) versus V0 for different values of magnetic parameter are presented in Fig. 8 for Pr = 0.7. It observed that for a fixed Pr, the surface heat transfer g0 (0) increases with magnetic parameter M for region V0 ∈ [0.0, 0.5) and decreases for region V0 ∈ (0.5, 1.0]. These profiles cross each other at V0 = 0.5. Table 1 illustrates the convergence of skin friction − f 00 (0) and heat transfer −g0 (0) after performing up to 35th -order approximation of functions f and g computed from deformation equations. As seen, the computation is terminated as soon as three consecutive values agree in their sixth decimal places. Table2 illustrates the variation of heat transfer rate g0 (0) for different Prandtl numbers such as Pr = 1.4, 7.0 for different values of V0 and M. The surface heat transfer rate increases with increase in Pr and V0 . 6 Conclusions The flow and heat transfer analysis is carried out on stagnation-point flow for an electrically conducting viscous fluid due to a continuous stretching surface. A series solution is derived as a function of the wall normal height η , for velocity f and temperature g using Homotopy Analysis Method. The coefficients contain all the other parameters such as V0 , M and Pr. Computations have been carried out to retain at least up to 10th degree polynomial on η to get the results accurate up to six decimal places. The Computer Algebra software Mathematica is used to perform these semi-analytical calculations. The effect of various parameters such as magnetic parameter M, ratio of stretching velocity V0 are analyzed on flow velocities, skin friction and heat transfer rates. The influence of Prandtl number Pr on the temperature profiles and heat transfer rates are also analyzed. It may be concluded that the effect of V0 on flow velocity is very significant in the sense that zero skin friction occurs at V0 = 0.5. The zero skin friction of this case is not surprising since the surface and the fluid moves with the same velocity. And the trend in the variation of velocity profiles as well as skin friction coefficients are just opposite for V0 < 0.5 and V0 > 0.5. The skin friction and heat transfer increases with increase in magnetic parameter for V0 < 0.5, but decreases for V0 > 0.5. The temperature profiles decreases for the increase in the values of V0 and M where as it increases as Prandtl number increases. It is concluded that the effect of the study parameters on flow variables have been computed with simplest calculation rather than a very heavy computation involved in any numerical methods. Hence analytical solutions are always advantageous over numerical methods.


151

Acknowledgements The author Shankar Rao Munjam gratefully acknowledge UGC- Rajiv Gandhi National Fellowship (RGN-SRF), Government of India for providing financial assistance. References [1] Sakiadis, BC. (1961), Boundary-layer behavior on continuous solid surfaces: II. The boundary layer on a continuous flat surface, AIChE Journal, 7(2), 221–225. [2] Crane, L.J. (1970), Flow past a stretching plate, Zeitschrift f angewandte Mathematik und Physik ZAMP, 21(4), 645–647. [3] Hiemenz, H. (1911), Die grenzschicht an einem in dengleich formizen flussigkeitsstrom eigetanch geraden kreiszlinder, Dingl Polytech. J, 326, 321–328. [4] Kumari, M. and Nath, G. (1999), Flow and heat transfer in a stagnation-point flow over a stretching sheet with a magnetic field, Mechanics Research Communications, 26(4), 469-478. [5] Kumari, M. and Nath, G. (1999), Development of flow and heat transfer of a viscous fluid in the stagnationpoint region of a three-dimensional body with a magnetic field, Acta Mechanica, 135(1-2), 1–12. [6] Chiam, T.C. (1994), Stagnation Point flow towards a Stretching Plate, Journal of the Physical Society of Japan. [7] Mahapatra, T.R., and Gupta, A.S. (2001), Magnetohydrodynamic stagnation-point flow towards a stretching sheet, Acta Mechanica, 152(1-4), 191–196. [8] Mahapatra, T.R., Nandy, S.K., and Gupta, A. S. (2009), Analytical solution of magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface, Applied Mathematics and Computation, 215(5), 1696–1710. [9] Mahapatra, T.R., Nandy, S.K., and Gupta, A.S. (2010), Dual solution of MHD stagnation-point flow towards a stretching surface, Engineering, (2), 299–305. [10] Ali, F.M., Nazar, R., Arifin, N.M., and Pop, I. (2011), MHD stagnation-point flow and heat transfer towards stretching sheet with induced magnetic field, Applied Mathematics and Mechanics, 32(4), 409–418. [11] Ali, F.M., Nazar, R., Arifin, N.M., and Pop, I. (2014), Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field, Applied Mathematics and Mechanics, 35(2), 155–166. [12] Ishak, A., Nazar, R., and Pop, I. (2007), Mixed convection on the stagnation point flow toward a vertical, continuously stretching sheet, Journal of Heat Transfer, 129(8), 1087–1090. [13] Ishak, A., Nazar, R., Arifin, N.M., Ali, F.M., and Pop, I. (2011), MHD stagnation-point flow towards a stretching sheet with prescribed surface heat flux, Sains Malaysiana, 40(10), 1193–1199. [14] Sharma, P.R., and Singh, G. (2009), Effects of variable thermal conductivity and heat source/sink on MHD flow near a stagnation point on a linearly stretching sheet, Journal of Applied fluid mechanics, 2(1), 13–21. [15] Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC press, Boca Raton: Chapman and Hall. [16] Liao, S.J. (2010), An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15(8), 2003–2016. [17] Liao, S.J. (2012), Homotopy Analysis Method in Nonlinear Differential Equations, Higher Education Press & Springer, Beijing & Heidelberg/New York/. [18] Oahimire, J.I., and Olajuwon, B.I. (2013), Hydromagnetic Flow Near a Stagnation Point on a Stretching Sheet with Variable Thermal Conductivity and Heat Source/Sink, International Journal of Applied Science and Engineering, 11(3), 331–341. [19] Rasekh, A., Farzaneh-Gord, M., Varedi, S.R., and Ganji, D.D. (2013), Analytical solution for magnetohydrodynamic stagnation point flow and heat transfer over a permeable stretching sheet with chemical reaction, Journal of Theoretical and Applied Mechanics, 51(3), 675–686. [20] Kazem, S., Sanaeikia, A., Ahmadvand, M., and Saberi, H. (2011), An RBF solution to a stagnation point flow towards a stretching surface with heat generation, In Computational Science and Engineering (CSE), IEEE 14th International Conference, 239–244. [21] Cheng, J. and Dai, S. (2010), A uniformly valid series solution to the unsteady stagnation-point flow towards an impulsively stretching surface, Science China Physics, Mechanics and Astronomy, 53(3), 521–526. [22] Sinha, A., and Misra, J.C. (2014), Effect of Induced Magnetic Field on Magnetohydrodynamic Stagnation Point Flow and Heat Transfer on a Stretching Sheet, Journal of Heat Transfer, 136(11), 112701. [23] Zhu, J., Zheng, L.C., and Zhang, X.X. (2009), Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis, Applied Mathematics and Mechanics, 30(4), 463–474.

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[24] Qi, D., and Hong-Qing, Z. (2009), Analytic solution for magnetohydrodynamic stagnation point flow towards a stretching sheet, Chinese Physics Letters, 26(10), 104701. [25] Akbar., N.S, Nadeem., S., Rizwan, UL Haq., and Shiwei Ye. (2014), MHD stagnation point flow of Carreau fluid toward a permeable shrinking sheet: Dual solutions, Ain Shams Engineering Journal, 5, 1233–1239. [26] Yadav, R.S. and Sharma, P.R. (2014), Effects of Heat Source/Sink on Stagnation Point Flow over A Stretching Sheet, International Journal of Engineering Research and Technology, 3(5), 1–8. [27] Malvandi, A., Hedayati, F., and Ganji, D.D. (2014), Slip effects on unsteady stagnation point flow of a nanofluid over a stretching sheet, Powder Technology, 253, 377–384. [28] Ibrahim, W., and Makinde, O.D. (2015), Double-Diffusive in Mixed Convection and MHD Stagnation Point Flow of Nanofluid Over a Stretching Sheet, Journal of Nanofluids, 4(1), 28–37. [29] Hayat, T., Asad, S., Mustafa, M., and Alsaedi, A. (2015), MHD stagnation-point flow of Jeffrey fluid over a convectively heated stretching sheet, Computers & Fluids, 108, 179–185. [30] Hayat, T., Ali, S., Awais, M., and Alhuthali, M.S. (2015), Newtonian heating in stagnation point flow of Burgers fluid, Applied Mathematics and Mechanics, 36(1), 61–68.



Longitudinal Dimensions of Polygon-shaped Planetary Waves Ranis N. Ibragimov1†and Guang Lin2 1 GE

Global Research 1 Research Circle, Niskayuna, NY 12309, USA Mathematics, and School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA 2 Department of

Submission Info Communicated by Lev Osctrovsky Received 9 July 2014 Accepted 1 March 2015 Available online 1 July 2015 Keywords Shallow water approximation Planetary hexagon-shaped waves Free boundary problem Atmospheric modeling

Abstract Polygon-shaped longitudinal large-scale waves are described by means of higher-order shallow water approximation corresponding to the Cauchy–Poisson free boundary problem on the stationary motion of a perfect incompressible fluid circulating around a circle. It is shown that there are four basic physical parameters, which exert an influence on a wave number (or wave length), which is one of the basic values used to characterize the planetary flow pattern in mid-troposphere. Some analogy with the jet-stream following hexagon-shaped path at Saturn’s north pole is observed.

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

1 Introduction Planetary waves are large-scale perturbations of the atmospheric dynamical structure that extend coherently around a full longitude circle. They are important because they have significant influence on the wind speeds, temperature, distribution of ozone, and other characteristics of the middle atmosphere structure. They also play an important role in global climate control and weather prediction [1–3]. In oceanographic applications, understanding of the atmospheric processes mechanisms have greatly increased due to microstructures measurements over the past two decades. In terms of mathematical modeling, the large-scale atmospheric dynamics is usually described by moving air masses on a sphere or circle by means of three and two dimensional Navier–Stokes or Euler equations a thin rotating spherical shell (see e.g. [4–12] ) or within the theory of shallow water approximation [13–20]. Particularly, a largescale two-dimensional modeling with the inclusion of a spherical shape can be associated e.g. with the eastward moving, wave of warm water, known as a Kelvin wave that can be seen traveling eastward along the equator as shown in the left Panel of Figure 1 (see also [21–23]). Another spectacular example of circulating waves is demonstrated on the right panel of Figure 1 showing a jet stream that follows a hexagon-shaped path at the north pole of Saturn. The hexagon was hidden in darkness † Corresponding



154

Ranis N. Ibragimov, Guang Lin /Journal of Applied Nonlinear Dynamics 4(2) (2015) 153–167

Fig. 1 Left: Sea-level height data from November 2009 showing the dynamics of warm water known as Kelvin waves that can be seen traveling eastward along the equator (black line) in Nov. 01, 2009 image. El Ninos form when trade winds in the equatorial western Pacific relax over a period of months, sending Kelvin waves eastward across the Pacific like a conveyor belt. Image credit: NASA/JPL. Right: Image from Cassini, made possible only as Saturn’s north pole emerged from winter darkness, shows new details of a jet stream that follows a hexagon-shaped path and has long puzzled scientists (source: http://saturn.jpl.nasa.gov/video/videodetails/?videoID=200)

during the winter of Saturn’s long year, a year that is equal to about 29 Earth years. But as the planet approached its August 2009 equinox and signaled the start of northern spring, the hexagon was revealed to Cassini’s cameras. This is the first time the whole hexagonal shape has been mapped out in visible light by Cassini, and these images show unprecedented details of Saturn’s high northern latitudes. The hexagon was originally discovered in images taken by Voyager spacecraft in the early 1980s. Since 2006, the Cassini Visual and Infrared Mapping Spectrometer (VIMS) instrument has been observing the hexagon at infrared wavelengths, but at lower spatial resolution than these visible light images. This image also shows another unexplained phenomena such as waves that can be seen traveling along hexagon. Scientists think the hexagon is a meandering jet stream at 77 degrees north latitude, but they don’t know what controls the path the stream takes. Multiple images acquired by the VIMS instrument over a 12-day period showed that the feature is nearly stationary and is likely an unusually strong pole-encircling planetary wave that extends deep into the atmosphere. Scientists had speculated that a large vortex seen outside the hexagon during the Voyager observations exerted forces on the jet stream making it adopt a hexagonal pattern in a manner similar to how jet streams on Earth divert around high-pressure systems. However, in these new images, the vortex is notably absent while the hexagon persists almost 30 years after it was first seen. The images were taken in visible light with the Cassini spacecraft wide-angle camera on Jan. 3, 2009. The images were obtained at a distance of approximately 764,000 kilometers (475,000 miles) from Saturn. The smallest resolved features at the latitude of the hexagon have a horizontal scale of approximately 100 kilometers. Recent laboratory experiments in [24] suggest that the observed Saturn’s North Polar Hexagon might result from the stabilization of a standing waves caused by the difference in angular velocity. However, because of the complex atmospheric structure in Saturn, the provided experiments do not provide the clear answers and these waves and the six-sided shape of the


155

jet stream remain a mystery up to the date. The information about this phonemenon can be fouind at http://saturn.jpl.nasa.gov/video/videodetails/?videoID=200. The primary focus of this paper is to a show that longitudinal large-scale waves within a central gravity field might also provide a similar polygon-shaped structure when looking from above the North Pole, as shown in Figure 1 by means of two-dimensional free boundary large-scale shallow water approximation describing a simple atmospheric motion around an equatorial plane (see also [25]). As has been discussed in [26] and [27], the mathematical model can be derived from the assumption that the atmosphere is approximated by a perfect fluid and its motion is irrotational and pressure on a free boundary is constant. It is also postulated that the fluid depth is small compared to the radius of the circle and the gravity vector is directed to the center of the circle. In the first approximation, shallow water equations represent the mathematical theory that can be used to investigate the fluid flows in channels (see e.g. [28–30]). However, as has been discussed in [27], this theory does not reveal the role the role of an undisturbed level of the fluid surface which is needed to determine the precision of the first approximation. A higher order approximation is derived in this work.

2 The model We introduce polar coordinates x = r cos θ , y = r sin θ and use the following notation: R is the radius of the Earth, θ is a polar angle, r is the distance from the origin, h = h0 + η (t, θ ) , where h0 is undisturbed level of atmosphere above the Earth and η (t, θ ) is the level of disturbance of a free boundary, as shown schematically in Figure 2. It is supposed in what follows that θ ∈ [0, 2π ] while r ∈ [R, h (t, θ )] . The → homogeneous gravity field − g is assumed to be a constant and directed to the center of the Earth. The restriction θ ∈ [0, 2π ] appears for the following reason: the velocity potential ϕ (ς ) can be introduced by the analyticity of the complex potential ϑ (ς ) = ϕ + iψ , where ς = reiθ is the independent complex variable and ψ (ς ) is the stream function. Correspondingly, the complex velocity d ϑ /d ς is a singlevalued analytic function of ς , although ϑ is not single-valued. In fact, when we turn around the bottom 2´π r = R once, ϕ increases by − ∂∂ψr (R, θ ) d θ which has a positive sign by the maximum principle (Hopf’s 0

lemma). Hence, if we remove the width of annulus region θ = 0, r ∈ [R, R + h0 ] , then at every point (r, θ ) , the complex potential ϑ (ς ) is a single-valued analytic function. → We start with the usual assumption that the velocity field − v = (vr , vθ ) satisfies the Euler’s equations and the no-leak condition vr = 0 on a solid bottom r = R. We also assume the kinematic condition on the free boundary. Namely, the velocity on the free boundary r = R + h (t, θ ) is tangential to the free boundary. We define the free boundary by equation f = r − h (θ ,t) = 0 so that the kinematic condition is written as ∂f − df = +→ v ∇ f = 0, (1) dt ∂t where

∇=

∂ 1 ∂ , ∂r r ∂θ

.

(2)

In what follows, it is assumed that the fluid motion is potential in the domain of the motion which allows to introduce the stream function ψ (t, r, θ ) via vr = −

1 ∂ψ , r ∂θ

vθ =

∂ψ . ∂r

(3)

So that the no-leak condition on the solid boundary can be written as ψ (R, θ ,t) = 0 whereas the kinematic condition (1) takes the form

156


h (θ,t) R g

θ

r

h0

0

Fig. 2 Schematic showing a longitudinal atmospheric motion circulating around the Earth.

∂h 1 ∂ψ 1 ∂ψ ∂h + + = 0. ∂t r ∂θ r ∂r ∂θ Since 1 vr = − r

R+h ˆ

R

∂ vθ dr, ∂θ

(4)

(5)

we can also write Eq. (4) at the free boundary r = R + h as the mass balance equation. i.e., 1 ∂ ∂h + ∂t R + h ∂θ

R+h ˆ

vθ dr = 0.

(6)

R

Following [27], we next define the average velocity u (θ ,t) as 1 u (θ ,t) = h

R+h ˆ

R

1 vθ (r, θ ,t) dr = ψ (R + h, θ ,t) . h

(7)

In terms of the average velocity, the kinematic condition (4) is written as

∂h 1 + ∂t r

R+h ˆ

R

1 ∂ψ ∂h ∂ vθ dr + = 0. ∂θ r ∂r ∂θ

Finally, the dynamic condition is obtained from the requirement that the pressure p is constant at the free boundary r = R + h (θ ,t). Thus, projecting of the impulse equation → 1 →2 1 ∂− v → + ∇( |− v | ) + ∇p = − g, ∂t 2 ρ

(8)


on the tangential vector

157

1 ∂h − → τ =( , 1) r ∂θ

to the free boundary yields 1 ∂ 1 2 1 1 ∂h ∂ p 1 ∂ p 1 ∂ h ∂ vr ∂ vθ 1 ∂ h ∂ 1 2 + + vr + v2θ + vr + v2θ + ( + ) = 0, r ∂θ ∂t ∂t r ∂θ ∂r 2 r ∂θ 2 ρ r ∂θ ∂r r ∂θ

(9)

where ρ is a constant fluid density. Thus, since p|r=R+h = const. and ψ is the harmonic function at the domain of the fluid motion, the model describing a longitudinal atmospheric motion around the Earth can be written as the following free boundary problem: 2 ∂ 2ψ ∂ψ 2∂ ψ = 0 (R < r < R + h), (10) + r +r 2 2 ∂θ ∂r ∂r

ψ (R, θ ,t) = 0,

(11)

ψ (R + h, θ ,t) = u (θ ,t) h,

(12)

1 ∂ h ∂ 2ψ 1 ∂ 1 ∂ψ 2 ∂ 2ψ ∂ψ 2 g ∂h − 2 ) ]+ + [ 2( ) +( = 0, (r = R + h) , (13) ∂ t ∂ r r ∂ θ ∂ t ∂ θ 2r ∂ θ r ∂ θ ∂r r ∂θ ∂h ∂ + (uh) = 0, (r = R + h). (14) r ∂t ∂θ One can check by direct differentiation that there exists an exact stationary solution to the model (10)–(14) given by Γ r (15) h0 = 0, ψ0 = − log( ), 2π R where Γ = const.is intensity of the vortex (source) localized at the center of the earth and is related with the the rotation rate of the earth (angular velocity Ω = 2π rad/day ≈ 0.73 × 10−4 s−1 ) by the equation Γ = 2π ΩR2 . The solution (15) corresponds to the singular constant flow with an undisturbed free surface with the vortex localized at the origin. However, the vortex is isolated since R represents a solid boundary. Thus the exact solution (15) can be visualized as a flow whose streamlines are concentric circles with the common center at the origin. Understanding of singular flows were conduced in [13,31,32] and [14]. As has been remarked in [33], the computational experiments in the latter papers provide a credible evidence to support the conclusion that singular solutions may exist on a stationary sphere in terms of shallow water approximation. We remark that, in terms of physical interpretation, the fluid particles at the North and South Poles spin around themselves at a rate Ω = 2π rad/ day, whereas fluid particles in the polar domain θ ∈ [θ0 , π − θ0 ] do not spin around themselves but simply translate provided π θ0 ∈ 0, 2 . Thus the physically possible atmospheric motion rotating around the poles correspond to the flows that are being translated along the equatorial plane (Ibragimov, [9, 10]). At certain extent, the above ansatz (15) can also be associated with such atmospheric phenomena as illustrated in Figure 3 which is used to show the NASA images of a polar vortex on Venus (Left panel) and clouds circling over Saturn’s north pole (Right panel). Particularly, it is believed that the polar vortex as shown in the left panel of Figure 3 is a very powerful whirlpool swirling steadily around the planet’s poles at all times. It might be caused by a gigantic hurricane with two calm, dark eyes. This double-eyed feature, dubbed the “dipole of Venus,” was thought to form when warm air from the planet’s equator rose and traveled toward the pole, where it cooled and sank to form a deep, swirling atmospheric pit. For decades, astronomers expected to find

158


Fig. 3 Left: Recent pictures of a polar vortex on Venus which is attributed to cloud formations on the palnet leaving an unexplained dark hole. It has been iscovered at Venus’ north pole by the Pioneer Venus spacecraft in 1979. Credit: ESA/Virtis/INAF-IASF/Obs. de Paris-LESIA Right: Picture taken by the Cassini spacecraft of clouds circling over Saturn’s north pole (source: http://www.wired.com/2010/09/venus-polar-vortex/)

a similar vortex at Venus’ south pole. While Venus itself rotates slowly, just once every 117 Earth days, its atmosphere whips around the planet once every four Earth days. This “super-rotating” atmosphere ought to form massive storms at both poles, astronomers reasoned [34, 35]. The image of the clouds circling over Saturn’s north pole shown in the right panel of Figure 3 are taken by the Cassini spacecraft from a distance of about 380,000 kilometers and represents the stunning detail in Saturn’s atmosphere. Clouds rise and sink and get stretched out, forming long valleys and ridges, streamers circling the planet’s pole. This vortex is over 2000 kilometers; that’s far bigger than a fully mature hurricane on Earth, but unlike a terrestrial cyclone, this may be a permanent feature in Saturn’s atmosphere (the source: http://www.wired.com/2010/09/venus-polar-vortex/; see also [36]).

3 Shallow water approximation It is useful to recast the model in nondimensional form by introducing the following dimensionless variables: R t θ = θ, r = R + h0 r, h = h0 , h, t = √ gh0 , u = gh0 u. ψ = h0 gh0 ψ We next introduce the parameter

(16)

h0 . (17) R Of course, water is shallow if the parameter ε is small. So, in the present model (10)–(14), the functions η (θ ,t) and ψ (r, θ ,t) are two unknown functions whereas the parameter ε is a given parameter. Although shallow water theory is usually related to the case when the water depth is small relative to the wavelengths of the waves, we find it more appropriate to choose the radius of the earth R as a

ε=


159

natural physical scale since, in the frame of the present model, we consider waves with wavelengths of the order of the radius of the Earth. The the dynamic condition (13) is then nondimensionalized as follows: 1 1 ∂ 2ψ ε2 ∂ 2ψ ∂ h ∂ ε2 ∂ψ 2 ∂ψ 2 ∂h − ) )+ + ( ( ) +( = 0. 2 2 ∂ t ∂ r (1 + ε h) ∂ t ∂ θ ∂ θ 2 (1 + ε h) ∂ θ (1 + ε h) ∂ θ ∂r (1 + ε h) ∂ θ

(18)

Following the Lagrange’s method we represent the stream function ψ by the following series expansion: ψ = ∑ε n ψ (n) . Then the Laplace equation (10) takes the form n

∂ 2 ψ (0) ∂ 2 ψ (1) ∂ 2 ψ (0) ∂ ψ (0) ) + ε ( + 2r + ∂ r2 ∂ r2 ∂ r2 ∂r 2 (0) ∂ 2 ψ (0) ∂ 2 ψ (2) ∂ 2 ψ (1) ∂ ψ (1) ∂ ψ (0) 2∂ ψ + r ) + 0(ε 3 ) = 0. + + 2r + r + +ε 2 ( ∂θ2 ∂ r2 ∂ r2 ∂ r2 ∂r ∂r

(19)

A comparison of the terms with the same order ε in equation (19) yields a recurrent system of differential equations for the determination of all functions ψ (n) , i.e. the Lagrange method consists in presentation of ψ as the solution of the Cauchy problem with boundary conditions (11)–(12) for ψ (0) and zero boundary conditions for ψ (1) and ψ (2) . Thus, up to the order ε 2 , the function ψ is determined as follows:

ψ = ur + ε (u

r r2 r2 r3 r r − uh ) + ε 2 (uh − uθ θ + uθ θ h2 − uh2 ). 2 2 4 6 6 4

(20)

Note that the unknowns u and h are related by the dynamic and conditions (18) and the kinematic condition (14) which is written in nondimensional form as follows: ∂ 1∂ 2 ε h + 2h + (uh) = 0. 2 ∂t ∂θ

(21)

(1 + ε h)−1 = 1 − ε h + ε 2h2 + 0 ε 3 ,

(22)

Using the Taylor series expansion and keeping the terms 0 ε 2 , we write the dynamic condition (18) as

∂ 2ψ 1 ∂ 2 ∂ ψ 2 ∂ψ 2 ∂ 2ψ ∂ h + ) ) − ε2 (ε ( ) +( ∂ t∂ r 2 ∂ θ ∂θ ∂r ∂ t∂ θ ∂ θ εh ∂ ∂ ψ 2 ∂h ∂h ) + ε h )(ε h − 1) + ( = 0. (23) 2 ∂θ ∂r ∂θ ∂θ Substituting ψ given by (20) into equation (23), we arrive at the following system of nonlinear shallow water equations (higher-order analogue of the Su - Gardner equations [37]): +(

5h u u2 ut − ht + 2huuθ − hθ + hhθ − 2h [ut + uuθ + hθ ]) 2 2 2 2 2 h h 1 2 +ε 2 ( ut − uθ θ t + uθ θ hht − ut θ hhθ + h2 uθ uθ θ 4 3 3 3 3 2 1 h2 + h uuθ + hhθ uuθ θ − uuθ θ θ + h2 hθ ) + o ε 3 = 0, 4 3 3 ∂ 1∂ 2 ε h + 2h + (uh) = 0. 2 ∂t ∂θ ut + uuθ + hθ + ε (

(24) (25)

160


Application of Lie Group Analysis ( [12, 38, 39]) shows that there exists an exact singular nonstationary one-parameter invariant solution (an invariant solution of a differential equation is a solution of the differential equation which is also an invariant curve (surface) of a group admitted by the differential equation. Such solutions can be found without determining its general solution.) of the model (24)–(25) at zeroth order epsilon, which can be written as

2θ + α, 3t

u=

h=(

θ − α )2 , 3t

(26)

where α is an arbitrary constant. Sophus Lie proposed for the first time to study the symmetries of differential equations and use them for constructing solutions at the end of nineteenth century. By a symmetry we mean a continuous group of transformations acting on the dependent and independent variables of the system of differential equations so that the system stays unchanged ( [40, 41]). Since the effects of rotation are not included in this work, the invariant solution (26) is different from the set of invariant solutions obtained in [40]. These solutions for different values of time t are plotted in Figure 4 versus the polar angle θ , in which we set α = 10. For example, the exact solution for h at t = 0.1 can be associated with a single wave oscillating around the equatorial plane, as shown in Figure 5. Finding the invariant solutions of the complete shallow water system (24)– (25) with ε = 0 will be the task of the forthcoming project. Unfortunately, any small perturbation of an equation breaks the admissible group of transformations and reduces the applied value of these ”refined” equations and group theoretical methods in general. Therefore, development of methods of group analysis stable with respect to small perturbations of differential equations has become vital. As is discussed in the Conclusion section, the methods of Approximate Group Analysis will be applied to the given model (24)–(25).

120

Exact solution

80 100

Exact solution

80

θ=π

U (t =0.1) H (t = 0.1) U (t = 0.3) H (t = 0.3)

U

60 40

H

20 0

0.2

0.4

0.6

0.8

1

time 60

40

20

0

0

1

2

3

θ

4

5

6

Fig. 4 Exac solution of the shallow water model, that results in the limiting case of the system system(24)–(25) at ε = 0.


161

r

h(θ,t) R θ

Fig. 5 Schematic presentation of the invariant solution of the zeroth-order model (26) for h(θ ,t) at t = 0.1.

4 Stationary waves In order to investigate the polygon-shaped waves having the structure similar to the hexagon as shown in Figure 1, we analyze the model (10)–(14) in the reference frame moving with a wave so that the unknown functions do not depend on time. In this case, the two-dimensional model (10)–(14) for two unknowns ψ (r, θ ) and h (θ ) > 0 is written in non-dimensional variables (16) as

ε2

2 ∂ 2ψ ∂ψ 2∂ ψ = 0 (0 < r < h) , + (1 + ε r) + ε (1 + ε r) 2 2 ∂θ ∂r ∂r

ψ (0, θ ) = 0,

ψ (h, θ ) = Q,

(27) (28)

ε2 ∂ψ 2 ∂ψ 2 ) + 2gh = 2b, (r = h), ( ) +( 2 (1 + ε r) ∂ θ ∂r

(29)

∂h ∂ + (uh) = 0, (r = h) , ∂t ∂θ

(30)

r

where b is the Bernoulli’s constant, Q = h0 u0 is the constant representing a flow rate and, as follows from (15), Γ ln (1 + ε ) . (31) u0 = 2π h0 We first observe that the model (27)–(30) can be reduced to the following relation evaluated at the free boundary:

∂ ∂θ

ˆh [(1 + ε r)( 0

∂ψ 2 ε2 ∂ ψ 2 ∂ψ 2 ε2 ∂ ψ 2 dh ) − ( ) + ) ] = −(1 + ε h)[( ( ) ] . 2 ∂r 1 + εr ∂ θ ∂r (1 + ε h) ∂ θ d θ

(32)

162


In the stationary case, for the equation (25), we have uh = Q = const. Using this relation, we can exclude u from (24), integrate (32) and substitute the expression for ψ given by (20) to arrive to the following first-order differential equation for the unknown h (θ ) : 2 1 2 2 dh 2 ε2 ε2 ε Q ( ) = − ε h4 + (ε b − 1)h3 + (2b + Q2 )h2 + ( Q2 − c)h + Q2 , 3 dθ 3 4 2

(33)

where c is an integrating constant. We remark that in the limiting case ε → 0 (which corresponds to the case of the flat bottom when R → ∞), we obtain the cubic Bouusinesq-Rayleigh equation [42]. We denote by h j ( j = 1, 2, 3, 4) the roots of the polynomial (33). Then the Viète theorem [43] yields the following relation between the parameters Q, c and b are the roots h j ( j = 2, 3, 4) : Q2 =

− 23ε + ε2 (h2 h3 + h2 h4 + h3 h4 ) − h2 − h3 − h4 2

ε − 23ε h2 h13 h4 + 34 H − 16

;

1 1 1 2ε ε c = Q2 [ + + + ] − h2 h3 h4 ; 2 h2 h3 h4 3 ε 1 b = Q2 H + (h2 h3 + h2 h4 + h3 h4 ) , 2 3 where H=

1 1 1 + + . h3 h4 h2 h4 h2 h3

(34)

(35) (36)

(37)

Additionally, a simple perturbation analysis shows that formulae (37) imply that h1 represents a nonphysical solution with the following asymptotic: h1 → ∞ as R → ∞. Namely, h1 = −

3 1 . 2ε h2 h3 h4

(38)

and the roots h2 , h3 and h4 are also the roots of the Bouusinesq-Rayleigh equation in the limiting case when ε → 0. The existence of non-trivial wave-like solutions corresponds to the case when all the roots of the polynomial (33) are real and have the values in the interval 0 < h2 h3 < h < h4 . Particularly, we also remark that, since, according to its physical meaning, h is positive and continuous function, the domain of the admissible solution is the interval [h3 , h4 ]. Implicitly, the shape of the free boundary is given by the quadrature

εQ θ=√ 3

ˆh h3

ds , (ε s + λ )(s − h2 ) (s − h3 ) (s − h4 )

(39)

where λ > 0 is a constant. Namely, according to the relation (38) for h1 ,

λ=

3 1 . 2 h2 h3 h4

(40)

We next introduce a small finite Jacobi amplitude a = h4 − h3 and assume that a is of order ε . In this approximation, the change of variable of integration s = h3 + ξ reduces the integral in (39) to the following asymptotic form:. ˆh h3

ds ˜ (ε s + λ )(s − h2 ) (s − h3 ) (s − h4 )

ˆξ 0

dξ , (ε h3 + λ ) (h3 − h2 ) ξ (1 − ξ )

(41)


163

which can now be evaluated in terms of elementary functions. As follows from (41), the polygon-shaped wave structure is described by a 2π −periodic nonlinear wave and the wavenumber n is determined by the relation √ 3 (ε h3 + λ )(h3 − h2 ) . (42) n= ε For example, in order to visualize the particular hexagon-shaped path as shown in in Figure 1, we set n = 6 and visualize the gravity waves by detecting the wave trougs and crests by locating the points of minimum and maximum of points of minimum of the integral in (39) as shown in Table 1. Table 1 trougs

θ =0

crests

π 6

θ=

θ= θ=

π 3 π 2

θ= θ=

2π 3 5π 6

θ =π

θ=

7π 6

θ=

θ=

4π 3 9π 6

θ= θ=

4π 3 11π 6

θ = 2π

Schematically, the resulting waves for n = 6 can be visualized as shown in Figure 6, where the scales are chosen arbitrarily but the points of trougs and crests correspond to values of θ in Table 1. In terms of the mathematical modeling presented here, the value R should not represent necessarily the radius of the planet, it can be just a radial scale satisfying the relation R/h0 0, y > 0}. Now, from (3), the activator-inhibitor with its density confined to a fixed open bounded domain Ω in RN with smooth boundary is expressed as the following reaction-diffusion system ⎧ 2 u ∈ Ω, t > 0, ⎪ ⎪ xt = d1 Δx + ρ (ρ1 y + x ) − xy, ⎪ ⎪ ⎨ y = d Δy + ρλ (ρ xy + x2 y) − λ y2 , u ∈ Ω, t > 0, t 2 2 (4) ⎪ ∂ x = ∂ y = 0, u ∈ ∂ Ω, t > 0, ⎪ ν ν ⎪ ⎪ ⎩ x(u, 0) = x0 (u) ≥ 0, y(u, 0) = y0 (u) ≥ 0, u ∈ Ω. In the above, Δ is the Laplaican operator on Ω, where d1 and d2 denote respectively diffusivity of prey and predator are kept independent of space and time. The no-flux boundary condition means that the statical environment Ω is isolated and ν is the outward unit normal to ∂ Ω. The initial values x0 (u), y0 (u) are assumed to be positive and bounded in Ω. Understanding the qualitative behavior of dynamical systems is essential for the real world applications. The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a steady point is known as the Hopf bifurcation. For the studies on Hopf bifurcation for reaction diffusion systems in various mathematical models such population ecology, physics and bio-chemistry etc (see [8–12]). Yi et al. [13] derived an explicit algorithm for determining the direction of Hopf bifurcation and stability of the bifurcating periodic solutions for a diffusive predator-prey system. In particular, they have shown the existence of multiple spatially non-homogeneous periodic orbits while the system parameters are all spatially homogeneous. Earlier Yi et al. [14] considered a Lengyel-Epstein diffusive predator-prey system of the chlorite-iodide-malonic acid reaction and they derived the precise conditions on the parameters so that the spatially homogeneous equilibrium solution and the spatially homogeneous periodic solution become Turing unstable. To the best of the authors knowledge, there is no work exists in the direction of stability and Hopf bifurcation of diffusive bio-chemical reaction of the morphogenesis process.

M. Sambath, K. Balachandran /Journal of Applied Nonlinear Dynamics 4(2) (2015) 181–195

183

2 Hopf bifurcation analysis In this section, we mainly focus on the existence of spatially homogeneous and non-homogeneous periodic solutions bifurcating from the Hopf bifurcation. The system (3) has a constant positive steady state (x∗ , y∗ ), see in [7], where (1 + ρρ1 − ρ2 ) + (1 + ρρ1 − ρ2 )2 + 4ρρ1 ρ2 ∗ , y∗ = ρ x∗ (ρ2 + x∗ ). x = 2 To cast our discussion into the frame work of the Hopf bifurcation theorem, we translate (3) into the following system by the transition xˆ = x − x∗ , yˆ = y − y∗ . For the sake of convenience, we still denote xˆ and yˆ by x and y respectively. Thus the reaction-diffusion system (3) becomes ⎧ u ∈ (0, l π ), t > 0, ⎪ xt − d1 xuu = f (λ , x, y), ⎪ ⎪ ⎪ ⎪ ⎨ yt − d2 yuu = g(λ , x, y), u ∈ (0, l π ), t > 0, (5) ⎪ xu (0,t) = xu (l π ,t) = 0, yu (0,t) = yu (l π ,t) = 0, t > 0, ⎪ ⎪ ⎪ ⎪ ⎩ x(u, 0) = x (u), y(u, 0) = y (u), u ∈ (0, l π ). 0 0 Define

f (λ , x, y) = ρ (ρ1 (y + y∗ ) + (x + x∗)2 ) − (x + x∗)(y + y∗ ), g(λ , x, y) = λ ρ (x + x∗)(y + y∗ )(ρ2 + (x + x∗)) − λ (y + y∗)2 ,

where f , g : R × R2 −→ R are C∞ smooth with f (λ , 0, 0) = 0. = g(λ , 0, 0) 2 Now we define the real-valued Sobolev space U = (x, y) ∈ [H (0, l π )]2 : (xu , yu )|u=0,lπ = 0 , and the complexification of U : UC := U ⊕ iU = {x1 + ix2 : x1 , x2 ∈ U }. The linearized operator of the steady state system (5) evaluated at (λ , 0, 0) is ⎞ ⎛ 2 d1 ∂∂u2 + A(λ ) B(λ ) ⎠, L(λ ) = ⎝ 2 d2 ∂∂u2 + D(λ ) C(λ ) with the domain DL(λ ) = UC where A(λ ) = fx (λ , 0, 0) = 2ρ x∗ − y∗ ,

B(λ ) = fy (λ , 0, 0) = ρρ1 − x∗ ,

C(λ ) = gx (λ , 0, 0) = λ ρ y∗ (ρ2 + 2x∗ ), D(λ ) = gy (λ , 0, 0) = λ (ρ x∗ (ρ2 + x∗ ) − 2y∗ ). The following condition is essential to guarantee that the Hopf bifurcation occurs: (H) There exists a number λ H ∈ R and a neighborhood O of λ H such that for λ ∈ O, L(λ ) has a pair of complex, simple, conjugate eigenvalues p(λ ) ± iω (λ ), continuously differentiable in λ , with p(λ H ) = 0, ω0 = ω (λ H ) > 0 and p (λ H ) = 0; all other eigenvalues of L(λ ) have non-zero real parts for λ ∈ O. Now we recall the Hopf bifurcation result appearing in [13] and apply it to the analysis of our model. It is well known that the eigenvalue problem −ϕ = μϕ , u ∈ (0, l π ); ϕ (0) = ϕ (l π ) = 0, has eigenvalues μn =

n2 l2

(n = 0, 1, 2, . . .), with corresponding eigenfunctions ϕn (u) = cos( nul ). Let ∞ nu φ an =∑ cos , b ψ l n n=0

184


be an eigenfunction of L(λ ) corresponding to an eigenvalue β (λ ), that is, L(λ )(φ , ψ )T = β (λ )(φ , ψ )T . Then, from a straightforward analysis, we obtain the following relation: an a Ln (λ ) = β (λ ) n , n = 0, 1, 2, . . . , bn bn where

⎛ Ln (λ ) = ⎝

2

−d1 nl2 + A(λ )

B(λ )

C(λ )

−d2 nl2 + D(λ )

2

⎞ ⎠.

It follows that eigenvalues of L(λ ) are given by the eigenvalues of Ln (λ ) for n = 0, 1, 2, . . . . The characteristic equation of Ln (λ ) is

β 2 − Tn (λ )β + Dn (λ ) = 0, n = 0, 1, 2, . . . , where

⎧ (d1 + d2 )n2 ⎪ ∗ ∗ ⎪ , ⎨ Tn (λ ) = 2ρ x − (1 + λ )y − l2 4 2 ⎪ ⎪ ⎩ D (λ ) = d d n − n [d (2ρ x∗ − y∗ ) − d λ y∗ ] + λ Θ, n 1 2 4 2 1 l l2

(6)

and 2

2

Θ = (ρ x∗ y∗ (2 + x∗ ) + 3y∗ ) − (y∗ + 4ρ x∗ y∗ + ρ 2 ρ1 ρ2 y∗ + 2ρ 2 ρ1 x∗ y∗ ). Therefore the eigenvalues are determined by Tn (λ ) ± Tn2 (λ ) − 4Dn (λ ) , n = 0, 1, 2, . . . . β (λ ) = 2 If the condition (H) holds, we see that, at λ = λ H , L(λ ) has a pair of simple purely imaginary eigenvalues ±iω0 if and only if there exists a unique n ∈ N ∪ {0} such that ±iω0 are the purely imaginary eigenvalues of Ln (λ ). In such case, denote the associated eigenvector by q = qn = (an , bn )T cos nlπ , with an , bn ∈ C, such that Ln (λ )(an , bn )T = iω0 (an , bn )T or L(λ H )q = iω0 q. We shall identify the Hopf bifurcation value λ H which satisfies the condition (H) taking the following form, if there exists n ∈ N ∪ {0} such that Tn (λ H ) = 0, Dn (λ H ) = 0, and T j (λ H ) = 0, D j (λ H ) = 0 for j = n,

(7)

and for the unique pair of complex eigenvalues p(λ ) ± iω (λ ) near the imaginary axis p (λ H ) = 0. It is 2 easy to drive from (6) that Tn (λ ) < 0 and Dn (λ ) > 0, if 1 ≥ 2/(ρ2 + x∗ ), which implies that (0, 0) is a locally asymptotically stable steady state of system (5). 2 If 1 < 2/(ρ2 + x∗ ), we define

λ0∗ =

2 − 1 > 0. ρ2 + x∗2

(8)

Hence the potential Hopf bifurcation point lives in the interval (0, λ0∗ ]. For any Hopf bifurcation λ H in (0, λ0∗ ], p(λ H ) ± iω (λ H ) are the eigenvalues of L(λ H ) where Tn (λ H ) , ω (λ H ) = p(λ ) = 2 H

Dn (λ H ) − p2 (λ H ),


185

1 p (λ H ) = Tn (λ H ) < 0. 2

(9)

and

From the above discussion, the determination of Hopf bifurcation point reduces to describing the set Λ1 = λ H ∈ (0, λ0∗ ] : for some n ∈ N ∪ {0}, (7) is satisfied , when a set of parameters (d1 , d2 , ρ , ρ1 , ρ2 ) are given. In the following we fix (d1 , d2 , ρ , ρ1 , ρ2 ) > 0 and choose l appropriately. First λ0H = λ0∗ is always an element of Λ1 for any l > 0 since T0 (λ0H ) = 0, T j (λ0H ) < 0 for any ( j ≥ 1), Dm (λ0H ) > 0 for any m ∈ N ∪ {0}. This corresponds to the Hopf bifurcation of spatially homogeneous periodic solution. Apparently λ0H is also the unique value for the Hopf bifurcation of the spatially homogeneous periodic solution for any l > 0. Hence in the following we look for spatially non-homogeneous Hopf bifurcation points. Note that, when λ < λ0∗ , it is easy to show that Tn (λ ) = 0 is equivalent to (d1 + d2 )n2 . y∗ l 2

λ = λ0∗ −

Substituting it into the second equation of (6), we have Dn (λ ) = −d12

(d1 + d2 )Θ n4 n2 − (d2 (2ρ x∗ − y∗ ) − d1 λ0∗ y∗ + ) + λ0∗ Θ. l4 l2 y∗

Let B0 =

(d1 + d2 )Θ + d2 (2ρ x∗ − y∗ ) − d1 λ0∗ y∗ , y∗

then Dn (λ ) > 0 if and only if n2 l2

0, that is, given the fixed N defined by (11), for every 0 < n ≤ N, d2 < ε (l, ρ , ρ1 , ρ2 , N), where 2

ε (l, ρ , ρ1 , ρ2 , N) :=

λ0∗ y∗ − Nl2

2

λ0∗ y∗ − Nl2

> 0.

(12)

Therefore Di (λnH ) > 0. To adopt the framework of [15], see also [13], we write the system (5) in the vector form dX = L(λ )X + F(λ , X ), dt where

F(λ , X ) :=

(13)

f (λ , x, y) − A(λ )x − B(λ )y , g(λ , x, y) −C(λ )x − D(λ )y

with X = (x, y)T ∈ X . At each λ = λnH , n = 0, 1, 2, . . . , N, (13) can be reduced to dX = L(λnH )X + Fn (X ), dt

(14)

where Fn (X ) = F(λ , X )|λ =λnH . Let ·, · be the complex-valued L2 inner product on Hilbert space UC defined as ˆ lπ (x¯1 x2 + y¯1 y2 )du, X1 , X2 = 0

with Xi = (xi , yi ∈ XC (i = 1, 2). Throughout this paper, we use f¯ to denote the conjugate of f . Then λ X1 , X2 = λ¯ X1 , X2 . Let L∗ (λnH ) be the adjoint operator of L(λnH ), that is, x, L(λnH )y = L∗ (λnH )x, y . Then L∗ (λnH ) is also defined on UC and ⎞ ⎛ ∂2 H H d1 ∂ u2 + A(λn ) C(λn ) ⎠. L∗ (λnH ) = ⎝ ∂2 H H d2 ∂ u2 + D(λn ) B(λn ) )T

From (H), we can chose q = (an , bn )T cos nul , q∗ = (a∗n , b∗n )T cos nul ∈ UC satisfying ¯ = 0; L∗ (λnH )q∗ = −iω0n q∗ , q∗ , q = 1, q∗ , q

(15)

here wn0 , q, q∗ will be given concretely by (32) in the sequel. We decompose U = U c ⊕U s with U c = {zq + z¯q¯ : z ∈ C}, U s = {x ∈ U : z ∈ q∗ , x = 1}. For any (x, y) ∈ U, there exist z ∈ C and w = (w1 , w2 ) ∈ U s such that x = zan cos nul + z¯a¯n cos nul + w1 , x w1 , or = zq + z¯q¯+ w2 y y = zbn cos nu + z¯b¯n cos nu + w2 . l

Now (14) can be reduced to the following system in (z, w) coordinates: z˙(t) = iω0n z + q∗ , Fn , w(t) ˙ = L(λnH )w + H(z, z¯, w),

(16)

l

(17)


187

where ¯ H(z, z¯, w) = Fn − q∗ , Fn q − q¯∗ , Fn q,

Fn = Fn (zq + z¯q¯+ w).

(18)

As in [15], we write Fn in the form 1 1 Fn (X ) = Q(X , X ) + C(X , X , X ) + O(|X |4), 2 6

(19)

where Q,C are symmetric multilinear forms. For simplicity, we write QXY = Q(X ,Y ), CXY Z = C(X ,Y, Z). For later use, we calculate Qqq , Qqq¯ and Cqqq¯ as follows nu cn en gn 2 nu 2 nu cos cos cos3 , , Qqq¯ = , Cqqq¯ = Qqq = dn fn hn l l l where (with the partial derivatives evaluated at (λnH , 0, 0)) ⎧ cn = fxx a2n + 2 fxy an bn + fyy b2n , ⎪ ⎪ ⎪ ⎪ ⎪ dn = gxx a2n + 2gxy an bn + gyy b2n , ⎪ ⎪ ⎪ ⎨ e = f |a |2 + f (a b¯ + a¯ b ) + f |b |2 , n xx n xy n n n n yy n 2 + g (a b 2 ¯ ⎪ = g |a | + a ¯ b ) + g f n xx n xy n n n n yy |bn | , ⎪ ⎪ ⎪ ⎪ ⎪ gn = fxxx |an |2 an + fxxy (2|an |2 bn + a2n b¯n ) + fxyy (2|bn |2 an + b2n a¯n ) + fyyy |bn |2 bn , ⎪ ⎪ ⎩ hn = gxxx |an |2 an + gxxy (2|an |2 bn + a2n b¯n ) + gxyy (2|bn |2 an + b2n a¯n ) + gyyy |bn |2 bn .

(20)

Let H(z, z¯, w) = then, by (18) and (19), we have

H02 2 H20 2 z + H11 zz + z + o (|z| · |w|); 2 2

(21)

¯ H20 = Qqq − q∗ , Qqq q − q¯∗ , Qqq q, ∗ ∗ ¯ H11 = Qqq¯ − q , Qqq¯ q − q¯ , Qqq¯ q.

It follows from Appendix A of [15] that system (17) possesses a center manifold. We can write w in the form: w=

w02 2 w20 2 z + w11 zz + z + o (|z|3 ). 2 2

(22)

By (21) and (22), together with L(λnH )w + H(z, z¯, w) = we have

dw ∂ w ∂ z ∂ w ∂ z¯ = + , dt ∂ z ∂ t ∂ z¯ ∂ t

−1 H20 , w20 = 2iω0n − L(λnH )

−1 w11 = − L(λnH ) H11 .

Actually, by [13], we have ⎧ 1 2nu c ⎪ n H −1 ⎪ + 1) n ], [(cos ⎪ ⎨ 2 2iω0 I − L(λn ) dn l w20 = ⎪ −1 c0 a0 a¯0 ⎪ ⎪ − q∗ , Qqq − q¯∗ , Qqq ], [ ⎩ 2iω0n I − L(λ0H ) b¯0 d0 b0

if n ∈ N, (23) if n = 0,

188

and


⎧ 1 2nu e ⎪ H −1 ⎪ + 1) n ], [(cos ⎪− L(λn ) ⎨ fn 2 l w11 = ⎪ a0 a¯ −1 e0 ⎪ ⎪ [ − q∗ , Qqq¯ − q¯∗ , Qqq¯ ¯0 ], ⎩− L(λ0H ) b0 f0 b0

if n ∈ N, (24) if n = 0,

−1 −1 Notice that the calculation of 2iω0n I − L(λnH ) and L(λnH ) in (23) and (24) is restricted to the . subspaces spanned by the eigen-modes 1 and cos 2nu l Now the reaction-diffusion system restricted to the center manifold can be given by gi j i j dz = iω0n z + q∗ , Fn = iω0n z + ∑ z z¯ + O(|z|4 ), dt i! j! 2≤i+ j≤3

(25)

where g20 = q∗ , Qqq , g11 = q∗ , Qqq¯ , g02 = q∗ , Qq¯q¯ , g21 = 2 q∗ , Qw11 q + q∗ , Qw20 q¯ + q∗ ,Cqqq¯ . The dynamics of (17) can be determined by the dynamics of (25). As in [15], we write the Poincare normal form of (13) (for λ in a neighborhood of λnH ) in the form M

z) j , z˙ = (p(λ ) + iω (λ ))z + z ∑ c j (λ )(z¯

(26)

j=1

where z is a complex variable, M ≥ 1 and c j (λ ) are complex-valued coefficients. As in [15], we have c1 (λ ) =

|g11 |2 |g02 |2 g21 g20 g11 (3p(λ ) + iω (λ )) + + + . 2 2 2(p (λ ) + ω (λ )) p(λ ) + iω (λ ) 2(p(λ ) + 3iω (λ )) 2

Thus i 1 g21 (g20 g11 − 2|g11 |2 − |g02 |2 ) + n 2ω0 3 2 1 1 i q∗ , Qqq . q∗ , Qqq¯ + q∗ , Qw11 q + q∗ , Qw20 q¯ + q∗ ,Cqqq¯ = n 2ω0 2 2

c1 (λnH ) =

(27)

with w20 and w11 in the form of (23) and (24) respectively. Then summarizing our analysis above and applying Theorem 2.1 in [13], we have the main result of this section on the existence of both spatially homogeneous and non-homogeneous periodic solutions bifurcating from Hopf bifurcation. First we consider the bifurcation direction and stability of the bifurcating (spatially homogeneous) periodic solutions. Theorem 1. For the system (1), the bifurcating (spatially homogeneous) periodic solutions bifurcating from λ = λ0H are locally asymptotically stable (resp. unstable) if Re(c1 (λ0H )) < 0 (resp.> 0). Furthermore the direction of Hopf bifurcation at λ0H is supercritical (resp. subcritical) if Re(c1 (λ0H )) < 0 (resp. >0). Proof. Following the notations and calculation in [13], we set q = (a0 , b0 )T = (1, q∗ = (a∗0 , b∗0 )T =

(2ρ x∗ − y∗ ) − iω0 T ) , ρρ1 − x∗

ρρ1 − x∗ ω0 + i(2ρ x∗ − y∗ ) ( , −i)T , 2ω0 π l ρρ1 − x∗


189

such that q∗ , q = 1 and q∗ , q ¯ = 0. Then, by direct computation, we get c0 = u1 + iv1 , d0 = u2 + iv2 , e0 = u3 , f0 = u4 , g0 = u5 + iv5 , h0 = u6 + iv6 , where 2ρ x∗ − y∗ 2ω0 ), v1 = , ∗ ρρ1 − x ρρ1 − x∗ 2λ ρ (ρ2 + 2x∗ )(2ρ x∗ − y∗ ) −2ω0 2λ ρ (ρ2 + 2x∗ ) , v = , u2 = 2λ ρ y∗ + 2 ρρ1 − x∗ ρρ1 − x∗ 2ρ x∗ − y∗ 2ρ x∗ − y∗ ), v3 = 2λ ρ (y∗ + (ρ2 + 2x∗ ) ), u3 = 2(1 − ∗ ρρ1 − x ρρ1 − x∗ 2ρ x∗ − y∗ −2mω0 θ . ), v5 = u5 = 6λ ρ ( ρρ1 − x∗ mu∗ (a + bu∗ )

u1 = 2(1 −

Then 1 [(u1 ω0 + v1 (2ρρ1 − y∗ ) − v2 (ρρ1 − x∗ )) q∗ , Qqq = 2ω0 +i(v1 ω0 − u1 (2ρ x∗ − y∗ ) + u2 (ρρ1 − x∗ ))], ∗ 1 [ω0 u3 + i(v3 (ρρ1 − x∗ ) − u3 (2ρ x∗ − y∗ ))], q , Qqq¯ = 2ω0 ∗ 1 [(u1 ω0 − v1 (2ρρ1 − y∗ ) + v2 (ρρ1 − x∗ )) q¯ , Qqq = 2ω0 +i(v1 ω0 + u1 (2ρ x∗ − y∗ ) − u2 (ρρ1 − x∗ ))], ∗ 1 [ω0 u3 + i(u3 (2ρ x∗ − y∗ ) − v3 (ρρ1 − x∗ ))], q¯ , Qqq¯ = 2ω0 ∗ ρρ1 − x∗ )(u5 + iv5 ). q ,Cqqq¯ = i( 2ω0

Direct computation gives

a0 a¯0 ∗ ∗ c0 − q , Qqq − q¯ , Qqq = 0, H20 = d0 b0 b¯0 a0 a¯0 ∗ ∗ e0 − q , Qqq¯ − q¯ , Qqq¯ ¯ = 0. H11 = f0 b0 b0

Then, by Yi et al. [13], it implies that w20 = w11 = 0; hence ∗ q , Qw20 q¯ = q∗ , Qw11 q¯ = 0. After elementary but lengthy computations, we obtain Re(c1 (λ0H )) = Re{ =

1 i ∗ q , Qqq . q∗ , Qqq¯ + q∗ ,Cqqq¯ } 2ω0 2

−1 {u3 ω0 [v1 ω0 − u1 (2ρ x∗ − y∗ ) + u2 (ρρ1 − x∗ )] 3 8ω0 +[v3 (ρρ1 − x∗ ) − u3 (2ρ x∗ − y∗ )][u1 ω0 + v1 (2ρρ1 − y∗ ) − v2 (ρρ1 − x∗ )]}.

It follows from (9) that p (λ0H ) < 0 and then, by Theorem 2.1 in [13], the periodic solutions bifurcating from λ = λ0H are locally asymptotically stable (resp. unstable) if Re(c1 (λ0H )) < 0 (resp.> 0). Furthermore the direction of Hopf bifurcation at λ0H is supercritical (resp. subcritical) if Re(c1 (λ0H )) < 0 (resp. >0).

190


From Theorem II and Section 3 in Chapter 1 of [15], under (H), we establish the main result of our work: Theorem 2. For any λnH , defined by (10), if there exists ε = ε (l, ρ , ρ1 , ρ2 , N) defined by (12), such that 0 < d2 < ε , then the system (5) undergoes a Hopf bifurcation at each λ = λnH (0 ≤ n ≤ N). With s sufficiently small, for λ = λ (s), λ (0) = λnH , there exists a family of T (s)− periodic continuously differentiable solutions (x(s)(u,t), y(s)(u,t)) and the bifurcating periodic solutions can be parametrized in the form x(s)(u,t) = s an e2π it/T (s) + a¯n e−2π it/T (s) cos nul + o(s2 ), (28) y(s)(u,t) = s bn e2π it/T (s) + b¯n e−2π it/T (s) cos nul + o(s2 ), where an , bn are given by (2.30); T (s) =

1 Re(c1 (λnH )) H 2π 2 4 H (1 + τ s ) + o(s ), τ = − (Im(c ( λ )) − ω (λn )), 2 2 1 n ω0n ω0n p (λnH )

and T (0) =

4π 4π Re(c1 (λnH )) H H τ = − (Im(c ( λ )) − ω (λn )). 2 1 n wn0 (ω0n )2 p (λnH )

If all eigenvalues (except ±iω0n ) of L(λnH ) have negative real parts, then the bifurcating periodic solutions are stable (resp. unstable) if Re(c1 (λnH )) < 0(resp. > 0). The bifurcation is supercritical (resp. subcritical) if − p (1λ H ) Re(c1 (λnH )) < 0(resp. > 0). n Moreover (i) The bifurcating periodic solutions from λ0H are spatially homogeneous which coincide with the periodic solutions of the corresponding ODE system. (ii) The bifurcating periodic solutions from λnH , n > 0, are spatially non-homogeneous. Proof. Following the bifurcation formula (pp. 28-32, [15]), we observe that (26) is rotationally invariant: if z is a solution, then so is zeiφ for any real number φ and the trajectories of (26) are circles with centers at z = 0. This simple geometry is reflected in efficient computation of the Maclaurin d z¯ expansions of λ (s) and T (s). Formating z¯dz dt + z dt from (26), we obtain M d(z¯ z) = 2z¯ z(p(λ ) + ∑ Re(c j (λ ))(z¯ z) j ). dt j=1

(29)

The right-hand side of (29) is zero if and only if z = 0 or M

z) j = 0. p(λ ) + ∑ Re(c j (λ ))(z¯

(30)

j=1

Since z(t) ≡ 0, (30) is a proper condition to ensure that the right-hand side of (29) is zero. In such case, z(t)¯ z(t) is a non-negative constant, denoted by s2 for some s ≥ 0. Thus (26) can be written as M

z˙ = iz[ω (λ ) + Im( ∑ c j (λ )s2 j )]. j=1

It follows from (31) that z(t) = se2π it/T (s) and, by expansion, we have T (s) =

1 Re(c1 (λnH )) H 2π 2 4 H (1 + τ s ) + o(s ), τ = − (Im(c ( λ )) − ω (λn )). 2 2 1 n ω0n ω0n p (λnH )

(31)


191

From (16), we get the solution of (5) in the form of (28). Since q and q∗ satisfy (15), it is easy to get 2ρ x∗ − y∗ − n2 d1 − iω0n T nu nu ) cos , = (1, ∗ l ρρ1 − x l ω0n nu ρρ1 − x∗ 2ρ x∗ − y∗ n2 d1 nu ∗ ∗ ∗ T ( + i( − 2 = ), −i)T cos . q = (an , bn ) cos n ∗ ∗ l 2l πω0 ρρ1 − x ρρ1 − x l (ρρ1 − x∗ ) l q = (an , bn )T cos

(32)

Now we consider the stability and bifurcation direction. Since p (λnH ) < 0, we only need to compute the sign of Re(c1 (λnH )). By (27), we have 1 1 (33) Re(c1 (λnH )) = Re q∗ , Qw11 q + Re q∗ , Qw20 q¯ + Re q∗ ,Cqqq¯ . 2 2 From [13], we get q∗ , Qqq = q∗ , Qqq¯ = 0. In order to calculate Re(c1 (λnH )), we have to calculate

q∗ , Qw11 q , q∗ , Qw20 q¯ and q∗ ,Cqqq¯ .

It is straightforward to compute −1 2iω0n I − L2n (λnH ) ⎛ ⎞ 4n2 n H ∗ ∗ ∗ ∗ ρρ1 − x ⎜ i2ω0 + d2 l 2 − λn (ρ x (ρ2 + x ) − 2y ) ⎟ 1 ⎜ ⎟, = ⎠ 2 α1 + iα2 ⎝ 4n H ∗ ∗ n ∗ ∗ λn ρ y (ρ2 + 2x ) i2ω0 + d1 2 − 2ρ x + y l −1 2iω0n I − L0 (λnH ) i2ω0n − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗) ρρ1 − x∗ 1 = , α3 + iα4 λnH ρ y∗ (ρ2 + 2x∗) i2ω0n − 2ρ x∗ + y∗

where 4n2 4n2 H ∗ ∗ ∗ − λ ( ρ x ( ρ + x ) − 2y ))(d − 2ρ x∗ + y∗ ) − 4(ω0n)2 2 1 n l2 l2 +λnH ρ y∗ (x∗ − ρρ1)(ρ2 + 2x∗ ),

α1 = (d2

4n2 − 2ρ x∗ + y∗ − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗)), l2 α3 = (−λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗)) (−2ρ x∗ + y∗ ) − 4(ω0n)2 + λnH ρ y∗ (x∗ − ρρ1)(ρ2 + 2x∗),

α2 = 2ω0n ((d1 + d2)

α4 = 2ω0n (−2ρ x∗ + y∗ − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗)).

By (23) and (24), when n ∈ N, we get ⎞ 4n2 n H ∗ ∗ ∗ ∗ ω + d − λ ( ρ x ( ρ + x ) − 2y ))c + ( ρρ − x )d (i2 n n ⎟ 2 2 2 1 0 n ⎜ 1 l ⎟cos 2n u ⎜ = ⎠ 2 ∗ ∗ ∗ 2(α1 + iα2 ) ⎝ l 4n mu v (b + v ) λnH ρ y∗ (ρ2 + 2x∗)cn + (i2ω0n + d1 2 + u∗ − )dn l θ i2ω0n − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗) cn + (ρρ1 − x∗ )dn 1 + , 2(α3 + iα4 ) λnH ρ y∗ (ρ2 + 2x∗ )cn + (i2ω0n − 2ρ x∗ + y∗ ) dn ⎛

w20

192


⎛

w11

⎞ 4n2 H ∗ ∗ ∗ ∗ (−d + λ ( ρ x ( ρ + x ) − 2y ))e − ( ρρ − x ) f n n⎟ 2 2 2 1 n −1 ⎜ l ⎜ ⎟cos 2n u = ⎠ 2 2α5 ⎝ l 4n −λnH ρ y∗ (ρ2 + 2x∗ )en − (d1 2 − 2ρ x∗ + y∗) fn l H λn (ρ x∗ (ρ2 + x∗ ) − 2y∗) en − (ρρ1 − x∗ ) fn 1 − , 2α6 −λnH ρ y∗ (ρ2 + 2x∗)en + (2ρ x∗ − y∗ ) fn

where α5 = (−d2

4n2 4n2 + λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗))(d1 2 − 2ρ x∗ + y∗ ) − (ρρ1 − x∗ )λnH ρ y∗ (ρ2 + 2x∗ ), 2 l l

α6 = λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗) (−2ρ x∗ + y∗ ) − (ρρ1 − x∗ )λnH ρ y∗ (ρ2 + 2x∗).

The direct computation yields fxx = 2; fxy = −1; fxxx = fxxy = fyy = fuvv = fyyy = 0;

(34)

gxx = 2λ ρ y∗ ; gyy = −2λ ; gxy = ρλ (ρ2 + 2x∗ ); gxxy = 2λ ρ .

Here and in the following we always assume that all the partial derivatives of f and g are evaluated at (λnH , 0, 0). It is easy to get Qw20 q¯ =

gxx ξ + gxy η + gxy b¯n ξ

Qw11 q =

fxx ξ + fxy η + fxy b¯n ξ

fxx ξ¯ + fxy η¯ + fxy bn ξ¯

nu 2nu cos + cos l l

nu 2nu cos + cos l l gxx ξ¯ + gxy η¯ + gxy bn ξ¯

fxx τ + fxy Λ + fxy b¯n τ

gxx τ + gxy Λ + gxy b¯n τ

¯ + fxy bn τ¯ fxx τ¯+ fxy Λ ¯ + gxy bn τ¯ gxx τ¯+ gxy Λ

cos

nu , l

cos

nu , l

where ξ=

1 4n2 [(i2ω0n + d2 2 − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗))cn + (ρρ1 − x∗ )dn ], 2(α1 + iα2 ) l

4n2 1 mu∗ v∗ (b + v∗) [λnH ρ y∗ (ρ2 + 2x∗ )cn + (i2ω0n + d1 2 + u∗ − )dn ], 2(α1 + iα2 ) l θ 1 [(i2ω0n − λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗))cn + (ρρ1 − x∗ )dn ], τ= 2(α3 + iα4 ) 1 [λ H ρ y∗ (ρ2 + 2x∗ )cn + (i2ω0n − 2ρ x∗ + y∗)dn ], Λ= 2(α3 + iα4 ) n

η=

−1 4n2 ξ¯ = [(−d2 2 + λnH (ρ x∗ (ρ2 + x∗ ) − 2y∗))en − (ρρ1 − x∗ ) fn ], 2α5 l −1 4n2 η¯ = [−λnH ρ y∗ (ρ2 + 2x∗ )en − (d1 2 − 2ρ x∗ + y∗ ) fn ], 2α5 l −1 H ∗ ∗ ∗ τ¯ = [(λ (ρ x (ρ2 + x ) − 2y ))en − (ρρ1 − x∗ ) fn ], 2α6 n ¯ = −1 [−λnH ρ y∗ (ρ2 + 2x∗ )en + (2ρ x∗ − y∗ ) fn ]. Λ 2α6


193

By (20) and (34), we have cn = fxx + 2 fxy bn , dn = gxx + 2gxy bn , en = fxx + 2 fxy Re(bn ), fn = gxx + 2gxy Re(bn ), gn = fxxx + (3Re(bn ) + Im(bn )i) fxxy , hn = gxxx + (3Re(bn ) + Im(bn )i)gxxy . Notice that, for any n ∈ N, ˆ

lπ 0

cos2

ˆ

nu l π = , l 2

0

lπ

cos

2nu nu l π cos2 = , l l 4

ˆ

lπ 0

cos3

nu 3 = lπ , l 8

so lπ q∗ , Qw20 q¯ = [a¯∗n ( fxx ξ + fxy η + fxy b¯n ξ ) + b¯∗n (gxx ξ + gxy η + gxy b¯n ξ )] 4 lπ + [a¯∗n ( fxx τ + fxy Λ + fxy b¯n τ ) + b¯∗n (gxx τ + gxy Λ + gxy b¯n τ )], 2 lπ ∗ ∗ q , Qw11 q = [a¯n ( fxx ξ¯ + fxy η¯ + fxy bn ξ¯) + b¯∗n (gxx ξ¯ + gxy η¯ + gxy bn ξ¯)] 4 lπ ¯ + fxy bn τ¯) + b¯∗n (gxx τ¯+ gxy Λ ¯ + gxy bn τ¯)], + [a¯∗n ( fxx τ¯+ fxy Λ 2 3 ∗ q ,Cqqq¯ = l π (a¯∗n gn + a¯∗n hn ). 8

Since l π a¯∗n = 1 + (

2ρ x∗ − y∗ d1 n2 ρρ1 − x∗ − 2 n )i, l π b¯∗n = − i, n ω0 l ω0 ω0n

it follows that 3 ω0n , Re q∗ ,Cqqq¯ = gxxy 8 ρρ1 − x∗

(35)

1 ω0n (ξI + 2τI )) Re q∗ , Qw20 q¯ = [ fxx (ξR + 2τR ) + fxy (ηR + 2ΛR − 4 ρρ1 − x∗ d1 n2 y∗ − 2ρ x∗ ) fxy (ξR + 2τR )] − 2 +( ∗ x − ρρ1 l (ρρ1 − x∗ ) 1 y∗ − 2ρ x∗ d1 n2 − 2 n )[ fxx (ξI + 2τI ) + fxy (ηI + 2ΛI ) + ( 4 ω0n l ω0

ω0n y∗ − 2ρ x∗ d1 n2 (ξR + 2τR ) fxy + ( ∗ + ) ∗ ρρ1 − x x − ρρ1 ρρ1 − x∗ −ω0n ρρ1 − x∗ {g [( ξ + 2 τ ) fxy (ξI + 2τI )] + xy R R 4ω0n ρρ1 − x∗

−

+ηI + 2ΛI + (ξI + 2τI )(

y∗ − 2ρ x∗ d1 n2 )] − x∗ − ρρ1 l 2 (ρρ1 − x∗ )

+gxx (ξI + 2τI )}, 1 ¯ + gxy (ξ¯ + 2τ¯)], Re q∗ , Qw11 q = [ fxx (ξ¯ + 2τ¯) + fxy (ξ¯ + 2Λ) 4

(36) (37)


194

where we have denoted ΓR = ReΓ and ΓI = ImΓ for Γ = ξ , η , τ , Λ. More precisely

α2 ω n α1 4n2 [(d2 2 − λnH [ρ x∗ (ρ2 + x∗ ) − 2y∗ ])cn + (ρρ1 − x∗ )dn ] + 2 0 2 cn , 2 2 l 2(α1 + α2 ) (α1 + α2 ) α1 ω n −α2 4n2 ξI = [(d2 2 − λnH [ρ x∗ (ρ2 + x∗ ) − 2y∗ ])cn + (ρρ1 − x∗ )dn ] + 2 0 2 cn , 2 2 l 2(α1 + α2 ) (α1 + α2 ) α2 ω0n d1 4n2 α1 H ∗ ∗ ∗ ∗ [ dn , ηR = λ ρ y ( ρ + 2x )c + ( + y − 2 ρ x )d ] + 2 n n l2 2(α12 + α22 ) n (α12 + α22 ) α1 ω0n −α2 d1 4n2 H ∗ ∗ ∗ ∗ [ dn , ηI = λ ρ y ( ρ + 2x )c + ( + y − 2 ρ x )d ] + 2 n n l2 2(α12 + α22 ) n (α12 + α22 ) α4 ω0n α3 H ∗ ∗ ∗ ∗ [(− cn , τR = λ [ ρ x ( ρ + x ) − 2y ])c + ( ρρ − x )d ] + 2 n 1 n n 2(α32 + α42 ) (α32 + α42 ) α3 ω n α4 [(−λnH [ρ x∗ (ρ2 + x∗ ) − 2y∗ ])cn + (ρρ1 − x∗ )dn ], τI = 2 0 2 cn − (α3 + α4 ) 2(α32 + α42 ) H ∗ α4 ω0n α3 ∗ ∗ ∗ dn , λ ρ y ( ρ + 2x )c + (y − 2 ρ x ) d + ΛR = 2 n n 2(α32 + α42 ) n (α32 + α42 ) H ∗ α3 ω n α4 ∗ ∗ ∗ λ ρ y ( ρ + 2x )c + (y − 2 ρ x ) d . ΛL = 2 0 2 dn − 2 n n n (α3 + α4 ) 2(α32 + α42 ) ξR =

Finally, substituting (35)–(37) into (33), we get the expression of Re(c1 (λnH )). The proof is completed. 3 Numerical results In this section, we present some numerical simulations to verify our theoretical analysis proved in the previous section by using MATLAB. In system (1), let Ω = (0, 40π ), l = 40, ρ = 0.1, ρ1 = 0.35, ρ2 = 0.07, d1 = 5 and d2 = 0.1. Then it follows from Theorem 2.2 that there exist four Hopf bifurcation points: λ0H = λ0∗ ≈ 0.9878, λ1H ≈ 0.9561, λ2H ≈ 0.8608, λ3H ≈ 0.7021. Hence we know, from Theorem 2.1, that the bifurcating periodic solutions from λ0H are locally asymptotically stable and the direction of Hopf bifurcation at λ0H is supercritical. Next we present numerical simulations near Hopf bifurcation points λ0H : for λ = 1.0127 and λ = 0.7691; the solution (x(t, u), y(t, u)) tends to a constant steady state (0.9675, 0.1004) and spatially homogeneous periodic solution respectively (see Fig. 1 and 2).

2

0.25

1.5

0.2 Predator

Prey

1 0.5 0

0.15 0.1 0.05

−0.5 −1 600

0 600 60

400

40

200 t

60

400

20 0

0

x

40

200 t

20 0

0

x

Fig 1. Numerical simulations showing the constant steady state of the system (1) is locally asymptotically stable with λ = 1.0127 > λ0H and initial value (sin(0.1u), 0.1).

3

0.4

2

0.3 Predator

Prey


1 0

195

0.2 0.1

−1 600

0 600 60

400

40

200 t

60

400 200

20 0

0

x

40

t

20 0

0

x

Fig 2. Numerical simulations showing a spatially homogeneous periodic of the system (1) emerges when λ across the first Hopf bifurcation point λ0H with λ = 0.7691 < λ0H and initial value (sin(0.1u), 0.1). Acknowledgment The second author is thankful to the DRDO, New Delhi for financial support to carry out this work. References [1] Berding, C. and Haken, H.(1981), Pattern formation in morphogenesis, Journal of Mathematical Biology, 14, 133–151. [2] Gierer, A. and Meinhardt, H. (1972), Atheory ofbiological pattern formation, Kybernetik, 12, 30–39. [3] Granero-Porati, M.I. and Porati, A. (1984), Temporal organization in a morphogenetic field, Journal of Mathematical Biology, 20, 153–157. [4] Huang, X. (2001), A Mathematical Model in Morphogenetic Processes, Lectures on Mathematical Biology (Universida Simon Bolivar, Caracas, 2001). [5] Huang, X. and Zhu, L. (2005), Limit cycles in a general Kolmogorov model, Nonlinear Analysis: Theory, Methods & Applications, 60, 1393–1414. [6] Huang, X., Wang, Y., and Su, H. (2007), Limit cycles in morphogenesis, Nonlinear Analysis: Real World Application, 8, 1341–1348. [7] Wang, S., Huang, X., Zhu, L., andVillasana, M. (2010), Hopf bifurcation and multiple limit cycles in biochemical reaction of the morphogenesis process, Journal of Mathematical Chemistry, 47, 739–749. [8] Du, L. and Wang, M. (2010), Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, Journal of Mathematical Analysis and Applications, 366, 473–485. [9] Sambath, M., Gnanavel, S., and Balachandran, K. (2014), Bifurcation analysis of a diffusive predator-prey model with predator saturation and competition, Applicable Analysis, 92, 2439–2456. [10] Sambath, M. and Balachandran, K. (2014), Bifurcations in a diffusive predator-prey model with predator saturation and competition response, Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.3106, 38, 785–798. [11] Sivakumar, M., Sambath, M. and Balachandran, K.(2015), Stability and Hopf bifurcation analysis of a diffusive predator-prey model with Smith growth, International Journal of Biomathematics, 8, 1550013 [18 pages]. [12] Zhao, Z. and Song, X. (2007), Bifurcation and complexity in a ratio-dependent predator-prey chemostat with pulsed input, Applied Mathematics-A Journal of Chinese Universities, 22, 379–387. [13] Yi, F., Wei, J., and Shi, J. (2009), Bifurcation and spatiotemporal patterns in a homogeneous diffusion predator-prey system, Journal of Differential Equations, 246, 1944–1977. [14] Yi, F., Wei, J., and Shi, J. (2008), Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Analysis: Real World Applications, 9, 1038–1051. [15] Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H. (1981), Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge. [16] Wang, X. and Wei, J. (2013), Diffusion-driven stability and bifurcation in a predator-prey system with Ivlevtype functional response, Applicable Analysis, 37, 752–775.


Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Recognition and Threat Level Estimation of FOD Based on Image Content Features and Experiment Analysis Gang Xiao† 1 , Yu Li1 , Xiao Yun1 , and Jinhua Xie2 1 School 2 AVIC

of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China Radar and Avionics Institute, Wuxi Jiangsu 214063, China Submission Info

Communicated by Albert C.J. Luo Received 22 October 2014 Accepted 17 February 2015 2015 Available online 1 July 2015 Keywords Foreign Object Debris (FOD) Multi-sensors recognition Threaten level model BP Neural Network Classifier

Abstract Foreign Object Debris (FOD) is debris or article alien, which may have fallen onto a runway or taxiway, would potentially cause damage to aircraft. FOD surveillance system is used to FOD surveillance system as a significant technology. However, FOD is a kind of relatively random and diverse target that makes it difficult to analyze and recognize. In the reality, the primary goal of FOD recognition cannot recognize FOD accurately, because airport staffs do not care much about what exactly the FOD is but how great the FOD could threat to airplane. Therefore, based on FOD image features definition and threat level model, we proposed a simple and efficient method to estimate the FOD threat level. By comparison with BP neural network model, the error of new FOD threats coefficient model are no more than 15%. There are five sections discussed in the paper. First, the FOD surveillance system was introduced briefly. Second, the features of FOD including color, texture and shape and related extraction algorithm had been discussed. The HSV color model, texture model and normal shape feature like area, perimeter and eccentricity which are all used to describe FOD targets. Fourteen typical FOD targets such as stone fragment, metal chip, piece of rubber, pieces of paper, plastic and small plant are selected as experiment objects. Thirdly, a new FOD threat coefficient model was proposed, in terms of the values of features based on the huge amount of experiment results and human subjective judgment. Next, the threat level classifier was constructed by the BP neural network. The classifier was trained by some experiment FOD targets with given threat level. The others FOD targets were tested. Finally, the recognition and estimation results indicated that the proposed threat level model were effective and efficient, satisfied the robustness and real-time requirement. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Foreign Object Debris (FOD) is debris or article alien, which may have fallen onto a runway or taxiway, would potentially cause damage to aircraft [1, 2]. Not until Concorde accident occurred in Paris Charles † Corresponding



198

Gang Xiao el al. /Journal of Applied Nonlinear Dynamics 4(2) (2015) 197–213

de Gaulle airport had the threat of FOD hazard been noticed and studied [3]. The FOD surveillance system is designed to detect and recognize FOD in the airport runway. It is very difficult to analyze and recognize FOD target precisely because of the randomness and diverseness [4]. Actually the issue that the airport staffs do care is not the FOD’s attribute but how great the FOD could threat the airplane during the taking off and landing operations. In this paper, we proposed an efficient method based on image content features to describe FOD target, used to estimate the FOD threat level. The real system with our method and algorithm can help airport staffs to analyze FOD target and thus avoiding FOD hazard. This paper is organized as follows. The brief introduction of FOD detection and recognition system and two sensors are presented in Section 2. The FOD targets image features from IR sensor are discussed in detail in Section 3. The FOD threat level model is proposed in Section 4. Several real FOD data, as the experimental objects, and BP neural network are used to test and compare new threat level model in section 5. Conclusion is in the final section.

2 FOD detection and multi-sensors recognition system 2.1

The multi-sensors recognition System

The FOD target detection system introduced in this paper utilizes two main sensors including millimeterwave radar and IR camera. Millimeter-wave radar scans runway and detects any small potential FOD targets and obtains their coordinates, then IR camera takes background IR images and target IR images. Computer vision technology will be used to analyses the image and detect target. In this paper, the FOD detection and recognition system is consists of four procedures. Firstly, system preparation including establishment of coordinate system and background image database will be completed before operation. Secondly, FOD target are detected by the millimeter-wave radar and get the target’s coordinate. These coordinate will be converted to specific IR camera positions. Thirdly, in FOD target detection algorithm, Hough transformation method is applied to detect the straight lines and localize the runway region, and background subtracting method is applied to detect FOD target in runway region. Then, the data fusion and interaction of millimeter-wave radar and IR camera will be done for confirm automatically. Finally, in FOD target recognition system, some features of FOD targets will be extracted and analyzed, and then be classified into two categories, metal or the others. The workflow of the multi-sensors FOD recognition system presented in this paper is shown in Fig. 1.

Fig. 1 the workflow of the FOD system.

In the last decade, microwave and mm-Wave system have found increasing commercial applications and gained importance in comfort and security applications like in automotive radar sensors in the 77 GHz frequency range. P. Feil [5] gave a scheme with a Broadband 78 GHz Sensor for FOD detection application. Such sensors are equipped with narrow beam antennas for a good lateral resolution and a reasonably high bandwidth for a high range resolution. Using this kind of imaging radar sensor with sufficiently high frequency, it was able to detect relatively small pieces of metal, concrete, stone, or even plastic on an otherwise quite flat surface, such as FOD on a runway or taxiway. The millimeter-wave radar is an anti-stealth imaging sensor with smaller geometric dimensions. It is suitable for detect a small FOD in short distance and install in the vicinity of the airport runway. It works in uninterrupted scanning state to automatic detect FOD target and awake warning. The


199

distance and coordinate information of detected FOD will be calculated based on airport the partition map and millimeter-wave radar location. The platform of optical IR camera was given this coordinate information to locate imaging area and focusing these FOD, which were used for identification FOD. The remote control center also can get these FOD coordinate information and image for assistant diagnosis. The measure distance R and its resolution ΔR are main two key performance indicators for millimeterwave radar [6]. The radar range resolution, ΔR is proportional to the pulse width with a matched filter, and is inversely proportional to the bandwidth. ΔR is defined as: cΔτ , 2 c , ΔR = 2B

ΔR =

(1) (2)

where ΔR is a unit of a length distance. c is the light speed. Δτ is the transmitted signal bandwidth. And B = 1/Δτ is pulse width, which is match to the received. The required radar range resolution is decided by the target measurement. When the FOD target size L is greater than or equal to ΔR, where L direct is parallel to or along to the direction of radar scanning field line, the millimeter-wave radar resolution is accept to detect FOD targets. The point targets or surface targets are shown as detected results. The second key performance indicator, measure distance R is defined as: Pt A2r σ 1 ]4 , 4πλ 2 Simin Pt G2 λ 2 σ 1 ]4 , Rmax = [ (4π )3 Simin Rmax = [

(3) (4)

where Rmax is the maximum distance. σ is the radar cross-section (RCS). Pt is the radar transmitter power. Ar is the effective area of the receiving antenna. G is the antenna gain power. Simin is the minimum detectable signal, which can be detected by radar. These two equations can be converted by the relationship of the antenna gain and the effective area, that is G = 4π A/λ 2 . But the effective loss of the equipment and the environmental noise does not consider factors in these two equations. By the way, the RCS parameter σ and the minimum recovery signal Simin be determined by a big experimental data. In the (3), (4) equations, all parameters are related to millimeter-wave radar, except RCS (σ ) which is depending on FOD targets. However, FOD targets are quite difference and wide range. There are different calculating methods for RCS (σ ) according to different attributes of FOD targets [4]. But the typical FOD objects are both on centimeter. Their sizes are larger than the wavelength of the millimeter wave radar. The RCS (σ ) is the geometric scattering optical projection area of FOD targets. Otherwise, there is a platform to install millimeter wave radar to overcome the limitation of canning bandwidth. When in automatic mode, the millimeter-wave radar scans to detect the runway in order to ensure the safety of the aircraft on the runway. The millimeter-wave radar to scan the entire runway time TS should be less than two times the aircraft pass a same runway. According to the frequency of flights taking off and landing at same runway, the rotational speed of platform Rθ with millimeter wave radar can be calculated by flights interval time Ti , which is taking off and landing on a same runway. Rθ can be defined as follows: Ti , (5) Rθ = 180◦ where Rθ is the horizontal angular velocity of platform. There is TS < Ti .

200

2.2


Data fusion of millimeter-wave radar and IR camera

Under the complex circumstance of some airports, especially for those busy ones, the landing and taking off operation occur frequently and airplanes slide down on the runway every a few minutes. Therefore, it is extremely high requirement for the FOD detect system to achieve quick real-time response of FOD target. Unfortunately, because of the large quantity of information of sequence image and long processing time of complicated detection algorithm, simple optical system is very difficult to satisfy these demands. In order to overcome the disadvantage of simple optical sensors and accelerate the detection speed of target, we unite millimeter-wave radar and IR camera together to establish the multi-sensor system. Firstly, some preparations and rules should be done. A coordinate system is established as shown in Fig. 2. The coordinate system includes airport runway and sensors, and the runway is divided into several areas which have been labeled by different marks, like A, B, C, etc. Then, a conversion of labeled mark (or the coordinate) to IR camera’s physical posture is created as showed in Table 1. The physical posture includes horizontal angle, vertical angle, zoom level, focal length, etc.

Fig. 2 the coordinate system of runway and sensors.

Table 1 table of millimeter-wave radar coordinate and camera posture hhhh Area hhh A B C hhhh Position hh h Sensors

θh1 Z1

θv1 f1

θh2 Z2

θv2 f2

θh3 Z3

θv3 f3

... ... ...

... ...

Secondly, we will discuss the workflow of data processing under the condition set above. Fig. 3 shows us the workflow of information fusion and data interaction, it consists five steps: Step 1: millimeter-wave radar scans the whole runway and detects any potential FOD targets. Step 2: if the radar detects FOD on the runway, turn to step3, otherwise turn to step 5. Step 3: calculating and obtaining the FOD’s coordinate and its belonging area mark. Step 4: according to the FOD’s coordinate and area’s mark, IR camera turns to the right position which aims at the central of FOD’s belonging area and takes original target IR image. Step 5: if there aren’t any reports from radar scanning, the IR camera turns to constant position which aims at each labeled area to take and also update each labeled area’s background IR images. 2.3

System design and its workflow

The FOD detection and surveillance system including camera system is implemented on a local IPC. The camera system used to capture the original images contains three sensors which are laser rangefinder, infrared camera and visible camera. The visible camera had a wide range zoom lens which can capture the fine images of FOD. The original visible images had a pixel size of 752*582 and the average pixel size of FOD is 20*20 which is good enough to extract the content features [7]. The pre-processing algorithm, feature extraction algorithm and threat level estimation model and method proposed in this


201

paper are realized by VC++ and OpenCV library in IPC. The overall system workflow with 4 basic unites is shown in the Fig. 4.

Fig. 3 The workflow of data fusion of millimeter-wave radar and IR camera.

Fig. 4 the procedures of threat level estimation of FOD.

3 The definition of FOD image feature The purpose of Intelligent FOD detection and surveillance system is not only to detect FOD target but also to analyze the threat and hazard automatically through various target recognition techniques. The system is required to send all FOD analysis results including FOD attribute and FOD hazard warnings to air traffic controllers in ATCT which may support the threat estimation and other corresponding procedures. Compare to other traditional recognition assignment like face, words and fingerprint, FOD appears more random and accidental and thus makes them very difficult to predict and recognize. 11 kinds of common FOD targets had been noticed in the FAA AC 150/5220-24, none of them had detailed size, material or color information. In another words, the accurate identification of single FOD target is meaningless in the real application.

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Based on many FOD hazard surveys and experiments, HSV color model and its histogram as color feature, Tamura texture as texture feature and Area, Perimeter, Eccentricity and Location as shape feature had been used to describe the FOD in this paper [12–15]. The relation between these features and FOD threat levels was discussed and quantified. As the result of no standards or agreements are used to determine the FOD, so the FOD threat level is seemed to be a subjective judgment and estimation and all the image features are corresponded with the human subjective cognition and feelings of FOD. This is also the core idea of the estimation method discussed in this paper. 3.1

Color feature

The color format of images captured by most cameras including ours is RGB model. As we know, the RGB model is easy to process by computer but difficult for human to understand. For example, most people cannot tell what is the color of one object whose RGB parameters are 200, 200 and 100. The RGB model does not correspond with the human’s understanding of color. Relatively the HSV model (Hue, Saturation and Value) is much closer to the human’s cognition and feeling [4]. So the HSV color histogram can be used to describe their appearance. The highest bin in the histogram can represent the main color of FOD with which respect to some inherent characteristic and attribute. For example, green FOD is more likely to be considered as plant and white FOD is more likely to be considered as plastic. Meanwhile, it is much easier to believe that the plant FOD is less likely to cause huge damage than plastic FOD. So the FOD and its threat level estimation had been classified by its color. The transition formulas from RGB to HSV are defined in (8): V = max(R, G, B), (6) (V − min(R, G, B)) × 255 if V = 0, (7) S= V ⎧ (−B + G) × 60 ⎪ ⎪ if R = V, ⎪ ⎪ ⎪ S ⎪ ⎪ ⎨ 180 + (−R + B) × 60 (8) H= if G = V, ⎪ S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 240 + (−G + R) × 60 if B = V. S According to the experiment and analysis, the histogram of hue, saturation and value had been divided into 10, 3 and 3 bins. The histogram was established by counting the number of pixels in each bin. And then the histogram was normalized into the region of [0, 1]. The histogram features can be obtained by the formula (9). hi refers to the sum number of pixels in each bin. A is the total number of pixels. h refers to the normalized histogram feature. hi = 3.2

hi A

i = 0, · · · , N − 1.

(9)

Texture feature

Texture, as an important visual feature that independent from the image color and intensity, could describe the homogeneous phenomena and inherent features of object surface. Texture feature contains large structure and organization information of object surface and their relation to the surrounding environment. In other words, texture represents the regularity and position rule of pixels. Texture feature had already applied in Computer Vision and Pattern Recognition technology for many years, but until now the definition of texture is widely discussed and no standard one is accepted. The human’s feeling of texture is a subjective psychological effects and analysis which often be described by static features. At the basis of psychological research about the human’s visual perception of texture, Prof. Tamura and his team proposed 6 parameters which are coarseness, contrast,


203

directionality, line likeness, regularity and roughness to describe texture feature [8, 9]. In this paper, the first three parameters also as the most important three the coarseness, contrast and directionality had been chosen as FOD image texture feature. The main reason we use Tamura texture to describe FOD is its analysis and definition of texture is also based on human’s subjective recognition and feeling which is conform to the core idea in this paper. The coarseness parameter is obtained by formulas (9)∼(13). Ak (x, y) =

x+2k−1 y+2k−1 −1

g(i, j) , 22k i=x−2k−1 j=y−2k−1

∑

∑

Ek,h (x, y) = |Ak (x + 2k−1 , y) − Ak (x − 2k−1 , y)|, Ek,v (x, y) = |Ak (x, y + 2k−1 ) − Ak (x, y − 2k−1 )|, Sopt (x, y) = 2k , Ek = Emax = max(E1,h , E1,v , . . . , E5,h , E5,v ), Fcrs =

1 m n ∑ ∑ Sopt (i, j). m × n i=1 j=1

(10)

(11) (12) (13)

Formula (9) is used to calculate the average intensity (Ak ) of pixels in one working mask which size is 2k *2k where k = 1, 2, . . . , 6. The g(i, j) refers to gray level of each pixel. Formula (11) is used to calculate the intensity difference (Ek,h , Ek,v ) both on horizontal and vertical direction. Compare all the intensity differences, we obtain the max difference and desired k value for optimized Sopt as it shows in (12). Finally, the coarseness parameter is obtained by Sopt in (13). The contrast parameter is obtained by the static distribution condition of intensity of pixels as it shown in formula (14) where α4 is defined as σμ44 . The μ4 refers to 4 order moment and σ refers to standard deviation. Parameter n is normally 14 . Fcom =

σ , (α4 )n

(14)

The directionality parameter is obtained by formulas (15) (16) (17). (|ΔH | + |ΔV |) , 2 ΔV π θ = tan−1 ( ) + , ΔH 2

|ΔG| =

(15)

n−1

HD (k) = Nθ (k)/ ∑ Nθ (i),

(16)

i=0

np

Fdir = 1 − rn p ∑

∑

p φ ∈Wp

(φ − φ p )2 HD (φ ).

(17)

The formula (3)–(10) is used to calculate the magnitude and direction of gradient of each pixel. ΔH and ΔV refer to horizontal and vertical weights convolved with mask shown in (18). ⎛ ⎞⎛ ⎞ −1 0 1 1 1 1 ⎝ −1 0 1 ⎠ ⎝ 0 0 0 ⎠. (18) −1 0 1 −1 −1 −1 The formula is used to calculate the probability histogram of edge direction angel. If the image had an obvious direction characteristic, the histogram HD (k) will reach one peak as ΔG is greater than

204


a threshold t. Threshold t will be discussed in the experiment analysis in Section 5. Finally, we can obtain the directionality parameter by calculate the sharpness of each peak histogram. In formula (17), n p refers to the number of peaks, Wp and φ p refers to width and middle of one peak. 3.3

Shape feature

Shape is an important feature and also the most sensitive one for human. People often guess, then analyze and finally recognize one object by its shape. So it is a wise choice to use shape feature to describe FOD targets and then analyzed them. There are two kinds of shape features. One is contour feature like perimeter and area. These two fundamental features are the most intuitive feeling for human and also the basis of other high-level shape features. Another one is district feature like recklikeness, circlelikeness and eccentricity. Recklikeness and circlelikeness can describe how much one object could fulfill its enclosing rectangle and circle which means they can tell us the shape of object’s contour. Does it have many bends and corners? Is the object a slim and straight one or a bend and squatty one? For eccentricity, it can describe the discreteness of surface of object. As shown in Fig. 5, a ball FOD, a mental slice FOD and a leaf branch FOD have distinguished shape characteristics including perimeter, area, rectlikeness, circlelikeness and eccentricity. These features can be used to analyze the attribute of FOD and its threat level.

Fig. 5 samples of experimental FOD

In this section, the unit of physical parameter of all shape features is dimensionless pixels. The perimeter parameter is summed the numbers of all the 8 adjacent pixels in contour and the area parameter is summed the numbers of all the pixels within the contour defined above. Contour extracted algorithm is not the main topic in this paper. The rectlikeness parameter is obtained by the formula (19). S0 refers to the area of FOD and SMER refers to the area of minimum bounding rectangle of FOD. The range of rectlikeness parameter is 0 to 1. For those slim and bend object, the rectlikeness parameter is very small and merely equal to zero. The circlelikeness parameter is obtained by the formula (20). P is the perimeter of FOD and S0 is the area of FOD. The Eccentricity parameter is obtained by the formula (21). A refers to the length of major axis of FOD and μi j refers to the central moments.

Fcircle Fecc 3.4

S0

. SMER P2 = , S0 (μ20 − μ02 )2 + 4μ11 . = A

Frect =

(19) (20) (21)

Location feature

The real location of FOD in the airport runway is the most important information that ATCT want to know and maybe is the conclusive term to raise the FOD hazard warning. The Multi-sensor surveillance system may have special equipment like laser rangefinders to obtain the distance and location information. But the FOD surveillance system without such device or the device is malfunctioning still should have the capability to get the location information of FOD from camera system once it was


205

pre-calibrated. The technology of obtaining FOD’s location parameter is the hardware’s assignment. In this paper, we assume the FOD appears randomly and the location feature will be given manually.

4 FOD threat level model 4.1

Weighted summation model

The threat level model proposed in this paper is a weighted summation method. The threat level coefficient of FOD is contributed by every single image feature coefficient. The formula is defined as (22) [16]. 12

CT = ∑ ωiCi .

(22)

i=1

The CT ,Ci , ωi refers to the threat level coefficient, each feature’s threat level coefficient and each feature’s weight which detailed in Table 2 Table 2 Weight of feature coefficient Feature

Hue

Satur

Value

Coare

Contr

Direc

Rect

Circle

Ecce

Loc

ωi

0.8

0.5

0.8

0.8

0.5

0.6

1.2

1.3

1.5

2.0

The weight of different feature in Table 2 is considered by two main rules. Firstly, the location of FOD or where the FOD appeared on the runway is still the most important things that people concerned. So the weight of threat level coefficient of location is the largest one. Secondly, the shape feature is much easier to get the attention, understand and analysis for humans meanwhile it is difficult to evaluate the threat level from the color and texture feature because they are vulnerable to the image noise, light intensity and other outside impact. After the threat level coefficient was calculated, the threat level was defined as Table 3. When all the ωi , the weight, take 1 the threat level coefficient will reach the top. So we equally define five threat levels. Level A means the most severe FOD threat. Table 3 Threaten level

4.2 4.2.1

Threaten level

A

B

C

D

E

coefficient

50-40

40-30

30-20

20-10

10-0

Definition of threat coefficient of image features Color feature

All color features had been normalized into [0, 1] region. The Hue feature has been divided into 6 intervals each of which represent one spectrum color range from red, yellow, green, cyan, blue to magenta. According to the FOD hazard reports in China airport from CAAC, most FODs reflecting red and blue spectrum color are plastic. Most green FODs are plants. The yellow FODs could be wood objects, painted airport devices and rusty metals dropped from airplane. The Saturation feature of most color FOD is higher than 0.5. At the same time, the saturation is low for most metal and PVC material objects which do not have any colors. The white FOD like PVC has higher numeral value in Value feature and the same to metal FOD because of the light reflection character. The lower district of one FOD could be the shallow. Under the discussion and analysis above,


206

all three threat level coefficients of color feature are defined in the Tab. 4 ∼ 6. Table 4 Threaten coefficient of color feature with Hue Hue

0-0.1

0.1-0.2

0.2-0.4

0.4-0.6

0.6-0.7

0.7-1

Color

Red

Yellow

Green

Cyan

Blue

magenta

Coefficient

3

5

1

1

3

3

FOD

plastic

wood, metal

plant

plant

plastic

plastic, metal

Table 5 Threaten coefficient of color feature with Saturation Saturation

0-0.2

0.2-0.5

0.5-1

Coefficient

5

4

2

FOD

metal, PVC

light color

deep color

Table 6 Threaten coefficient of color feature with Value

4.2.2

Value

0-0.4

0.4-0.8

Coefficient

2

5

0.8-1 3

FOD

metal, PVC

light color

deep color

Texture feature

Texture is an inherent feature of object which contains large information of structure and organization of surface and their relation to the surrounding environment. The difference of texture between rocks, metal, wood, textile and plant are extremely large. We use the texture feature to determine the FOD’s materiel which would play a significant role to evaluate the FOD threat level. The surface of textile and wood is rough so the coarseness parameter is higher compare with the smooth surface like metal and plastic. The contrast feature is varying under different light conditions. Normally, the metals which have good reflection character and PVCs which have white surface reflect medium contrast appearance. For the directionality, the experimental data shows metals and PVCs have better directionality than others. Tables 7 ∼ 9 provides the definition of coefficient of texture feature. Table 7 Threaten coefficient of texture feature with Coarseness Coarseness

0-5

5-8

Coefficient

2

5

8-15 3

FOD

others

metal, PVC

textile, wood

Table 8 Threaten coefficient of texture feature with Contrast Contrast

5-15

15-25

25-35

Coefficient

2

2

3

FOD

others

metal, PVC

textile, wood


207

Table 9 Threaten coefficient of texture feature with Directionality

4.2.3

Directionality

0-0.2

0.2-0.5

0.5-1

Coefficient

5

4

2

FOD

others

metal, PVC

textile, wood

Shape feature

Only the two features, area and perimeter, hardly can describe the detail and real shape of one FOD. For example, the metal slice chip caused Concorde accident in Paris airport 2000 has very long perimeter but small area and bending structure. As the result, we introduced rectlikeness, circle likeness and eccentricity to describe the shape and according to these we evaluate the threat level. The definition of coefficient of shape feature is shown in Tables 10 ∼ 12. Table 10 Threaten coefficient of shape feature with Rectlikeness Rectlikeness

0-0.2

0.5-0.9

0.9-1

Coefficient

2

1

1

FOD

irregular shape

rectangle FOD

circle or ball FOD

Table 11 Threaten coefficient of shape feature with Circle likeness Circle likeness

13-19

19-50

50-∼

Coefficient

1

2

3

FOD

circle or ball FOD

rectangle FOD

irregular Shape

Table 12 Threaten coefficient of shape feature with Eccentricity

4.2.4

Eccentricity

0-30

30-50

Coefficient

1

2

50-∼ 3

FOD

circle or ball FOD

metal, PVC

irregular Shape

Location feature

In order to define the coefficient of location feature, we divide the runway into 3 regions as it shown in Fig.6. According to the real location of FOD on the runway, we define that if the FOD appeared in the red areas which are landing and taking off area the coefficient of location feature is the highest. For other areas like yellow one and green one the coefficient decreased. The definition of coefficient of location feature is shown in Table 13. Table 13 Threaten coefficient of shape feature Location

RED AREA

YELLOW AREA

GREEN AREA

Coefficient

5

4

2

Area

landing and taking off area

sliding area

taxiway


208

Fig. 6 Threaten level division of airport runway.

5 Experiment analysis In this paper, we introduced 14 common FOD objects shown in Fig.7 and Table 14 as our experimental data to test our model. The images of these FOD were captured by our camera introduced in section 2 and had been enhanced and pre-processed before extracting features. As the location, all the location feature of experimental FOD was given randomly. Table 14 Attribute of experimental FOD No.

Attribute

No

Attribute

1

tennis ball

8

tapeline

2

metal hammer

9

plastic wheel

3

metal slice

10

PVC pipeline

4

sharp metal cylinder

11

PVC pipe

5

metal key

12

rocks

6

screw

13

rubber

7

leaf

14

wood

Fig. 7 Images of experimental FOD.

5.1

Experimental data of millimeter-wave radar

The typical experimental data of millimeter-wave radar are shown as Fig. 7. There are partly typical FOD objects in this scanner area among 14 types of FOD targets. All the location feature of experimental FOD was given randomly. The FOD targets are detected in area A by the millimeter-wave radar and get the target’s coordinate. These coordinate will be converted to specific IR camera positions.


209

In this experiment, there are 120 scan cycles. The typical 8 frames are selected and shown as Fig. 8. There are false targets in this serial scan as those moving and active week targets, such as live birds. These moving and active week targets were detected only one or two frames in radar imaging. So these false targets can be ignored by compared with those fixed FOD targets. Because of the limitation of the measure distance R and its resolution ΔR of millimeter-wave radar, we just can get the detected targets coordinate. It is hard to recognize or distinguishes what kind of FOD in millimeter-wave radar images.

Fig. 8 Experimental data of millimeter-wave radar.

5.2

The feature of experimental FOD

As it shown in Table 15, the color FODs has both high values in Saturation and Value, the metal object has very low saturation value and the PVC objects have low Saturation but extreme high Value. Table 15 Threaten coefficient of color feature No

1

2

3

4

5

6

7

8

9

10

11

12

13

14

H

.206

.871

.504

.267

.667

.439

.259

.306

.788

.402

.294

.090

0

.106

S

.682

.021

.151

.039

.035

.014

.675

.592

.173

.035

.114

.337

0

.541

V

.867

.396

.426

.613

.353

.576

.478

.725

.118

.953

.925

.604

.831

.812

Note: H- Hue, S- Saturation,V- Value. “.206” means “0. 206”.All numbers are less than 1.

Table 16 is the results of texture feature of all 14 experimental FOD. As we can see, the textile surface FOD like No. 1, tennis ball, has higher Coarseness value as well as the No. 14 FOD – the wood. Table 16 Threaten coefficient of texture feature No.

1

2

3

4

Coarseness Contrast Directionality

5

6

12.1544

8.8346

11.1992

28.5546

29.0773

16.3426

0.4828

0.2353

0.4548

7

9.5822

7.0995

5.9733

8.4400

9.7082

12.8860

7.2669

12.0838

0.5665

0.7196

0.2331

0.4165

No.

8

9

10

11

12

13

14

Coarseness Contrast

7.8605 20.0859

10.2463 35.0501

8.1355 29.9555

8.8867 25.0292

8.6429 9.4969

7.9762 13.8789

10.0533 26.5410

Directionality

0.1389

0.5773

0.4971

0.3245

0.5358

0.3712

0.3834


210

On the contrast, the lower result of coarseness of metal and PVC object just conform to their smooth surface. The result of location and shape feature are combined and shown in Table 17. The location feature is set randomly. Because of shooting angles of our camera to experimental FOD are almost perpendicular to the FOD surface, the recklikeness feature is nearly close to 1. However, the circlelikeness and eccentricity feature gives a good expression to the detailed shape of FOD. Table 17 Threaten coefficient of location feature and shape feature

5.3

No.

1

2

3

4

5

6

7

Location

A

B

C

A

B

C

A

Rectlikeness Circlelikeness

1 13.61

0.98 57.13

0.99 147.62

1 38.29

0.89 38.80

1 38.77

0.97 123.04

Eccentricity

22.4

38.6

93.4

32.4

27.4

29.3

79.4

No.

8

9

10

11

12

13

14

Location

B

C

A

B

C

A

B

Rectlikeness

1

1

1

1

0.97

1

0.99

Circlelikeness

32.47

19.70

78.05

27.27

54.57

80.38

39.36

Eccentricity

24.3

21.4

37.7

20.3

41.6

69.4

31.9

Threat level coefficient and threat level estimation

Based on the weighted summation model discussed in last section and the results of all extract features, we obtain the final FOD threat level coefficient and threat level. The threat level coefficient is shown in Table 18 and Fig. 9 which provides a sequence image of 14 FOD sorted by their threat level from high to low. Top 5 is a piece of leaf, a piece of rubber, a stick of PVC pipeline, a hammer and a block of woods. Table 18 Result of threaten level No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Threat level coefficient

29.6

39.6

36.3

32.5

28.9

27.3

42.8

33.3

25.5

40.6

29.5

32

40.7

37.7

Threat level

C

B

B

B

C

C

A

B

C

A

C

B

A

B

(7)

(13)

(8)

(4)

(10)

(12)

(1)

(2)

(11)

(14)

(5)

(6)

(3)

(9)

Fig. 9 Images of experimental FOD sorted by threat level.

No. 14 FOD, a piece of leaf, has highest threat level which is contributed by its location feature


211

founded in RED AREA and complicated contour shape character. The No. 3 FOD, a chip of metal, seems should be the highest threat as we see. But it appeared at the Yellow AREA so the final threat level coefficient was decreased after the weighted summation. The result is the same with our subjective estimation. The threat level of experimental FOD obtained by the proposed model is effective. 5.4

Comparison with BP neural network model

In order to verify the effect of proposed model, we introduce the artificial BP neural network to evaluate the threat level of experimental FOD which will provide a comparison between the proposed model in this paper and BP neural network models [10–11]. The BP neural network classifier was constructed within 3 layers, 10 inputs which are the image features and 1 output which is threat level coefficient. There are 4 nodes in each layer. The transfer functions of input and output layer are tansig function and training function is Levenberg-Marquardt method [17]. A survey with several questionnaires that ask random people to tell their subjective feeling about the threat level of experimental FOD is pre-accomplished. The results of threat level of this survey are used as the expectation outputs of BP neural network in training procedure. One quarters of experimental FOD will be used as training samples and the rest as test samples. Based on the BP neural network classifier, the fourteen kinds of FOD feature data were used to test threat level as in Table 14 and Table 18. There are three-quarters of the training sample, 11 training samples, and 3 test samples in different experiment. In first experiment, the No. 11, 12, 13 FOD were selected as test samples, the others for training samples. These results of FOD threaten level estimation shown in Figure 10. In this figure, the error of threats coefficient are no more than 15%, the estimated threat level has a same level to the real threat. In the same way, the No. 1,4,10 FOD were selected as test samples, the others for training samples. The experimental results are shown in Figure 11. These two experimental results show that the FOD threat model is effective. In Fig. 10, Fig. 11, we can understand it is a scatterplot. The horizontal coordinate indicates the FOD sample, including test samples and training samples from No.1 to No.14. It is easy to analysis and read the tendency by compared with test value is true value with difference dash line and real-line

Fig. 10 Experiment result of FOD threaten level estimation (No. 11, 12, 13 as test samples, others as training samples).

212


while ordinate coordinate indicates threaten coefficient. A general result, three FOD feature data were selected as test samples randomly, with BP neural network and the proposed threat model. The error of threats coefficient are not greater than 15%. The threat level of estimate is closed to the real threat level. However, the average training samples costing time and the average test samples costing time are 5ms and 22ms respectively, with VC++ and OpenCV library in IPC. In conclusion, the results demonstrate that the proposed model had good effect and can meet the real-time requirement.

Fig. 11 Experiment result of FOD threaten level estimation (No. 1, 4, 10 as test samples, others as training samples).

6 Conclusion In this paper, a FOD recognition method and FOD threat level estimation model is proposed and discussed in detail. The proposed model is to stimulate the human’s subjective cognition mode and judgment to FOD, at the same time the content image features of FOD are used to describe the mode. The coefficient of each features were defined based on carefully analysis to large FOD data. After that, a weighted summation model is established and applied to calculate and estimate the threat level of 14 common experimental FOD. An artificial BP neural network classifier is introduced to verify the performance of new model. The experiment result shows the proposed method can recognize FOD and obtain FOD threat level effectively. And the proposed model has low computational complexity and high real-time execution which could easily implement to real application.

Acknowledgments The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helps to improve the quality of this paper greatly. This paper was supported by the National Basic Research Program of China (2014CB744903), National Natural Science Foundation of China (60904096), China Aviation Science Fund (2012ZC15005), China Aviation Support Program (61901060202), the China AVIC industrial projects (CXY2012SJ37) and China Scholarship Council (201208310570).


213

References [1] Jim Patterson, Jr (2008), Foreign Object Debris (FOD) detection research, International Airport Review, 23(2), 22–27. [2] FAA. AC 150/5220-24(2009), Airport Foreign Object Debris Detection Equipment, 1–13. [3] Steffi Baumgarten (2014), Case studies international-Incidents at international airports, ZRH Safety Newsletter FOD hazards, 2014(4), 10. [4] Beasley, Patrick. Tarsier (2007), Unique radar for helping to keep Debris off Airport Runways, the Future of Civil Radar 2006, The Institution of Engineering and Technology Seminar, London, 12–22. [5] Feil P. (2008), Foreign Objects Debris Detection (FOD) on Airport Runways Using a Broadband 78 GHz Sensor, Proceedings of the 38th European Microwave Conference, 1608–1611. [6] Ding, Lufei et al (2009), Radar Principles (fourth edition), Electronic Industry Press, Beijing. [7] Ferri, M., Giunta, G., Banelli, A., and Neri, D.(2009), Millimetre Wave Radar Applications to Airport Surface Movement Control and Foreign Object Detection, Proceedings of the 6th European Radar Conference. Roma, 1–8. [8] Rafael, C., Gonzalez, Richard E. Woods (2006), Digital Image Processing (Second Edition), Publishing House of Electronic Industry, 290–344. [9] Smith, J. and Chang, S. (1995), Tools and techniques for colour image retrieval, SPIE Proceedings: Storage & Retrieval for Image and Video Databases IV, 2670, 426–437. [10] Charl Coetzee, et al. (1998), PC Based Number Plate Recognition Systems, In Proc. IEEE International Symposium on Industrial Electronics, 605–610. [11] Lippmann, R. (1989), Pattern Classification Using Neural Networks, IEEE Communication on Mag, 7, 47-64. [12] Veltkamp, R. and Tanase, M. (1999), State-of-the-art in shape matching, Technical Report UU-CS-1999-27, Utrecht University, the Netherlands, 105–110. [13] Huang, T.S. and Yong, R. (1997), Image retrieval: past present and future, International Symposium on Multimedia Information Processing, Taiwan, 45–55. [14] Gunsel, B. and Tekalp, A. (1998),Shape similarity matching for query by example, Pattern Recognition, 3l (7), 931–944. [15] Persoon, E. and Fu, K.S. (1986), Shape discrimination using Fourier descriptors, IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 388–397. [16] Yu Li, Gang Xiao (2011), Airport runway foreign object detection system design and research, Laser and infrared, 41(8), 909–915. [17] Jorge J. Moré (1978), The Levenberg-Marquardt algorithm: Implementation and theory, Lecture Notes in Mathematics, 630, 105–116.

Book review Chaos, Information Processing and Paradoxical Games, The Legacy of John S Nicolis Edited by: Gregoire Nicolis (University of Brussels, Belgium), Vasileios Basios (University of Brussels, Belgium) World Scientific, 468pp, February 2015, Hardback: 9789814602129 The volume provides a self-contained survey of the mechanisms presiding information processing and communication. The main thesis is that chaos and complexity are the basic ingredients allowing systems composed of interesting subunits to generate and process information and communicate in a meaningful way. Emphasis is placed on communication in the form of games and on the related issue of decision making under conditions of uncertainty. Biological, cognitive, physical, engineering and societal systems are approached from a unifying point of view, both analytically and by numerical simulation, using the methods of nonlinear dynamics and probability theory. Epistemological issues in connection with incompleteness and self-reference are also addressed. Key Features • Highlights the universality, relevance and interdisciplinary dimension of chaos and complexity • Brings together topics and issues that have so far been addressed independently of each other and establishes unexpected connections • Provides complementary coverage of problems of concern as viewed by different well-established experts Contents: Preface Part I. Glimpses at Nonlinear Dynamics & Chaos, pp. 1 Chapter 1. Bohmian Trajectories in the Scattering Problem, by G Contopoulos, N. Delis and C. Efthymiopoulos, pp. 3-25. Chapter 2. Scaling Properties of the Lorenz System and Dissipative Nambu Mechanics, by Minos Axenides and Emmanuel Floratos, pp. 27-41. Chapter 3. Extreme Events in Nonlinear Lattices, by G. P. Tsironis, N. Lazarides, A. Maluckov and Lj Hadžievski, pp. 43-62. Chapter 4. Coarse Graining Approach to Chaos, by Donal MacKernan, pp. 63-86. Chapter 5. Fractal Parameter Space of Lorenz-like Attractors: A Hierarchical Approach, by Tingli Xing, Jeremy Wojcik, Michael A Zaks and Andrey Shilnikov, pp. 87-104. Part II. Chaos and Information, pp. 105 Chapter 6. Quantum Theory of Jaynes' Principle, Bayes' Theorem, and Information, by Hermann Haken, pp. 107-116. Chapter 7. Information Processing with Page–Wootters States, by Stam Nicolis, pp. 117-126. Chapter 8. Stochastic Resonance and Information Processing, by C. Nicolis, pp. 127-139. Chapter 9. Selforganization of Symbols and Information. By Werner Ebeling and Rainer Feistel, pp. 141-184. Part III. Biological Information Processing, pp. 185. Chapter 10. Historical Contingency in Controlled Evolution, by Peter Schuster, pp. 187-220. Chapter 11. Long-Range Order and Fractality in the Structure and Organization of Eukaryotic Genomes, by Dimitris Polychronopoulos, Giannis Tsiagkas, Labrini Athanasopoulou, Diamantis Sellis and Yannis Almirantis, pp. 221-252. Chapter 12. Towards Resolving the Enigma of HOX Gene Collinearity, by Spyros Papageorgiou, pp. 253-273. Part IV. Complexity, Chaos & Cognition, pp. 275. Chapter 13. Thermodynamics of Cerebral Cortex Assayed by Measures of Mass Action, by Walter J. Freeman, pp. 277-300. Chapter 14. Describing the Neuron Axons Network of the Human Brain by Continuous Flow Models, by J. Hizanidis, P. Katsaloulis, D. A. Verganelakis and A. Provata, pp. 301-318. Chapter 15. Cognition and Language: From Apprehension to Judgment - Quantum Conjectures, by F. T. Arecchi, pp. 319-343. Chapter 16. Dynamical Systems++ for a Theory of Biological System, by Kunihiko Kaneko, pp. 345-354. Chapter 17. Logic Dynamics for Deductive Inference - Its Stability and Neural Basis, by Ichiro Tsuda, pp. 355-373. Part V. Dynamical Games and Collective Behaviours, pp. 375. Chapter 18. Microscopic Approach to Species Coexistence Based on Evolutionary Game Dynamics, by Celso Grebogi, YingCheng Lai and Wen-Xu Wang, pp. 377-382. Chapter 19. Phase Transitions in Models of Bird Flocking, by H. Christodoulidi, K. van der Weele, Ch G. Antonopoulos and T. Bountis, pp. 383-398. Chapter 20. Animal Construction as a Free Boundary Problem: Evidence of Fractal Scaling Laws, by S. C. Nicolis, pp. 399-409. Chapter 21. Extended Self Organised Criticality in Asynchronously Tuned Cellular Automata, by Yukio-Pegio Gunji, pp. 411-430. Part VI. Epilogue, pp. 439. A Posthumous Dialogue with John Nicolis: IERU, by Otto E Rössler, pp. 433-437. Part VII. Appendix: Selected References from John Nicolis' Bibliography, pp. 441-448. Index pp. 449-455. Readership: Graduate students, researchers, and academics from various fields interested in chaos, information processing and complexity science.

Book review Newtonian Nonlinear Dynamics for Complex Linear and Optimization Problems Authors: Luis Vázquez and Salvador Jiménez Publisher: Springer Science-Business Media New York 2013, ISBN 978-1-4614-5911-8 In a Pinball Machine, the player tries to score points by manipulating a metal ball on a playing field inside a glass covered case. The objectives of the game are to score as many points as possible, to earn free games and to maximize the time spent playing by earning extra balls and keeping balls in play as long as possible. Apart from the new challenging features, the good old pinball playing field is essentially a planar surface inclined upwards from 3 to 7 ◦, away from the player, and includes multiple targets and scoring objectives. The ball is put into play by the use of the plunger which propels upwards the ball. Once the ball is in play, it tends to move downwards towards the player, although the ball can move in any direction, sometimes unpredictably, as the result of contact with objects on the playing field or by the player actions. To return the ball to the upper part of the playing field, the player makes use mainly of one or more flippers. The game ends whenever the ball crosses downwards the “flippers barrier”. The Pinball Machine provides a simple mechanical example of the linear optimization problem, basically in a surface embedded in the three-dimensional space. In all pinball games, the play with every ball finishes when that ball reaches the minimum gravitational potential energy immediately after the flippers barrier. On the other hand, the playing field where the ball moves is, effectively, a convex planar region. The duration of the ball motion is always finite, even considering the human interaction. This fact indicates that the minimum of the Objective Function (in this case, the Potential Gravitational Energy) is always attained by the motion of the ball. This example suggests us to associate the solutions of some optimization problems to the motion of Newtonian particles. At the same time, this example is a bridge that allows us to construct algorithms for linear / nonlinear optimization problems and unconstrained extrema by applying to them the numerical algorithms used to simulate the equations of motion for a Newtonian particle. These are the motivation and the objective of this monograph. The framework of this monograph wants to be constructive: we want to present some methods and their features that show how Newton’s equation for the motion of one particle in classical with finite difference methods can create a mechanical scenario within which we may solve some basic, though complex, problems. We, thus, apply these ideas to solve linear systems and eigenvector problems, as well as programming, both linear and nonlinear, in different dimensions, in the spirit of the suggestive books by Mordecai Avriel (“Nonlinear Programming. Analysis and Methods”) and John T. Betts (“Practical Methods for Optimal Control and Estimation Using Nonlinear Programming”). For this latter case, the goal of the monograph is to show a breakthrough analysis method of optimization by combining the features of the motion of a Newtonian classical particle and finite difference numerical algorithms associated with the equation of motion. Many challenging questions remain open, but a new, fresh and feasible approach to solve them is shown. This unified numerical and mechanical approach is new and it represents a simple but useful tool not yet fully exploited. This monograph is intended for a broad public: undergraduate and graduate students or researchers who are confronted in their work with linear systems and eigenvalue or optimization problems and who are open to new perspectives in the way these problems can be addressed. To help the reader to explore these ideas, the authors propose a list of related exercises at the end of each chapter. The basic mechanical equations and assumptions are presented in Chap. 1: basic laws for the motion of a particle under Newton’s second law, in one and several dimensions, with and without dissipation. Different cases, depending on the acting potential, are presented. Also two numerical schemes are analysed to simulate the corresponding equations of the motion. All this material should be thoroughly used in the sequel as basic building blocks with which to construct methods to solve the proposed problems, ranging from linear algebra to nonlinear programming. In Chap. 2 it is proposed a new iterative approach to solve systems of linear equations. The new strategy integrates the algebraic basis of the problem with elements from classical mechanics and the finite difference method. The approach defines two families of convergent iterative methods. Each family is characterized by a linear differential equation, and the methods are obtained from a suitable finite difference scheme to integrate the associated differential equation. These methods are general and depend on neither the matrix dimension nor the matrix structure. The basic features of each method are studied. As a consequence, it is presented a general method to determine whether a given square matrix is singular or not. In Chap. 3 the developed methods are applied to several examples. These are compared with other similar characteristics, such as Jacobi, Gauss–Seidel, and Steepest Descent Methods and several aspects are discussed about choosing the parameter values for the numerical methods. In Chap. 4 the computation of eigenvectors and eigenvalues of matrices is developed. For a general square matrix, not necessarily symmetric, it is constructed a family of dynamical systems whose state converges to eigenvectors which correspond to eigenvalues with smallest and biggest real part. The convergence and several numerical tests are analysed. Besides, the authors extend the application of the method to the effective computation of all eigenvalues with intermediate real part. Some examples and comparisons with the Power Methods are presented in Chap. 5. Some ways to enhance the linear convergence of the method are developed by combining it with two different quadratic methods. In Chaps. 6 and 7 the above ideas are applied to solve the so-called programming problems. Chapter 6 is devoted to the classical linear programming problem. A new iterative process is proposed to approach the solution of the Primal Problem associated with the linear programming problem: maxZ = CTx, with some linear constraints. The method is based on translating the problem to the motion of a Newtonian particle in a constant force field. The optimization of the objective function is related to the search for the minimum of the particle’s potential energy. Several solution strategies which depend on the number of dimensions are developed and also illustrated through different examples. The monograph comes to an end in Chap. 7, which is devoted to the classical quadratic programming: the previous method is extended to the case of optimizing a quadratic objective function with linear constraints as well as to the case of a linear function with quadratic constraints. The method can also be extended to the general case of a nonlinear objective function with linear constraints. Professor Dumitru Baleanu, Institute of Space Sciences Magurele-Bucharest, Romania

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Aims and Scope The interdisciplinary journal publishes original and new research results on applied nonlinear dynamics in science and engineering. The aim of the journal is to stimulate more research interest and attention for nonlinear dynamics and application. The manuscripts in complicated dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in nonlinear dynamics and engineering nonlinearity. Topics of interest include but not limited to • • • • • • • • • • • •

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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]

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Journal of Applied Nonlinear Dynamics Volume 4, Issue 2

June 2015

Contents Study of the Effect of the Coupling in a Dispersion-managed Dual Core Optical Fiber Using the Collective Variables Approach Roger Bertin Djob, Aurèlien Kenfack-Jiotsa, and Timolèon Crèpin Kofane…………......

101－116

Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems Y. Boukal, M. Darouach, M. Zasadzinski, and N.E. Radhy…………….…………………

117－130

A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth Wesson and Richard Rand…...............................................................................

131－140

Stagnation Point Solution Due to a Continuously Stretching Surface with Applied Magnetic Field using HAM R. Seshadri and S.R. Munjam…….….…………………………...………………..………

141－152

Longitudinal Dimensions of Polygon-shaped Planetary Waves Ranis N. Ibragimov and Guang Lin…..……………..…...………………………………

153－167

Conservation Laws of a Gardner Equation with Time-dependent Coefficients M.S. Bruzon, M.L. Gandarias, and R. de la Rosa………...……………..……………...…

169－180

Influence of Diffusion on Bio-chemical Reaction of the Morphogenesis Process M. Sambath and K. Balachandran …………………………………..…….……………..

181－195

Recognition and Threat Level Estimation of FOD Based on Image Content Features and Experiment Analysis Gang Xiao, Yu Li, Xiao Yun, and Jinhua Xie……...….…………………….……………..

197－213

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