Discontinuity, Nonlinearity, and Complexity

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Jun 2, 2015 - Centre de Physique Théorique, Aix-Marseille Université, CPT. Campus de Luminy ...... [23] Hirsch, M.W. (1997), Differential Topology, Springer, New York. [24] Cox, D., Little, ...... Email: maurice.courbage@univ-paris- diderot.fr.
Volume 4 Issue 2 June 2015

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

An Interdisciplinary Journal of

Discontinuity, Nonlinearity, and Complexity

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Continued on back materials

An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 2, June 2015

Editors Valentin Afraimovich Xavier Leoncini Lev Ostrovsky Dimitry Volchenkov

L&H Scientific Publishing, LLC, USA

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Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 111–119

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

Poincar´e Recurrences in the Circle Map: Fibonacci Stairs V.S. Anishchenko†, N.I. Semenova, and T.E. Vadivasova Saratov State University - 83 Astrakhanskaya str., Saratov, Russia, 410012 Submission Info Communicated by Dimitri Volchenkov Received 17 September 2014 Accepted 2 October 2014 Available online 1 July 2015 Keywords Poincare recurrences Circle map Afraimovich-Pesin dimension Rotation number

Abstract We show that the dependence of the mimimal Poincar´e return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the “Fibonacci stairs”. The rigorous result for the Afraimovich-Pesin dimension equality αc = 1 is confirmed for irrational rotation numbers with the measure of irrationality μ = 2. It is shown that some transcendental number are Diophantine and have the measure μ = 2. It is also confirmed that the gauge function 1/t cannot be applied for Liouvillian numbers. All the obtained features hold for both the linear and the nonlinear circle map. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The Poincar´e recurrence is one of the fundamental features pertaining to the time evolution of dynamical systems. Recurrence, according to Poincar´e, implies that practically any phase trajectory leaving a point x0 of the phase space will pass infinitely many times arbitrarily close to its initial state as it evolves in time. Poincar´e termed this type of motion in dynamical systems stable in the Poisson sense [1]. The statistics of Poincar´e recurrences in the global approach has been a topic of research in recent years [2–4]. The local approach idea of the Poincar´e recurrrence theory consists in calculating Poincar´e recurrences to some ε -vicinity of the given initial state [5, 6]. In the framework of the global approach, recurrence times are considered in all covering elements of the whole set and their statistics is then studied. The main characteristic of the Poincar´e recurrence statistics in the global approach is the dimension of return times, which has been introduced in [4] and called as the Afraimovich-Pesin dimension (AP dimension). It has been established that the statistics of return times in the global approach depends on the topological entropy hT . Poincar´e recurrences have been studied theoretically for mixing sets with hT > 0 [2–4] and the theoretical results have been confirmed by numerical simulation [7–9]. The situation is different in the case of ergodic sets without mixing, i.e., when hT = 0. There are practially no works devoted to studies of the properties of Poincar´e recurrences for this case. Some important rigorous results have been obtained in [2–4] for the shift circle (the linear circle map) with an irrational rotation number. In this work, we aim to study numerically the Poincar´e recurrence statistics in the linear and the nonlinear circle map with different irrational rotation † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.001

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numbers and to reveal new peculiarities of Poincar´e recurrences. It is widely used in nonlinear dynamics to study quasiperiodic oscillations with two independent frequencies. 2 Basic rigorous results In the global approach, the whole set of phase trajectories of a dynamical system is covered with cubes (or balls) of size ε  1. A minimal time of the first recurrence of a phase trajectory in the ui -vicinity τinf (ui ) is calculated for each covering element ui (i = 1, 2, . . . , m). Then the mean minimal return time is defined over the whole set of covering elements ui [9–11] as follows: τinf (ε ) =

1 m ∑ τinf (ui ). m i=1

(1)

It has been shown in [2] that in the general case d

τinf (ε ) ∼ φ −1 (ε αc ),

(2)

where d is the fractal dimension of the considered set, and αc is the AP dimension of a return time sequence. The gauge function φ (t) in (2) can be given by one of the following forms: 1 φ (t) ∼ , φ (t) ∼ exp(−t), φ (t) ∼ exp(−t 2 ), . . . . t

(3)

The appropriate choice of φ (t) depends on the topological entropy hT , as well as on the multifractality of the considered set if such a property exists. It has been shown in [2, 4] that for hT = 0, the gauge function has an asymptotic form φ (t) ∼ 1/t and the following expression holds d

τinf (ε ) ∼ ε − αc , ε  1 .

(4)

For chaotic systems, we have hT > 0 and φ (t) ∼ exp(−t), and thus, the following law is valid [10]: τinf (ε ) ∼

d ln ε , ε  1 αc

(5)

3 Model We consider an example of a minimal set, which is generated by the circle map [12]: Θn+1 = Θn + Δ + K sin Θn ,

mod 2π ,

(6)

where Δ and K are the parameters of the map. The circle maps of type (6) are reference models of a wide class of dynamical systems with quasiperiodic behaviour with two independent frequencies whose ratio defines the rotation number. We first study the shift circle (6) for K = 0. In this case, the ergodic set {Θn } is characterized by zero topological entropy hT = 0, the rotation number ω = Δ/2π is irrational, and the distribution p(θ ) is uniform on the interval 0 ≤ Θn ≤ 2π . For example, the ergodic set {Θn } can be realized in a nonautonomous oscillator if its stroboscopic section is considered through the period of an external force. It has been shown in [2, 4] that the AP dimension of the set {Θn } essentially depends on the rotation number and the methods of its definition. An irrational rotation number ω can be approximated by a ratio of two integers. Additionally, the rate of irrational number approximations can be different. In terms of the convergence rate of rational approximations, irrational numbers can be divided into Diophantine and Liouvillian numbers.

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The approximation error obeys the inequality:  m  C  ω −  < μ , n n

113

(7)

where m/n is a convergent, which is formed by pairs of numbers from the natural number set, 2 ≤ μ < ∞ is the measure of irrationality, and C is a constant. An irrational rotation number ω is said to be Diophantine if the upper bound of values of μ for each of which the inequality (7) has infinitely many solutions is finite. Otherwise, ω is a Liouvillian number. We consider the case of Diophantine approximations of ω . Following [2, 4], (7) can be written as follows:  1 m   ω − (8)  ≥ [ν (ω )+1+δ ] ,  n n where ν (ω ) = sup{v(ω )} is a maximal rate of Diophantine approximations for an irrational number over all possible pairs of m and n, and δ > 0. It has been proven for (8) that τinf (ε ) ∼ ε

− ν (1ω )

, or lnτinf (ε ) ∼ −

1 ln ε . ν (ω )

(9)

Here we take into account that the fractal dimension d = 1. When (9) is compared with (5), it is appearant that in this case ν (ω ) = αc , i.e., the AP dimension coincides with the rate of approximations of the irrational rotation number ω . Furthermore, the approximation (gauge) function is φ (t) ∼ 1/t. 4 Numerical results We now calculate the dependence τinf (ε ) for the set generated by the circle map (6) for K = 0 and with the rotation number ω equal to the golden ratio: √ 1 √ ω = Δ/2π = ( 5 − 1) ≈ 0.618 . . . , Δ = π ( 5 − 1). (10) 2 Since a point rotates uniformly on a circle, there is only one value of τinf (ε ) which is independent of the initial state Θ = Θ0 for any interval ε . Hence, we do not need to calculate the mean value (1). Calculation results are shown in Fig. 1. The ln τinf (ε ) dependence looks like a step function, which can be referred to as the ”Fibonacci stairs”. It has the following features: 1. A sequence of τinf (ε ) increases with decreasing ln ε and fits the basic Fibonacci series (. . ., 8, 13, 21, 34, 55, 89, 144, . . .). Each subsequent number is the sum of the previous two. The values of τinf are indicated in Fig. 1,a. 2. When ε is varied within any of the stairs steps, three return times τ1 < τ2 < τ3 can be distinguished. Additionally, τ3 = τ1 + τ2 and τ1 = τinf . These return times agree the basic Fibonacci series. This property follows from Slater’s theorem [13]. 3. It has been established numerically and proven theoretically that lengths and heights of all steps of the Fibonacci stairs (see Fig. 1,a) have a universal property. The length of the ith step can be defined as follows: (11) Δi ε = ln εi − ln εi+1 = const = − ln ω , i 1, where εi and εi+1 are the values of ε on the boundaries of the corresponding step. ¿From (11) it follows that εi+1 = εi ω . Similarly, for the height of the ith step we have: Δi τinf (ε ) = ln τinf (εi+1 ) − ln τinf (εi ) = − ln ω , i 1.

(12)

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The index i in (11) and (12) grows with decreasing ε (see Fig. 1,a). Thus, the height and length of all the stairs steps shown in Fig. 1,a are universal and equal to − ln ω , where ω is the rotation number (see Appendix). To calculate τinf (ε ) a specially selected discrete sequence of εk [13, 14] can be used as vicinities of returns. We consider the points on the circle, which are formed when one iterates from the initial point Θ0 for rational approximations of the rotation number ωk = mk /nk , k = 1, 2, . . ., where mk /nk is a convergent of a continued fraction. We choose an interval εk which includes only the initial point Θ0 . The boundaries of this interval are determined by denominators of the kth and the (k + 1)th convergent, while its size and the minimal return time to this interval can be found as follows [14]:

εk = 2π (

1 1 + ), τinf = nk . nk nk−1

(13)

Using this method and applying simple transformations, it can be easily shown that (11) is valid. Thus, the set of τinf (εk ) can be obtained by calculating only the denominators nk of rational approximations. The corresponding data are shown in Fig. 1,b. This dependence is a straight line with unit slope. From (9) and (4) it follows that in this case

ν (ω ) = αc = 1.

(14)

Thus, both the AP dimension and the rate of rotation number approximations are equal to 1 in the case of the golden ratio. √ Similar results are obtained for the silver ratio ω = 2 + 1. The universal properties (11) and (12) of the Fibonacci stairs are also valid here. The differences are that the sequence of minimal return times for the silver ratio obeys the Pell law nk+1 = 2nk + nk−1 and forms the series: . . ., 29, 70, 169, 408, 985, . . . [15]. Calculating the sequence of vicinities, we plot the dependence ln τinf (Δi ε ), which is also a straight line with slope −1. Thus, as for the golden ratio, the AP dimension and the rate of approximation ν (ω ) are equal to 1 for the silver ratio. The universal geometry of the Fibonacci stairs is valid only for the golden and silver ratios and results from their expansion in a continued fraction. This universal property is violated for other irrational numbers. The golden and silver ratios are algebraic Diophantine numbers with the measure of irrationality μ = 2 (7). It has been proven that any algebraic irrational number has μ = 2 [16]. This means that expressions (8), (9) and ν (ω ) = αc = 1 are valid for any number which is a root of the real equation: CN ω N + · · · +C1 ω +C0 = 0.

(15)

√ We take ω 3 − 2 = 0, ω = 3 2 and perform calculations as in the previous two cases. The numerical results depicted in Fig. 2 show that a simple geometry of the Fibonacci stairs is broken down. The lengths and heights of the steps are different when the size of the ε -vicinity is varied. However, calculated both by averaging directly the Fibonacci stairs and by estimating theoretically the intervals Δi ε (11), the AP dimension is αc = ν (ω ) = 1.0 ± 0.01. We are now interested in transcendental or Liouvillian irrational numbers. Transcendental numbers have the measure of irrationality μ ≥ 2 and Liouvillian numbers have μ → ∞. It is known that all Liouvillian numbers are transcendental but the opposite is generally false. Let us consider an example of a transcendental number ω = π . Our calculations show that the equality ν (ω ) = αc = 1.0 is also valid in this case. This means that π is a Diophantine number with the measure of irrationality μ = 2. Calculation results of the dependence of ln δ on ln n, where δ = ln |ω − mn |, are shown in Fig. 3(a) for different values of the rotation number. The value of μ is defined as a slope of the linear approximation of the obtained dependences. This gives us μ = 2 for the first three cases (curves 1–3 in Fig. 3(a)). Thus, the data in Fig. 3(a) confirm the fact that π is a Diophantine number, although it is transcendental. Similar results have been obtained for transcendental numbers ω = e, ln 2 and corroborated the property described for ω = π .

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115

√ Fig. 1 (a) Dependence of the minimal return time τinf (ε ) for ω = ( 5 − 1)/2, K = 0, and Δ = 2πω in the map (6); (b) dependence τinf (εk ) calculated using (13).

√ Fig. 2 Dependence τinf (ε ) for ω = 3 2 and K = 0 in the map (6). Curve 1 shows the Fibonacci stairs, curve 2 is a linear approximation of the curve 1, curve 3 shows calculation results using (13).

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√ Fig. 3 (a) The measure of irrationality μ for ω = ( 5 − 1)/2 (curve 1), ω = π (curve 2), theoretical approximation ln δ = −2 ln n (curve 3), and ω = λ (curve 4); (b) dependence τinf (ε ) for ω = λ .

Finally, we consider the case of a Liouvillian irrational number. It can be given, for example, by ∞

ω = λ = ∑ 10−i! .

(16)

i=0

As follows from Fig. 3(a) (curve 4), μ → ∞ for ω = λ . The dependence of ln τinf on ln ε is shown in Fig. 3,b for ω = λ . It can be seen that the function φ (t) ∼ 1/t cannot be considered in this case as a gauge function because the points indicated in Fig. 3(b) do not lie on a straight line. Unfortunately, we were not able to find an appropriate form of the gauge function φ (t) for ω = λ due to the finite computer accuracy. We have only found that (8) and (9) do not hold for Liouvillian irrational numbers. This fact confirms the theoretical results [2, 4]. 5 Nonlinear case We now consider the map (6) for 0 < K < 1, when the system is nonlinear. For K > 0, the distribution density p(Θ) is not uniform on the interval 0 ≤ Θ ≤ 2π . In this case, the whole area must be divided into intervals ui and τinf (ui ) is calculated for each partition element. Then the mean value τinf (ε ) is found using (1). We choose the rotation number ω to be equal to the √ golden ratio. When K is varied, ω also changes. Therefore, for different fixed K, Δ is chosen so that ω = ( 5 − 1)/2. Calculation results for lnτinf (ε ) are shown in Fig. 4 for three different values of the parameter K.

V.S. Anishchenko et al. / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 111–119

117

√ Fig. 4 Dependence lnτinf (ε ) for the map (6) at ω = ( 5 − 1)/2 and for different values of K: K = 0 (curve 1), K = 0.1 (curve 2), and K = 0.6 (curve 3)

. As can be seen from Fig. 4, as K increases, the Fibonacci stairs is gradually smoothed and tends to a straight line. The mean value of the slope of all the three plots is −1. This implies that for the nonlinear circle map (6), the AP dimension αc also coincides with the rate ν (ω ) and is equal to 1. We note an important point. It is known that in the case of an irrational rotation number, the nonlinear circle map (6) can be transformed (or reduced) to the linear circle map ψn+1 = ψn + Δ by using a suitable nonlinear change of variable ψ = g(θ ) [12, 17]. This change must satisfy the property g(θ + 2π ) = 2π + g(θ ). The result described above for the AP dimension αc = 1, which has been obtained both for the linear and the nonlinear circle map, testifies that the AP dimension is an invariant with respect to the nonlinear change of variables. 6 Conclusions The detailed study has first revealed that the dependence of the minimal return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the “Fibonacci stairs”. It has been shown numerically that for Diophantine numbers with the measure of irrationality μ = 2, the AP dimension αc coincides with the rate of Diophantine approximations ν (ω ) (8) and is equal to 1. This conclusion is valid for the circle map (6) in both the linear (K = 0) and the nonlinear (K > 0) case. Our results confirm the theoretical conclusion [2, 4] and indicate that the gauge function φ (t) ∼ 1/t enables one to estimate the AP dimension for minimal sets on the circle. Since Diophantine numbers form a set of full measure on any interval, the function 1/t can be the most appropriate for irrational shifts on the circle. However, this function φ (t) ∼ 1/t cannot be used in the case of Liouvillian rotation numbers with μ → ∞. The established laws may be useful to diagnose a quasiperiodic mode in dynamical systems, including experimental data analysis. 7 Appendix We prove the third universal property of the Fibonacci stairs, which is expressed by (11) and (12). Several ways can be used to prove the validity of (11) and (12). We give one of them. Taking into account that the rotation number for the golden ratio is ω = nk−1 /nk (k 1), we transform (13) [14] into

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εk = 2π (

nk−1 nk nk−1 + nk−2 ω + ) = 2π ( + ). nk nk−1 nk nk−1 nk−1 nk nk−1

(17)

The Fibonacci numbers satisfy the property nk = nk−1 + nk−2 , so from (17) we have

εk = 2πω (

1 nk−1

+

1 2π 2π 2π . )+ = ωεk + , or εk = nk nk nk nk (1 − ω )

(18)

Calculating Δk ε = ln(εk /εk+1 ) with (18) gives Δk ε = − ln ω . This implies that the lengths of the Fibonacci stairs steps are universal in the limit of ε → 0. The proof of (12) follows from Fig. 1,a. Since the minimal return times for the golden ratio are the Fibonacci numbers, we have Δi τinf = ln(ni+1 /ni ) = − ln ω , i 1. The validity of (11) and (12) can be proven similarly for the silver ratio. In this case, it must be taken into account that the sequence of growing minimal return times obeys the Pell law. It follows from the given proof that the universality of (11) and (12) strictly holds for ε → 0. However, from an experimental standpoint, the lengths and heights of the steps differ by the value that is less than 10−3 . The reason of this is a high rate of approximation of an irrational number (7). Acknowledgments The reported study was partially supported by RFBR, research project No. 15-02-02288. N.I. Semenova gratefully acknowledges the Dynasty Foundation. References [1] Nemytskii, V.V. and Stepanov, V.V. (1989), Qualitative Theory of Differential Equations, Dover Publications, New York. [2] Afraimovich, V. (1997), Pesin’s dimension for Poincar´e recurrences, Chaos, 7, 12–20. [3] Afraimovich, V.V. and Zaslavsky, G. (1997), Fractal and multifractal properties of exit times and Poincar´e recurrences, Physical Review E, 55, 5418–5426. [4] Afraimovich, V., Ugalde, E., and Urias, J. (2006), Fractal Dimension for Poincar´e Recurrences, Elsevier. [5] Kac, M. (1947), On the notion of recurrence in discrete stochastic processes, Bulletin of the American Mathematical Society, 53, 1002–1010. [6] Hirata, M., Saussol, B., and Vaienti, S. (1999), Statistics of Return Times: A General Framework and New Applications, Communications in Mathematical Physics, 206, 33–55. [7] Penn´e, V., Saussol, B., and Vaienti, S. (1997), Fractal and statistical characteristics of recurrence times, Talk at the Conference isorder and Chaos, Rome, September 1997, preprint CPT. [8] Anishchenko, V.S., Astakhov, S.V., Boev, Ya.I., Biryukova, N.I., and Strelkova, G.I. (2013), Statistics of Poincar´e recurrences in local and global approaches, Communications in Nonlinear Science and Numerical Simulation, 18, 3423–3435. [9] Anishchenko, V.S. and Astakhov, S.V. (2013), Poincar´e recurrence theory and its applications to nonlinear physics, Physics – Uspekhi, 56(10), 955–972. [10] Afraimovich, V., Lin, W.W., and Rulkov, N.F. (2000), Fractal dimension for Poincar´e recurrences as an indicator of synchronized chaotic regimes, International Journal of Bifurcation and Chaos, 10, 2323–2337. [11] Anishchenko, V., Khairulin, M., Strelkova, G., and Kurths, J. (2011), Statistical characteristics of the Poincare return times for a one-dimensional nonhyperbolic map, The European Physical Journal B, 82, 219–225. [12] Pikovsky, A., Rosenblum, M., and Kurths, J. (2002), Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press. [13] Slater, N.B. (1967), Gaps and steps for the sequence n mod 1, Proceedings of Cambridge Philosophical Society, 4, 1115–1123. [14] Buric, N., Rampioni, A., and Turchetti, G. (2005), Statistics of Poincar´e recurrences for a class of smooth circle maps, Chaos, Solitons and Fractals, 23, 1829–1840. [15] Bicknell, M. (1975), A primer on the Pell sequence and related sequences, Fibonacci Quarterly, 13, 345–349.

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[16] Roth, K. (1955), Rational Approximations to Algebraic Numbers, Mathematika, 2, 1–20. [17] Denjoy, A. (1932), Sur les courbes d´efinies par les e´ quations diff´erentielles a` la surface du tore, Journal de Math´ematiques Pures et Appliqu´ees, 11, 333–375.

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A Semi-analytical Prediction of Periodic Motions in Duffing Oscillator Through Mapping Structures Albert C.J. Luo1† and Yu Guo2 1 Department of Mechanical and Industrial

Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA 2 McCoy School of Engineering, Midwestern State University, Wichita Falls, TX 76308, USA Submission Info Communicated by Valentin Afraimovich Received 12 September 2014 Accepted 11 November 2014 Available online 1 July 2015 Keywords Duffing oscillator Discrete implicit maps Bifurcation trees Periodic motions

Abstract In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction For solutions of periodic motions in nonlinear dynamical systems, analytical and numerical techniques are adopted. The analytical methods include the method of averaging, perturbation methods, harmonic balance method, and generalized harmonic balance method. Through the analytical methods, one can obtain the analytical expressions of approximate solutions of periodic motions in dynamical systems. The numerical methods are based on discrete maps obtained by discretization of differential equations for dynamical systems. The discrete maps include explicit and implicit maps. The explicit maps can be directly used to obtain numerical solutions of differential equations for dynamical systems, but the computational errors will be accumulated in numerical results. Once the iteration times become large, the numerical results may not be adequate for numerical solutions of dynamical systems. In this paper, one of specific implicit maps will be used to develop mapping structures for periodic motions. Based on the mapping structures, the methodology for analytical prediction of † Corresponding

author. Email address: [email protected], [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.002

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periodic motions in nonlinear dynamical systems will be presented. The Duffing oscillator will be investigated as a sample problem to show how to use the method presented in this paper for an analytical prediction of the bifurcation trees of periodic motions in nonlinear dynamical systems. Before discussion of this methodology, a brief history of existing techniques for periodic motions in nonlinear systems is discussed. The analytical methods for periodic motions are discussed first. Such an issue started in 1788 from Lagrange [1] for investigating the three-body problem as a perturbation of the two-body problem by the method of averaging. In the end of the 19th century, Poincare [2] further developed the perturbation theory for the motions of celestial bodies. In 1920, van der Pol [3] used the method of averaging to determine the periodic solutions of oscillation systems in circuits. Until 1928, the asymptotic validity of the method of averaging was not proved. Fatou [4] gave the proof of the asymptotic validity through the solution existence theorems of differential equations. In 1935, Krylov and Bogoliubov [5] further developed the method of averaging for nonlinear oscillations in nonlinear vibration systems. In 1964, Hayashi [6] presented the perturbation methods including averaging method and principle of harmonic balance. In 1969, Barkham and Soudack [7] extended the Krylov-Bogoliubov method for the approximate solutions of nonlinear autonomous second-order differential equations (also see, Barkham and Soudack [8]). In 1987, a generalized harmonic balance approach was used by Garcia-Margallo and Bejarano [9] to determine approximate solutions of nonlinear oscillations with strong nonlinearity. Rand and Armbruster [10] used the perturbation method and bifurcation theory to determine the stability of periodic solutions. In 1989, Yuste and Bejarano [11] used the elliptic functions rather than trigonometric functions to improve the Krylov-Bogoliubov method. In 1990, Coppola and Rand [12] used the averaging method with elliptic functions to determine approximation of limit cycle. In 2012, Luo [13] developed an analytical method for analytical solutions of periodic motions in nonlinear dynamical systems. Luo and Huang [14] applied such a method to a Duffing oscillator for approximate solutions of periodic motions, and Luo and Huang [15] gave the analytical bifurcation trees of period-m motions to chaos in the Duffing oscillator. The analytical bifurcation trees of period-m motion to chaos in the Duffing oscillator with twin-well potentials were presented in Luo and Huang [16, 17]. With extensive applications of computers, numerical computations become very popular to obtain numerical results for differential equations through discretization. Once are obtained the discrete maps for dynamical systems, discrete dynamical systems can be used to investigate nonlinear dynamics of dynamical systems. Based on nonlinear maps, one discovered the existence of chaotic motions in nonlinear dynamical systems through iteration of discrete maps. In 2005, Luo [18,19] presented a mapping dynamics of discrete dynamical systems which is a more generalized symbolic dynamics. The systematical description of mapping dynamics in discontinuous dynamical systems was presented in Luo [20]. The discrete maps can be arbitrary implicit and explicit functions rather than explicit maps in numerical iterative methods only. From discrete mapping structures, periodic motions in discrete dynamical systems can be predicted analytically, and the stability and bifurcation analysis of periodic motions in nonlinear dynamical systems can be completed. Such an idea was applied to discontinuous dynamical systems in Luo [20, 21]. In 2014, Luo [22] developed a semi-analytical method to determine periodic motions in nonlinear dynamical systems through discrete implicit maps. Herein, this new method will be used to analytically predict the bifurcation trees of periodic motions in the Duffing oscillator. In this paper, periodic motions in the Duffing oscillator will be investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, periodic motions will be predicted through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motions will be performed. The bifurcation trees of periodic motions will be also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed. The harmonic amplitude spectrums will be presented to show how many harmonic terms for each periodic motion can provide a good approximation, and the corresponding analytical expression of periodic motions can be given approximately.

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2 Methodology Periodic motions in dynamical systems will be presented herein. If a nonlinear system has a periodic motion with a period of T = 2π /Ω, then such a periodic motion can be expressed by a discrete points through discrete mappings of continuous dynamical systems. The method is stated through the following theorem. From Luo (2014), we have the following theorem. Theorem 1. Consider a nonlinear dynamical system x˙ = f(x,t, p) ∈ R n ,

(1)

where f(x,t, p) is a Cr -continuous nonlinear vector function (r ≥ 1). If such a dynamical system has a periodic motion x(t) with finite norm x and period T = 2π /Ω, then there are a set of values of time tk (k = 0, 1, . . . , N) with (N  1) on a time interval (t0 , T + t0 ) and a set of points xk , such that x(tk ) − xk  ≤ εk with a small εk ≥ 0 and (2) f(x(tk ),tk , p) − f(xk ,tk , p) ≤ δk , with a small δk ≥ 0. Furthermore, there exists a vector function gk with gk (xk−1 , xk , p) = 0(k = 1, 2, . . . , N),

(3)

that determines a general implicit mapping Pk : xk−1 → xk (k = 1, 2, . . . , N). The particular form of the function gk is determined by a particular computational scheme which one uses to study the system in Eq. (1). Consider a mapping structure as P = PN ◦ PN−1 ◦ · · · ◦ P2 ◦ P1 : x0 → xN ; (4) with Pk : xk−1 → xk (k = 1, 2, . . . , N). For xN = Px0 , if there is a set of nodes points x∗k (k = 0, 1, . . . , N) computed by gk (x∗k−1 , x∗k , p) = 0, (k = 1, 2, . . . , N)

(5)

x∗0 = x∗N ,

then the points x∗k (k = 0, 1, . . . , N) are approximations of points x(tk ) of the periodic solution. In the neighborhood of x∗k , with xk = x∗k + Δxk , the linearized equation is given by Δxk = DPk · Δxk−1 with gk (x∗k−1 + Δxk−1 , x∗k + Δxk , p) = 0

(6)

(k = 1, 2, . . . , N). The resultant Jacobian matrices of the periodic motion are DPk(k−1)···1 = DPk · DPk−1 · · · · · DP1 , (k = 1, 2, . . . , N);

(7)

DP ≡ DPN(N−1)···1 = DPN · DPN−1 · · · · · DP1 , where



∂ xk DPk = ∂ xk−1

 (x∗k−1



∂ gk =− ∂ xk ,x∗ ) k

−1 (x∗k−1 ,x∗k )



∂ gk ∂ xk−1

 (x∗k−1 ,x∗k )

.

(8)

The eigenvalues of DP and DPk(k−1)···1 for such a periodic motion are determined by |DPk(k−1)···1 − λ¯ In×n | = 0, (k = 1, 2, . . . , N); |DP − λ In×n | = 0.

(9)

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Thus, the stability and bifurcation of the periodic motion can be classified by the eigenvalues of DP(x∗0 ) with o m o ([nm 1 , n1 ] : [n2 , n2 ] : [n3 , κ3 ] : [n4 , κ4 ]|n5 : n6 : [n7 , l, κ7 ])

(10)

o where n1 is the total number of real eigenvalues with magnitudes less than one (n1 = nm 1 + n1 ), n2 is the total m o number of real eigenvalues with magnitude greater than one (n2 = n2 + n2 ), n3 is the total number of real eigenvalues equal to +1; n4 is the total number of real eigenvalues equal to −1; n5 is the total pair number of complex eigenvalues with magnitudes less than one, n6 is the total pair number of complex eigenvalues with magnitudes less than one, n7 is the total pair number of complex eigenvalues with magnitudes equal to one.

(i) If the magnitudes of all eigenvalues of DP are less than one, the approximate periodic solution is stable. (ii) If at least the magnitude of one eigenvalue of DP is greater than one, the approximate periodic solution is unstable. (iii) The boundaries between stable and unstable periodic motion give bifurcation and stability conditions. To explain how to approximate the periodic motion in an n-dimensional nonlinear dynamical system, consider an n1 ×n2 plane (n1 +n2 = n), as shown in Fig. 1. N-nodes of the periodic motion are chosen for an approximate solution with a certain accuracy x(tk ) − xk  ≤ εk (εk ≥ 0) and |f(x(tk ),tk , p) − f(xk ,tk , p) ≤ δk (δk ≥ 0). Letting δ = max{δk }k∈{1,2,...,N} and ε = max{εk }k∈{1,2,...,N} be small positive quantities prescribed, the periodic motion can be approximately described by a set of specific implicit mappings Pk with gk (xk−1 , xk , p) = 0(k = 1, 2, . . . , N) with a periodicity condition xN = x0 . Based on the approximate mapping functions, the nodes of the trajectory of periodic motion are computed approximately, which is depicted by a solid curve. The exact solution of the periodic motion is described by a dashed curve. The node points on the periodic motion are depicted with short lines. The symbols are node points on the exact solution of the periodic motion. The discrete mapping Pk is developed from the differential equation. With the control of computational accuracy, the nodes of the periodic motion can be obtained with a good approximation. From the stability and bifurcation analysis, the period-1 motion under period T = 2π /Ω, based on the set of discrete implicit mapping Pk with gk (xk−1 , xk , p) = 0(k = 1, 2, . . . , N), is stable or unstable. If the perioddoubling bifurcation occurs, the periodic motion will become a periodic motion under period T = 2T , and such a periodic motion is called the period-2 motion. Due to the period-doubling, 2N nodes of the period-2 motion will be employed to describe the period-2 motion. Thus, consider a mapping structure of the period-2 motion with 2N implicit mappings. P = P2N ◦ P2N−1 ◦ · · · ◦ P2 ◦ P1 : x0 → x2N ; with Pk : xk−1 → xk (k = 1, 2, . . . , 2N).

(11)

For x2N = Px0 , there is a set of points x∗k (k = 0, 1, . . . , 2N) computed by the following implicit vector functions gk (x∗k−1 , x∗k , p) = 0, (k = 1, 2, . . . , 2N) x∗0 = x∗2N .

(12)

After period-doubling, the period-1 motion becomes period-2 motion. The nodes points increase to 2N points during two periods (2T ). The period-2 motion can be sketched in Fig. 2. The node points are determined through the discrete implicit mapping with a mathematical relation in Eq.(12). On the other hand, T = 2T =

2(2π ) 2π Ω = ⇒ω = . Ω ω 2

(13)

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125

Fig. 1 Period-1 motion with N-node points. Solid curve: numerical results with N-nodes marked by short lines, and dashed curve: expected exact results with N-nodes marked by symbols. The local shaded area is a small neighborhood of the exact solution at the kth node. The symbols are node points on the exact solution of the periodic motion.

Fig. 2 Period-2 motion with 2N-nodes. Solid curve: numerical results. The symbols are node points on the periodic motion.

During the period of T , there is a periodic motion, which can be described by node points xk (k = 1, 2, . . . , N ). Since the period-1 motion is described by node points xk (k = 1, 2, . . . , N) during the period T , due to T = 2T , the period-2 motion can be described by N ≥ 2N nodes. Thus the corresponding mapping Pk is defined as (2)

(2)

Pk : xk−1 → xk (k = 1, 2, . . . , 2N), and

(2)∗

(2)∗

gk (xk−1 , xk (2)∗

x0

(2)∗

= x2N .

, p) = 0, (k = 1, 2, . . . , 2N),

(14)

(15)

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In general, for period T = mT , there is a period-m motion which can be described by N ≥ mN. The corresponding mapping Pk (m) (m) Pk : xk−1 → xk (k = 1, 2, . . . , mN), (16) and

(m)

(m)∗

(m)∗

gk (xk−1 , xk (m)∗

, p) = 0, (k = 1, 2, . . . , mN),

(17)

(m)∗

= xmN .

x0

From the above discussion, the period-m motion in a nonlinear dynamical system can be described through (mN + 1) nodes for period mT . As in Luo (2014), the corresponding theorem is presented as follows. Theorem 2. Consider a nonlinear dynamical system in Eq. (1). If such a dynamical system has a period-m motion x(m) (t) with finite norm x(m)  and period mT (T = 2π /Ω), then there are a set of values of time tk (k = (m) (m) 0, 1, . . . , mN) with (N  1) on a time interval [t0 , mT +t0 ] and a set of points xk , such that x(m) (tk )−xk  ≤ εk with a small εk ≥ 0 and (m) (18) f(x(m) (tk ),tk , p) − f(xk ,tk , p) ≤ δk , with a small δk ≥ 0. Furthermore, there exists a vector function gk with (m)

(m)

gk (xk−1 , xk , p) = 0,

k = 1, 2, . . . , mN,

(m)

(19)

(m)

that determines a general implicit mapping Pk : xk−1 → xk (k = 1, 2, . . . , mN). The particular form of the function gk is determined by a particular computational scheme which one used to study the system in Eq. (1). Consider a mapping structure as (m)

(m)

P = PmN ◦ PmN−1 ◦ · · · ◦ P2 ◦ P1 : x0 → xmN ; (m)

(m)

with Pk : xk−1 → xk (m)

(m)

(m)∗

(k = 0, 1, . . . , mN) computed by

(m)∗

, p) = 0,

For xmN = Px0 , if there is a set of points xk (m)∗

gk (xk−1 , xk (m)∗

(20)

(k = 1, 2, . . . , mN).

(k = 1, 2, . . . , mN),

(21)

(m)∗

= xmN ,

x0 (m)∗

then the points xk (k = 0, 1, . . . , mN) are approximations of points x(m) (tk ) of the periodic solution. In the (m)∗ (m) (m)∗ (m) neighborhood of xk , with xk = xk + Δxk , the linearized equation is given by (m)

Δxk

(m)

= DPk · Δxk−1 , (m)∗

(m)

(m)∗

with gk (xk−1 + Δxk−1 , xk

(m)

(22)

+ Δxk , p) = 0,

(k = 1, 2, . . . , mN). The resultant Jacobian matrices of the periodic motion are DPk(k−1)···1 = DPk · DPk−1 · . . . · DP1 ,

(k = 1, 2, . . . , mN);

(23)

DP ≡ DPmN(mN−1)···1 = DPmN · DPmN−1 · . . . · DP1 , where

 DPk =

(m)

∂ xk

(m) ∂ xk−1



 =− (m)∗ (m)∗ (xk−1 ,xk )

∂ gk (m)

∂ xk

−1 

 ∂ gk  (m)  ∂ xk−1 

. (m)∗

(m)∗

(xk−1 ,xk

)

(24)

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150 (m)∗

The eigenvalues of DP(x0

127

) and DPk(k−1)···1 for such a periodic motion are determined by |DPk(k−1)···1 − λ¯ In×n | = 0,

(k = 1, 2, . . . , mN);

(25)

|DP − λ In×n | = 0. (m)∗

Thus, the stability and bifurcation of the periodic motion can be classified by the eigenvalues of DP(x0 o m o ([nm 1 , n1 ] : [n2 , n2 ] : [n3 , κ3 ] : [n4 , κ4 ]|n5 : n6 : [n7 , l, κ7 ]).

) with (26)

(i) If the magnitudes of all eigenvalues of DP(m) are less than one (i.e., |λi | < 1, i = 1, 2, . . . , n), the approximate period-m solution is stable. (ii) If at least the magnitude of one eigenvalue of DP(m) is greater than one (i.e., |λi | > 1, i ∈ {1, 2, . . . , n}), the approximate period-m solution is unstable. (iii) The boundaries between stable and unstable period-m motions give bifurcation and stability conditions. 3 Discretization and mapping structures Consider the Duffing oscillator as:

x¨ + δ x˙ − α x + β x3 = Q0 cos Ωt.

(27)

The state equation of the above equation in state space is x˙ = y and y˙ = Q0 cos Ωt − δ x˙ + α x − β x3 .

(28)

The differential equation in Eq. (28) can be discretized by a midpoint scheme for the time interval t ∈ [tk−1 ,tk ] to form a map Pk (k = 1, 2, . . .) as Pk : (xk−1 , yk−1 ) → (xk , yk ) ⇒ (xk , yk ) = Pk (xk−1 , yk−1 ),

(29)

with the implicit relation as 1 xk = xk−1 + h(yk−1 + yk ), 2 1 1 1 1 yk = yk−1 + h[Q0 cos Ω(tk−1 + h) − δ (yk−1 + yk ) + α (xk−1 + xk ) − β (xk−1 + xk )3 ]. 2 2 2 8 3.1

(30)

Period-1 motions

To predict the periodic solution in such a Duffing oscillator analytically, consider a mapping structure as P = PN ◦ PN−1 ◦ . . . ◦ P2 ◦ P1 : (x0 , y0 ) → (xN , yN ),  

(31)

N−actions

with

P1 : (x0 , y0 ) → (x1 , y1 ) ⇒ (x1 , y1 ) = P1 (x0 , y0 ), P2 : (x1 , y1 ) → (x2 , y2 ) ⇒ (x2 , y2 ) = P2 (x1 , y1 ), .. . PN−1 : (xN−2 , yN−2 ) → (xN−1 , yN−1 ) ⇒ (xN−1 , yN−1 ) = PN−1 (xN−2 , yN−2 ), PN : (xN−1 , yN−1 ) → (xN , yN ) ⇒ (xN , yN ) = PN (xN−1 , yN−1 ).

(32)

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For tk = t0 + kh with given t0 and h, from Eq. (30), the corresponding algebraic equations are ⎫ ⎪ ⎬

1 x1 = x0 + h(y0 + y1 ), 2

for P1 ; 1 1 1 1 ⎪ ⎭ 3 y1 = y0 + h[Q0 cos Ω(t0 + h) − δ (y0 + y1 ) + α (x0 + x1 ) − β (x0 + x1 ) ], 2 2 2 8 .. . ⎫ 1 ⎪ ⎪ xk = xk−1 + h(yk−1 + yk ), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ 3 ⎪ yk = yk−1 + h[Q0 cos Ω(tk−1 + h) − δ (yk−1 + yk ) + α (xk−1 + xk ) − β (xk−1 + xk ) ], ⎪ ⎪ ⎬ 2 2 2 8 .. . ⎪ ⎪ ⎪ 1 ⎪ ⎪ xN = xN−1 + h(yN−1 + yN ), ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ 3 yN = yN−1 + h[Q0 cos Ω(tN−1 + h) − δ (yN−1 + yN ) + α (xN−1 + xN ) − β (xN−1 + xN ) ]. ⎭ 2 2 2 8

(33)

With periodicity conditions, we have (xN , yN ) = (x0 , y0 ).

(34)

From Eqs. (33) and (34), values of nodes for the discretized Duffing oscillator can be determined by 2(N + 1) equations. Such a periodic solution can be sketched in Fig. 3. The node points are depicted by the circular symbols, labeled by xk = (xk , yk )T (k = 0, 1, 2, . . . , N), and the initial and final points are equal for periodicity. The mappings are depicted through the curves with arrows. Once the node points of the period-1 motion x∗k (k = 0, 1, 2, . . . , N) are obtained, the stability of period-1 motion can be discussed by the corresponding Jacobian matrix. Consider a small perturbation in the neighborhood of x∗k , xk = x∗k + Δxk , (k = 0, 1, 2, . . . , N). For the mapping structure in Eq. (31), we have ΔxN = DPΔx0 = DPN · DPN−1 · . . . · DP2 · DP1 Δx0 ,  

N -muplication with



 ∂ x1 Δx0 , Δx1 = DP1 Δx0 ≡ ∂ x 0 (x∗ ,x∗ ) 1 0   ∂ x2 Δx1 , Δx2 = DP2 Δx1 ≡ ∂ x1 (x∗ ,x∗ ) 2 1 .. .   ∂ xN−1 ΔxN−2 , ΔxN−1 = DPN−1 ΔxN−2 ≡ ∂ xN−2 (x∗ ,x∗ ) N−1 N−2   ∂ xN ΔxN−1 ; ΔxN = DPN ΔxN−1 ≡ ∂ xN−1 (x∗ ,x∗ ) N

where  DPk =

and

(35)

∂ xk ∂ xk−1



∂ xk ⎢ ∂ xk−1 =⎢ ⎣ ∂y k (x∗k ,x∗k−1 ) ∂ xk−1



(36)

N−1

⎤ ∂ xk ∂ yk−1 ⎥ ⎥ for k = 1, 2, . . . , N ∂ yk ⎦ ∂ yk−1 (x∗k ,x∗k−1 )

(37)

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129

Fig. 3 Period-1 motion with N-nodes of the Duffing oscillator. The mapping structures are depicted through single mappings with the arrowed curves. The circular symbols represent the node points of the period-1 motion.

∂ xk ∂ xk−1 ∂ xk ∂ yk−1 ∂ yk ∂ xk−1 ∂ yk ∂ yk−1

1 ∂ yk = 1+ h , 2 ∂ xk−1 1 ∂ yk = h(1 + ), 2 ∂ xk−1 1 1 = −2(1 + δ h + Δh)−1 Δ, 2 2 1 1 1 1 = (1 + δ h + Δh)−1 (1 − δ h − Δh); 2 2 2 2

(38)

with

1 Δ = h[−4α + 3β (xk−1 + xk )2 ]. 8 To measure stability and bifurcation of period-1 motion, the eigenvalues are computed by |DP − λ I| = 0,

where



∂ xN DP = ∂ x0

 (x∗N ,x∗N−1 ,...,x∗0 )

= DPN · · · · · DP2 · DP1 =

(39)

(40) 1



k=N



∂ xk ∂ xk−1

 (x∗k ,x∗k−1 )

.

(41)

Owing to the 2-dimensional mapping, there are two eigenvalues. From Luo (2012), the stability of period-1 motions can be given as follows: (i) If the magnitudes of two eigenvalues are less than one (i.e., |λi | < 1, i = 1, 2), the period-1 motion is stable. (ii) If one of two eigenvalue magnitudes are greater than one (i.e., |λi | > 1, i ∈ {1, 2}), the period-1 motion is unstable. For the bifurcation conditions, we have the following statements. (i) If λi = 1, i ∈ {1, 2} and |λ j | < 1, j ∈ {1, 2} but j = i, the saddle-node bifurcation of period-1 motion occurs.

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(ii) If λi = −1, i ∈ {1, 2} and |λ j | < 1, j ∈ {1, 2} but j = i, the period-doubling bifurcation of period-1 motion occurs. For the stable period-doubling bifurcation, the period-doubling periodic motion will be observed. (iii) If |λ1,2 | = 1 with λ1,2 = α ±iβ , the Neimark bifurcation of period-1 motion occurs. For the stable Neimark bifurcation, the quasiperiodic motion relative to the period-1 motions will be observed. To measure the variation characteristics of node point xk with the initial condition x0 , we have |DPk(k−1)···1 − λ (k) I| = 0, where

 DPk(k−1)···1 =

∂ xk ∂ x0



(42) 1

(x∗k ,x∗k−1 ,...,x∗0 )

= DPk · · · · · DP2 · DP1 = ∏



l=k

∂ xl ∂ xl−1

 (x∗l ,x∗l−1 )

.

(43)

The dynamics characteristics of xk in the neighborhood of x∗k varying with the initial condition of x0 in the neighborhood of x∗0 can be discussed as follows: (k)

(i) If the magnitudes of two eigenvalues are less than one (i.e., |λi | < 1, i = 1, 2), the node point xk in the neighborhood of x∗k with variation of x0 will approach to x∗k for the period-1 motion. (k)

(ii) If one of two eigenvalue magnitudes are greater than one (i.e., |λi | > 1, i ∈ {1, 2}), the node point xk in the neighborhood of x∗k with variation of x0 will move away from x∗k for the period-1 motion. 3.2

Period-m motions

Once the period-doubling bifurcation of the period-1 motions occurs, the period-2 motions will appear. If the period-doubling bifurcation of the period-2 motion occurs, the period-4 motions will appear, and so on. In addition, other periodic motions will exist. In general, to predict the period-m motions in such a Duffing oscillator analytically, consider a mapping structure as follows (m)

(m)

(m)

(m)

(m)

(m)

P = PmN ◦ PmN−1 ◦ · · · ◦ P2 ◦ P1 : (x0 , y0 ) → (xmN , ymN ),  

mN -actions with

(m)

(m)

(m)

(m)

(m)

(44)

(m)

Pk : (xk−1 , yk−1 ) → (xk , yk ) ⇒ (xk , yk ) = Pk (xk−1 , yk−1 ),

(45)

(k = 1, 2, . . . , mN). From Eq. (30), the corresponding algebraic equations are (m) xk (m) yk

⎫ ⎪ ⎪ ⎬

1 (m) (m) h(y + yk ), 2 k−1

=

(m) xk−1 +

=

(m) yk−1 + h[Q0 cos Ω(tk−1 +

⎪ 1 1 1 1 (m) (m) (m) (m) (m) (m) ⎪ h) − δ (yk−1 + yk ) + α (xk−1 + xk ) − β (xk−1 + xk )3 ], ⎭ 2 2 2 8

(k = 1, 2, . . . , mN).

for Pk

(46)

The corresponding periodicity conditions are (m)

(m)

(m)

(m)

(xmN , ymN ) = (x0 , y0 ).

(47)

From Eqs. (46) and (47), values of nodes at the discretized Duffing oscillator can be determined by 2(mN + 1) (m)∗ equations. Once the node points xk (k = 1, 2, . . . , mN) of the period-m motion is obtained, the stability of

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

131

period-m motion can be discussed by the corresponding Jacobian matrix. For a small perturbation in vicinity of (m)∗ (m) (m)∗ (m) xk , xk = xk + Δxk , (k = 0, 1, 2, . . . , mN). we have (m)

(m)

(m)

ΔxmN = DPΔx0 = DPmN · DPmN−1 · . . . · DP2 · DP1 Δx0 .  

mN -muplication with

 (m) Δxk

=

(m) DPk Δxk−1



(m)

∂ xk



(m) ∂ xk−1

(k = 1, 2, . . . , mN), where

⎡  DPk =

(m)

∂ xk

(m)

∂ xk−1

 (m)∗

(xk

(m)∗

,xk−1 )

(m)

∂ xk

⎢ (m) ⎢ ∂x ⎢ k−1 = ⎢ (m) ⎢ ∂ yk ⎣ (m)

∂ xk−1

(m)

∂ xk

(48)

(m)

(m)∗

(xk

(m)∗

Δxk−1 ,

(49)

,xk−1 )



(m) ⎥ ∂ yk−1 ⎥ ⎥ (m) ⎥ ∂ yk ⎥ ⎦ (m) ∂ yk−1

for k = 1, 2, . . . , mN. (m)∗

(xk

(50)

(m)∗

,xk−1 )

To measure stability and bifurcation of period-m motion, the eigenvalues are computed by |DP − λ I| = 0, where

 DP =

(m)

∂ xmN

(m)

∂ x0

(51)

 = DPmN · · · · · DP2 · DP1 = (m)∗

(m)∗

(m)∗

(xmN ,xmN−1 ,...,x0

1





k=mN

)

(m)

∂ xk

(m)

∂ xk

 . (m)∗

(xk

(52)

(m)∗

,xk−1 )

Similarly, the stability and bifurcation conditions are the same as for the period-1 motion. 4 Analytical predictions From the foregoing section, the node points of periodic motions for the Duffing oscillator can be computed, and the set of node points of periodic motions with N + 1 points per period T = 2π /Ω are defined as

∑ = {(xk , yk )|tk = t0 + kT /N;t0 = 0; T = 2π /Ω; k = 0, 1, 2, . . .}.

(53)

The periodicity of period-m motion is (xk , yk ) = (xk+mN , yk+mN ). From the analytical prediction of the node points of periodic motion, the FFT can provide the harmonic amplitudes and phases, which will be presented in this section. To avoid presenting all node points of periodic motions, the node points relative to the initial condition point for each period are collected in the Poincare mapping section for period-m motions (m = 1, 2, . . .), as defined by Σm = {(x mod (k,N) , y mod (k,N) )|tk = t0 + kT /N;t0 = 0; T = 2π /Ω; k = 0, 1, 2, . . .},

(54)

which will be used to present the periodic motions. In this section, analytical predictions of both the bifurcation trees of period-1 motions to chaos and period-3 motions to chaos in the Duffing oscillator will be presented, and the corresponding stability and bifurcation analysis will be completed through the eigenvalue analysis of discrete mapping structures of periodic motions. Consider the system parameters

δ = 1.0,

α = 5.5,

β = 20.0,

Q0 = 10.0.

(55)

132

4.1

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

Analytical bifurcation trees

For a global of view, analytical predictions of the periodic motion in the Duffing oscillator are illustrated in Fig. 4. Analytical predictions provide a complete view of the stable and unstable periodic motions. The eigenvalue analysis gives the bifurcation and stability of the periodic motions in the Duffing oscillator. In Fig. 4, the prediction of complete bifurcation trees of the period-1 motion to chaos is presented through the period-1 to period-4 motions. In addition, the bifurcation tree of period-3 motion is included to show the coexisting periodic motions. The solid and dashed curves depict the stable and unstable motions, respectively. The solution pairs of asymmetric motions are presented with black and red colors, respectively. The symbols ‘SN’ and ‘PD’ represent the saddle node and period doubling bifurcations, respectively. The prediction of displacement x mod (k,N) and velocity y mod (k,N) of the periodic nodes varying with excitation frequency Ω are presented in Fig. 4 (a) and (b), respectively. The symmetric and asymmetric periodic motions are labeled by “S” and “A”, respectively. The period-1, period-2, period-4 and period-3 motions are labeled by P-1, P-2, P-4 and P-3, respectively. The asymmetric period-1 motions appear from the saddle-node bifurcations of the symmetric period-1 motions. The period-2 motions appear from the period-doubling bifurcations of the asymmetric period1 motions, and period-4 motion appear from the period-doubling bifurcation of the period-2 motion. Such period-2 and period-4 motions are asymmetric. The period-3 motions possess the symmetric and asymmetric motions. The asymmetric period-3 motions appears from the symmetric period-3 motion The real and imaginary parts, magnitudes of eigenvalues for all periodic motions are also illustrated in Fig. 4 (c)-(e), respectively. In Fig. 4 (c), the saddle-node bifurcations are given by λi = 1 and |λ j | < 1 (i, j ∈ {1, 2} but j = i), and the perioddoubling bifurcations are given by λi = −1 and |λ j | < 1(i, j ∈ {1, 2} but j = i). For unstable periodic motions, one of the two eigenvalues experiences |λi | > 1(i ∈ {1, 2}). For the bifurcation trees of period-1 to period-4 motion, the frequency range lies in Ω ∈ (0, ∞). However, period-3 motions lie in Ω ∈ (1.5, 1.8) and (4.0, 8.0). To make clear illustrations, the bifurcation trees of the period-1 to period-4 motions are presented in Fig. 5 (a)-(d). The symmetric period-1 motion exist for Ω ∈ (0, ∞). The unstable symmetric period-1 motions are in the range of Ω ∈ (1.016, 1.23), (1.50, 2.63) and (4.528, ∞), pertaining to the asymmetric period-1 motions. In addition, there are two segments of unstable symmetric period-1 motions, associated with multiple co-existing solutions with jumping phenomena, and the corresponding frequency ranges are Ω ∈ (1.46, 1.513) and (3.96, 5.98). The corresponding saddle-node bifurcations for jumping phenomena in the multiple solutions ranges are Ω ≈ 1.46, 1.513, 3.96, 5.98. The asymmetric period-1 motions are generated from the symmetric period-1 motions with saddle-node bifurcations. The bifurcation points are at Ω ≈ 1.016, 1.23, 1.50, 2.63, 4.528, 6.73. A pair of two asymmetric period-1 motions will be produced, and the two asymmetric period-1 motions are in Ω ∈ (1.016, 1.23) for the first branch, (1.50, 2.63) for the second branch and (4.528, ∞) for the third branch. The stable asymmetric period-1 motions are in the ranges of Ω ∈ (1.016, 1.23) for the first branch, Ω ∈ (1.50, 1.517) and (1.97, 2.63) for the second branch, (4.528, 4.88) and (7.27, ∞) for the third branch. The unstable asymmetric period-1 motions are in the range of Ω ∈ (1.517, 1.97) for the second branch, and Ω ∈ (4.88, 7.27) for the third branch. From the two asymmetric period-1 motions, the period-2 motions will be generated through the period-doubling bifurcation. The period-doubling bifurcation points of the asymmetric period-1 motions are at Ω ≈ 1.517, 1.97, 4.528, 7.27, which are also the saddle-node bifurcations of the period-2 motion. The period-2 motions exist in the range of Ω ∈ (1.517, 1.97) for the second branch, and Ω ∈ (4.88, 7.27) for the third branch. The stable period-2 motions are in Ω ∈ (1.517, 1.521) and (1.90, 1.97) for the second branch, and in Ω ∈ (4.88, 4.97) and (6.58, 7.27) for the third branch. The unstable period-2 motions are in Ω ∈ (1.52, 1.90) for the second branch and Ω ∈ (4.97, 6.58) for the third branch. The period-doubling bifurcations of period-2 motions are Ω ≈ 1.52, 1.90 for the second branch and Ω ≈ 4.97, 6.58 for the third branch, and they are the saddle-node bifurcation for the period-4 motions. The period-4 motions are in the range of Ω ∈ (1.52, 1.90) for the second branch and Ω ∈ (4.97, 6.58) for the third branch. For the third branch, the stable period-4 motions are in Ω ∈ (4.97, 5.03) and Ω ∈ (6.49, 6.58), and the unstable period-4 motions are in Ω ∈ (5.03, 6.49). The period-doubling bifurcations of period-4 motion in the third branch are at Ω ≈ 5.03, 6.49, which is the saddle-node bifurcation for period-8 motion. Thus, the period-8 motions exists for Ω ∈ (5.03, 6.49).

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

133

8.0

P-1 S

1.0

Periodic Node Velocity, ymod(k,N)

Periodic Node Displacement, xmod(k,N)

1.5

P-3 0.5 P-2 0.0

S

A

P-1

P-2

A

P-1

-0.5

-1.0 0.0

2.0

4.0

6.0

8.0

A

4.0

P-1 A

S P-1

0.0

P-2 P-3

-4.0 0.0

10.0

S

P-2 2.0

Excitation Frequency, :

4.0

6.0

8.0

10.0

Excitation Frequency, :

(a)

(b)

SN

1.0

0.0

-1.0

0.0

PD

2.0

4.0

6.0

8.0

Eigenvalue Imaginary Part, ImO1,2

Eigenvalue Real Part, ReO1,2

0.8

0.4

0.0

-0.4

-0.8 0.0

10.0

2.0

Excitation Frequency, :

4.0

6.0

8.0

10.0

Excitation Frequency, :

(c)

(d)

Eigenvalue Magnitude, |O1,2|

1.0

0.5

0.0

0.0

2.0

4.0

6.0

8.0

10.0

Excitation Frequency, :

(e) Fig. 4 A global view of the analytical prediction of bifurcation trees of period-1 and period-3 motions to chaos varying with excitation frequency Ω: (a) Periodic node displacement xmod(k,N) . (b) Periodic node velocity ymod(k,N) . (c) Real part of eigenvalues. (d) Imaginary part of eigenvalues. (e) Magnitude of eigenvalues. (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

Continuously, we can obtain period-16 motions to chaos. Because the stable motions for period-8 or higher order periodic motions exists for the short range of excitation frequency, the bifurcation tree of period-1 motion to chaos will not be computed anymore further.

134

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150 9.0

1.0

PD

SN

P-1

SN

0.5

SN

S

SN

P-2 PD

0.0

A

P-1

P-2

SN SN

-0.5

2.0

PD

A

PD

PD PD

SN

-1.0 0.0

Periodic Node Velocity, ymod(k,N)

Periodic Node Displacement, xmod(k,N)

1.5

P-2

4.0

6.0

8.0

0.6

6.0

S PD

P-1

PD

P-2

P-4 PD PD PD

SN

0.0

SN

SN

SN

1.2

S

0.8 1.48

1.2

PD

P-2

1.0

0.4 1.0

PD

P-1

0.6

A

SN PD PD

2.0

4.0

1.6

Excitation Frequency, :

(c)

4.0

3.0

3.6

1.8

PD

6.0

8.0

10.0

P-1

2.0

PD

SN PD

PD

PD

PD

SN

PD

PD 3.2 1.45

1.53

A

-1.0

SN

SN

-3.0 1.0

PD

P-2

1.0

PD

1.52

1.4

P-1

SN

SN

SN

Periodic Node Velocity, ymod(k,N)

Periodic Node Displacement, xmod(k,N)

P-2 PD

S SN

8

PD

(b)

SN

0.8

PD

6

P-2

A

5.0

SN

4

Excitation Frequency, :

1.4

1.0

PD

0.0

(a)

1.2

P-2

SN

Excitation Frequency, :

A

PD PD

PD

SN

3.0

-3.0 0.0

10.0

PD PD

P-1

SN

P-2

PD

S 1.2

1.4

1.6

1.8

2.0

Excitation Frequency, :

(d)

Fig. 5 Analytical prediction of bifurcation trees of period-1 motions to chaos: (a) Periodic node displacement xmod(k,N) . (b) Periodic node velocity ymod(k,N) ; A zoomed view: (c) Periodic node displacement xmod(k,N) , (d) Periodic node velocity ymod(k,N) . (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

To clearly illustrate the bifurcation trees of period-3 motion to chaos, the symmetric and asymmetric period3 motions are presented in Fig. 6. The period-3 motions have two branches. The symmetric period-3 motions are in Ω ∈ (1.523, 1.772) for the first branch and Ω ∈ (4.30, 7.89) for the second branch. The stable symmetric period-3 motions are in the ranges of Ω ∈ (1.523, 1.526) and (1.695, 1.772) for the first branch and, Ω ∈ (4.30, 4.39) and Ω ∈ (6.69, 7.89) for the second branch. The unstable symmetric period-3 motions are in Ω ∈ (1.526, 1.695) for the first branch and Ω ∈ (4.39, 6.69) for the second branch, which are also for the asymmetric period-3 motions. The four saddle-node bifurcations are at Ω ≈ 1.523, 1.772, 4.30, 7.89 for the stable and unstable symmetric period-3 motions, which will not generate the asymmetric period-3 motions. The other four saddle-node bifurcations of the symmetric period-3 motions at Ω ≈ 1.526, 1.695, 4.39, 6.69 are not only for the stable and unstable symmetric period-3 motions but also for appearance of the asymmetric period-3 motions. The stable asymmetric period-3 motions are in the ranges of Ω ∈ (1.526, 1.528) and (1.678, 1.695) for the first branch and in Ω ∈ (4.39, 4.417) and Ω ∈ (6.414, 6.69) for the second branch. The unstable asymmetric period-3 motions are in Ω ∈ (1.528, 1.678) for the first branch and Ω ∈ (4.417, 6.414) for the second branch, which are also for the asymmetric period-3 motions, which are also for period-6 motions. The period-doubling bifurcations of asymmetric period-3 motions are at Ω ≈ 1.528, 1.678 for the first branch and Ω ≈ 4.417, 6.414 for the second branch, which ate also the saddle-node bifurcation for period-6 motions.

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150 SN

Zoom

SN

SN

SN

A S

SN

P-3

0.8

S PD SN

A

0.0 P-3

PD

ZOOM SN

SN

SN

P-3

S

A 2.0

SN

SN

PD

0.0

PD

P-3

1.5

1.7

5.0

6.0

7.0

-2.0 1.5

8.0

Excitation Frequency, :

1.7

5.0

SN SN PD

6.0

7.0

8.0

Excitation Frequency, :

(a)

(b) PD SN

SN

SN SN PD

PD SN

SN

4.0 Periodic Node Velocity, ymod(k,N)

Periodic Node Displacement, xmod(k,N)

S

A

-0.8

1.4

SN

4.0 Periodic Node Velocity, ymod(k,N)

Periodic Node Displacement, xmod(k,N)

1.6

135

1.1 P-3 A 0.8

S

S

0.5 1.5

1.6

1.7

P-3

Excitation Frequency, :

S

0.0

-2.0 1.5

1.8

A

2.0

1.6

1.7

1.8

Excitation Frequency, :

(c)

(d)

Fig. 6 Analytical prediction of bifurcation trees of period-3 motions to chaos: (a) Periodic node displacement xmod(k,N) . (b) Periodic node velocity ymod(k,N) ; A zoomed view: (c) Periodic node displacement xmod(k,N) , (d) Periodic node velocity ymod(k,N) . (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

4.2

Frequency-Amplitude Characteristics

From discrete mapping structures, the node points of periodic motions are computed. Consider the node points (m) (m) (m) of period-m motions as xk = (xk , yk )T for k = 0, 1, 2, . . . , mN in the Duffing oscillator. The approximate expression for period-m motion is determined by the Fourier series as M j j (m) x(m) (t) ≈ a0 + ∑ b j/m cos( Ωt) + c j/m sin( Ωt). m m j=1

(56)

(m)

There are 2M + 1 unknown vector coefficients of a0 , b j/m , c j/m . To determine such unknowns, at least we have (m)

the given nodes xk (k = 0, 1, 2, . . . , mN) with mN + 1 ≥ 2M + 1. In other words, we have M ≤ mN/2. The node (m) points xk on the period-m motion can be expressed by the finite Fourier series as for tk ∈ [0, mT ] (m)

x(m) (tk ) ≡ xk

(m)

mN/2 j j (m) = a0 + ∑ b j/m cos( Ωtk ) + c j/m sin( Ωtk ) m m j=1 mN/2

= a0 + ∑ b j/m cos( j=1

j 2kπ j 2kπ ) + c j/m sin( ), m N m N

(k = 0, 1, . . . , mN − 1),

(57)

136

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

where T= (m)

a0

=

2π = NΔt; Ω

Ωtk = ΩkΔt =

2kπ , N

1 mN−1 (m) ∑ x , N k=0 k

⎫ 2 mN−1 (m) 2 jπ ⎪ ⎪ ), ⎪ b j/m = ∑ x cos(k mN k=1 k mN ⎬ ( j = 1, 2, . . . , mN/2) 2 mN−1 (m) 2 jπ ⎪ ⎪ ) ⎪ c j/m = ∑ x sin(k ⎭ mN k=1 k mN and

(m)

(m)

(m)

a0 = (a01 , a02 )T ,

b j/m = (b j/m1 , b j/m2 )T ,

c j/m = (c j/m1 , c j/m2 )T .

(58)

(59)

The harmonic amplitudes and harmonic phases for period-m motion are A j/m1 = A j/m2 =

 

b2j/m1 + c2j/m1 , b2j/m2 + c2j/m2 ,

c j/m1 , b j/m1 c j/m2 ϕ j/m2 = arctan . b j/m2

ϕ j/m1 = arctan

(60)

Thus the approximate expression for period-m motion in Eq. (54) is determined by (m)

x(m) (t) ≈ a0 +

mN/2



j=1

j j b j/m cos( Ωt) + c j/m sin( Ωt). m m

(61)

The foregoing equation can be expressed as 

x(m) (t) y(m) (t)



⎫ ⎧ j ⎪ ⎪ ⎪ A j/m1 cos( Ωt − ϕ j/m1 ) ⎪ mN/2 ⎨ ⎬ m 1 01 ≈ ≡ + ∑ ⎪ ⎩ x(m) (t) ⎭ ⎩ a(m) ⎭ j=1 ⎪ ⎪ ⎭ ⎩ A j/m1 cos( j Ωt − ϕ j/m2 ). ⎪ 2 02 m ⎫ ⎧ ⎨ x(m) (t) ⎬

⎧ ⎫ ⎨ a(m) ⎬

(62)

For simplicity, only the excitation frequency-amplitude curves for displacement x(m) (t) will be presented. Similarly, the frequency-amplitudes for velocity y(m) (t) can also be determined. Thus the displacement can be expressed as mN/2 j j (m) (m) (63) x (t) ≈ a0 + ∑ b j/m cos( Ωt) + c j/m sin( Ωt), m m j=1 and (m)

x(m) (t) ≈ a0 + where A j/m =



mN/2



j=1

j A j/m cos( Ωt − ϕ j/m ), m

(64)

c j/m . b j/m

(65)

b2j/m + c2j/m ,

ϕ j/m = arctan

To discuss nonlinear behaviors of period-m motion for the Duffing oscillator, the frequency-amplitude for displacement will be presented as follows. The acronyms SN and PD are the saddle-node and period-doubling bifurcations for period-m motions, respectively. In all plots, the unstable and stable solutions of period-m motions are represented by the dashed and solid curves, respectively.

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

4.2.1

137

Bifurcation trees of period-1 motion to chaos

The bifurcation trees of period-1 motion to chaos will be presented through the period-1 to period-4 motions will (m) be presented in Fig. 7. The given parameters are listed in Eq. (44). The constant term a0 (m = 1, 2, 4) is presented in Fig. 7 (i) for the solution center on the right side of the y-axis. The bifurcation tree is clearly observed. (m)L (m)R For the solution center on the left side of the y-axis, we have a0 = −a0 . For the symmetric period-m (m) (m) motion, we have a0 = 0, labeled by “S”. However, for asymmetric period-m motion, a0 = 0, labeled by “A”. For the symmetric period-1 motion to an asymmetric period-1 motion, the saddle-node bifurcation will occur. The saddle-node bifurcations are at Ω ≈ 1.016, 1.23, 1.50, 2.63, 4.528. For such saddle-node bifurcations, the asymmetric periodic motions appear, and the symmetric motions are from the stable to unstable solution or from the unstable to stable solution. The saddle-node bifurcations for symmetric motion jumping points are at Ω ≈ 1.46, 1.513, 3.96, 5.98. The symmetric period-1 motion are only from the stable to unstable solution or from the unstable to stable solution. When the asymmetric period-1 motion experiences a period-doubling bifurcation, the period-2 motion will appear and the asymmetric period-1 motion is from the stable to unstable solution. The frequencies of Ω ≈ 1.517, 1.97, 4.528, 7.27 are not only for the period-doubling bifurcations of the asymmetric period-1 motions but also for the saddle-node bifurcations of the period-2 motion. When the period-2 motion possesses a period-doubling bifurcation, the period-4 motion appears and the period-2 motion is from the stable to unstable solution. The frequencies of Ω ≈ 1.52, 1.90, 4.97, 6.58 are for the period-doubling bifurcations of period-2 motions and for the saddle-node bifurcation for the period-4 motions. The frequencies of Ω ≈ 5.03, 6.49 are for the period-doubling bifurcations of period-4 motions and for the saddle-node bifurcation for the period-8 motions. All period-2 and period-4 motions are on the branches of asymmetric period-1 motions, and the centers of the periodic motions are on the right side of the y-axis. In Fig. 7 (ii), the harmonic amplitude A1/4 is presented. For period-1 and period-2 motions, A1/4 = 0. The saddle-node bifurcations are at Ω ≈ 4.97, 6.58 for period-1 motion, and the period-doubling bifurcations are at Ω ≈ 5.03, 6.49. The bifurcation points are clearly observed, and the quantity of the harmonic amplitude for period-4 motion is A1/4 ∼ 7 × 10−2 . In Fig. 7 (iii), the harmonic amplitude A1/2 for period-4 and period-2 motions are presented. For the second branch, only the period-2 motion is presented because the stability range of period-4 motion is very small and more discrete nodes are needed to obtain such a period-4 motion. For the third branch, the bifurcation trees for period-2 to period-4 motions are clearly illustrated. The period-doubling bifurcations are at Ω ≈ 5.03, 6.49 for the third branch. The saddle-node bifurcations of the period-2 motion are at Ω ≈ 1.517, 1.97, 4.528, 7.27 for the second and third branches. The quantity level of the harmonic amplitude A1/2 is A1/2 ∼ 1.5× 10−1 . In Fig. 7 (iv), the harmonic amplitude A3/4 is presented, which is similar to the harmonic amplitude A1/4 . The quantity level of such harmonic amplitude is A3/4 ∼ 1.5× 10−2 . The other harmonic amplitude Ak/4 (k = 4l + 1, 4l + 3, l = 1, 2 . . .) will not be presented herein for reduction of abundant illustrations. In Fig. 7 (v), the primary harmonic amplitudes A1 versus excitation frequency Ω are presented for the period-1 to period-4 motion. The bifurcation trees are clearly observed. The entire skeleton of frequency-amplitude for the symmetric period-1 motion is presented, and the asymmetric period-1 motions and the relative period-2 and period-4 motions are attached to the symmetric period-1 motion. The quantity level of the primary amplitude is A1 ∼ 1.8 for all period-1 to period-4 motions. The bifurcation points are presented as before. In Fig. 7 (vi), the harmonic amplitude A3/2 is presented. The bifurcation trees are similar to the harmonic amplitude A1/2 . The quantity levels of A3/2 and A1/2 are almost same. That is, A3/2 ∼ 0.1 and A1/2 ∼ 0.15. To reduce abundant illustrations, Ak/2 (k = 2l + 1, l = 2, 3, . . .) will not be presented any more. In Fig. 7 (vii), (ix), (xi) and (xiii), the harmonic ampli(m) tude Ak (k = 2l, l = 1, 2, . . . , 4) are presented, which are similar to constant term a0 . The bifurcation trees have the similar structures for the different harmonic amplitudes but the corresponding quantity levels of harmonic amplitudes are different. That is, A2 ∼ 0.6, A4 ∼ 0.24, A6 ∼ 0.16, A8 ∼ 0.036 are for the first and second branches. However, for the third branch, we have A2 ∼ 0.01, A4 ∼ 10−4 , A6 ∼ 10−6 , A8 ∼ 10−9 . In Fig. 7 (viii), (x), (xii) and (xiv), the harmonic amplitude Ak (k = 2l + 1, l = 1, 2, . . . , 4) are presented, similar to the primary harmonic amplitude A1 . The bifurcation trees are different for the different harmonic amplitudes and the corresponding

138

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

quantity levels of harmonic amplitudes are different. That is, we obtain A3 ∼ 0.6, A5 ∼ 0.32, A7 ∼ 0.24, A9 ∼ 0.1 for Ω < 5. However, for Ω ≥ 5, we have A3 ∼ 10−3 , A5 ∼ 10−5 , A7 ∼ 10−6 , A8 ∼ 10−9 . To avoid abundant illustrations, the harmonic amplitudes of A21 and A61 are presented. For Ω > 1, A21 < 10−5 and A61 < 10−10 . 7.5e-2

1.2

P-4

A PD PD

(m)

Constant, a0

PD

PD

PD

P-2

P-2

0.4

PD PD

P-1

PD

P-1

PD

P-4

PD

0.0

5.0e-2

PD

2.5e-2 PD

S

SN SN SN

0.0

Harmonic Amplitude, A1/4

0.8

SN

SN

2.0

SN

4.0

6.0

8.0

0.0

10.0

0.0

2.0

4.0

6.0

(i)

8.0

10.0

P-4

PD

P-2

Harmonic Amplitude, A3/4

Harmonic Amplitude, A1/2

0.015

PD

P-2

P-4

PD PD

10.0

(ii)

0.15

0.05

8.0

Excitation Frequency, :

Excitation Frequency, :

0.10

SN

SN

PD

0.010

0.005

PD

PD

PD

0.00 0.0

SN

SN

SN

2.0

4.0

SN

PD

PD

SN

SN 0.6

A SN PD

PD P-4

SN

0.0 0.0

S PD

SN

A PD

1

2

PD PD

4.0

6.0

8.0

PD

P-2 0.01

PD

P-2 PD

P-4

PD

A S

2.0

Harmonic Amplitude, A3/2

SN

SN

6.0

0.1

1.0

1.2 SN

4.0

(iv)

S

SN

2.0

(iii) SN

SN

SN

SN

0.000 0.0

10.0

Excitation Frequency, :

P-1

0.6

8.0

Excitation Frequency, :

1.8

Harmonic Amplitude, A1

SN

6.0

10.0

0.001 0.0

SN

SN

2.0

SN

4.0

SN

6.0

Excitation Frequency, :

Excitation Frequency, :

(v)

(vi)

8.0

(m)

10.0

Fig. 7 Frequency-amplitude characteristics for bifurcation trees of period-1 to period-4 motions: (i) a0 (m = 1, 2, 4). (ii)-(xii) Aklm (m = 4, k = 1, 2, 3, 4; 6, 8, 12, . . ., 36, 84, 244); (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150 0.6

0.005

0.4

PD

A

PD

0.000

PD

S

0.0

8

PD

SN

SN SN SN

4

P-1

P-2

0.0

A

S

P-1

1e-3

PD

0.2 SN

1e-5

P-2

SN

SN

A

6.0

8.0

0.0 0.0

10.0

SN

2.0

6.0

SN

PD

SN SN

0

PD

5

10

PD

P-2

A P-1

PD

Harmonic Amplitude, A5

5e-5

0.24

1e-5

SN

SN

PD

1e-6

PD PD

1e-7

0.16

SN

4

8

P-1

PD

P-2 0.08

PD SN

SN SN SN

SN

2.0

SN

4.0

SN

PD

6.0

8.0

0.00 0.0

10.0

S

PD SN

2.0

Excitation Frequency, :

SN

4.0

0.16

PD

0.24

PD

1e-7

PD 1e-8

PD

P-2

0.04

SN

5

10

P-1

PD

SN 1.0 SN

PD PD

2.0

3.0

PD

1e-9

0.16

PD

1e-10

SN

1e-11

4

PD SN

4.0

0.00 0.0

7

P-1

0.08

SN

P-2 PD

S SN

SN

1e-8

Harmonic Amplitude, A7

A

0.08

10.0

1e-6

PD

1e-7

1e-9

8.0

(x)

1e-6

0.12

PD

6.0

Excitation Frequency, :

(ix)

Harmonic Amplitude, A6

10.0

0.32

0.16

0.00 0.0

8.0

(viii) PD

Harmonic Amplitude, A4

PD

PD

4.0

Excitation Frequency, :

0.24

0.0

10

PD

(vii)

0.00

5

SN

PD

SN

4.0

PD

S P-1

Excitation Frequency, :

0.08

PD

1e-4

0.4

PD

S

PD

SN

2.0

SN

SN

PD

PD

0.2

0.6

P-1 P-4 P-2

PD

Harmonic Amplitude, A3

Harmonic Amplitude, A2

0.010

139

PD

1.0

SN

2.0 Excitation Frequency, :

Excitation Frequency, :

(xi)

(xii) Fig. 7 Continued.

3.0

4.0

140

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

1e+0

0.036

0.024 PD 0

5

P-1

10

0.012 P-2

0.000 0.0

1.0 SN SN

SN SN SN

1e-2

1e-14

PD

4

7

P-1

P-2

1e-3

SN

PD

SN 3.0

2.0

0.0

4.0

1.0

2.0

3.0

4.0

Excitation Frequency, :

(xiii)

(xiv)

0.09

0.018 4e-13

4e-15

2e-13

0.06

0

SN 5

10

0.03

1.0

2.0

3.0

4.0

Harmonic Amplitude, A61

Harmonic Amplitude, A21

PD

PD 1e-13

PD

Excitation Frequency, :

0.00 0.0

PD

1e-12

1e-4

PD

SN

1e-11

1e-1

PD

PD

PD

1e-10 SN

PD

Harmonic Amplitude, A9

Harmonic Amplitude, A8

3e-9

1e-9

0.012 0

5

10

0.006

0.000 0.0

1.0

2.0

Excitation Frequency, :

Excitation Frequency, :

(xv)

(xvi)

3.0

4.0

Fig. 7 Continued.

For Ω < 1, A21 ∼ 0.1 and A61 ∼ 10−2 . From the above discussion, the periodic motion, For Ω > 1, we can use about 80 harmonic terms to approximate period-1, period-2 and period-4 motions. For Ω < 1 but not close to zero, we can use 240 harmonic terms to approximate period-1, period-2, and period-4 motions. For Ω ≈ 0, the infinite harmonic terms should be adopted to approximate the periodic motions. 4.2.2

Period-3 motions

The bifurcation trees of period-3 motion to chaos will be presented through the period-3 motions will be presented in Fig. 8. Since the period-6 motion has a short stable solution, the bifurcation tree may not be very nice. (3) The given parameters are still listed in Eq. (44). The constant term a0 is presented in Fig. 8 (i) for the solution (3)L (3)R center on the right side of the y-axis. For the solution center on the left side of the y-axe, a0 = −a0 . For the (3) symmetric period-3 motion, a0 = 0, also labeled by “S”. However, for asymmetric period-3 motion, we have (3) a0 = 0, also labeled by “A”. For the symmetric period-3 motion to an asymmetric period-3 motion, the saddlenode bifurcation will occur. The two closed branches of period-3 bifurcations are in range of Ω ∈ (1.5, 1.8) and Ω ∈ (4.0, 8.0). The four saddle-node bifurcations at Ω ≈ 1.523, 1.772, 4.30, 7.89 are for the stable and unstable symmetric period-3 motions only. However, the other four saddle-node bifurcations of the symmetric period-3 motions at Ω ≈ 1.526, 1.695, 4.39, 6.69 are not only for the stable and unstable symmetric period-3 motions but also for appearance of the metric period-3 motions. The period-doubling bifurcations of asymmetric period-3 motions are at Ω ≈ 1.528, 1.678 for the first branch and Ω ≈ 4.417, 6.414 for the second branch, which ate also

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

SN

0.12 SN

SN

SN

Harmonic Amplitude, A1/3

(3)

Constant, a0

0.04 A PD

PD

SN

5.0

6.0

7.0

0.6

P-3 0.3

S

PD

SN

A

PD

SN

PD

P-3

0.0 1.5

8.0

1.7

5.0

(i) SN

A

Harmonic Amplitude, A1

Harmonic Amplitude, A2/3

SN

P-3

PD

PD

PD

PD

SN

SN

SN

5.0

6.0

7.0

A

PD

0.80

S SN SN PD

A 0.35

S

P-3

PD SN SN 0.00 1.5 1.7

0.15 1.5

8.0

1.7

5.0

SN

SN

PD

P-3

0.400 Harmonic Amplitude, A3

Harmonic Amplitude, A2

0.016 A

0.008 PD

0.000 1.5

SN S

SN

1.7

SN

SN

PD

PD

SN

S 5.0

SN

6.0

Excitation Frequency, :

(v)

8.0

0.200

7.0

8.0

A PD SN

SN

0.003

0.000 1.5

SN

SN

S

PD

S A

P-3

PD SN

7.0

(iv)

SN

P-3

6.0

Excitation Frequency, :

(iii) A

PD SN

S

Excitation Frequency, :

0.024 SN

8.0

SN

SN

SN

0.88 SN

SN

SN

0.12

A

7.0

(ii)

P-3

0.06

6.0

Excitation Frequency, :

Excitation Frequency, :

0.18 SN

PD

A

SN

S

SN SN

1.7

SN

S

PD

0.08

0.00 1.5

SN

SN

SN

SN

A

P-3

PD SN

0.9

141

P-3 1.7

5.0

PD SN

6.0

7.0

8.0

Excitation Frequency, :

(vi) (m)

Fig. 8 Frequency-amplitude characteristics for stable and unstable period-3 motions: (i) a0 (m = 3). (ii)-(xii) Ak/m (m = 3, k = 1, 2, 3; 6, 9, . . ., 27, 60, 61, 180, 181); (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

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Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

0.012

SN

SN

SN

SN

0.15

P-3

SN PD

2.0e-4

Harmonic Amplitude, A5

Harmonic Amplitude, A4

A PD

0.008

0.004

4

PD

0.000 1.5

8

P-3

SN

SN

PD SN

SN S

0.0

1.7

6.0

SN

SN

P-3 S

0.10

A PD 0

PD

0.05

7.0

0.00 1.5

8.0

1.7

SN

SN

A

SN

0.06 SN

Harmonic Amplitude, A7

PD

P-3

PD

S

0

4 SN

SN

SN8

PD

SN

PD 4e-7

P-3

0.04

A

PD

S

SN

PD

PD

1.7

5.0

6.0

7.0

0.02

PD

PD

1.7

5.0

(ix) SN

SN

1.6e-2

6e-8

SN

SN SN

Harmonic Amplitude, A9

Harmonic Amplitude, A8

A PD

P-3 2.2e-3

PD

S

0

4

SN

SN

SN8

1.2e-2

0.0

1.5

SN

SN

5e-9

P-3

A

S

8.0e-3

S PD 0

4.0e-3

SN

1.7

PD

PD

5.0

6.0

7.0

8.0

0.0

Excitation Frequency, :

A

PD

1.5

SN 8

4 SN

P-3

SN

PD SN

8.0

PD

PD

P-3 4.4e-3

7.0

(x)

SN

PD

6.0

Excitation Frequency, :

Excitation Frequency, :

A

SN 8

4 SN

P-3

SN

0.00 1.5

8.0

PD

S 0

A PD

6.6e-3 SN

8.0

SN

SN

P-3

0.008

0.000 SN 1.5

7.0

Excitation Frequency, :

(viii)

3e-6

Harmonic Amplitude, A6

6.0

PD

A

0.004

SN 8

PD

5.0

(vii) SN

SN

4 SN

SN

Excitation Frequency, :

0.012 SN

SN PD

4e-5

P-3

PD

5.0

SN

SN

1.7

PD

PD

5.0

6.0

Excitation Frequency, :

(xi)

(xii) Fig. 8 Continued.

7.0

8.0

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

SN

2.4e-5 SN

SN

SN

4.5e-5

PD

0

-2e-15

8.0e-6

Harmonic Amplitude, A61/3

Harmonic Amplitude, A20

1.6e-5

PD

SN

SN 4

8

PD

PD

0.0

SN PD

P-3

A

SN

SN

1.5

PD

PD

1.7

5.0

6.0

SN

SN

2e-15

7.0

2e-15

P-3 PD

S A

-2e-15

1.5e-5

P-3

PD

1.5

1.7

SN

3.6e-14 SN

Harmonic Amplitude, A181/3

Harmonic Amplitude, A60

8.0e-15

PD

PD

-2e-15

SN

SN 4

5

6

7

8

PD

0.0

PD SN S SN1.7 1.5

8.0

SN

SN

5.0

P-3 2.4e-14

PD

6.0

7.0

8.0

PD

0

SN PD

S 1.2e-14

A

-2e-15

PD

P-3

PD

7.0

2e-15

2e-15

-4e-15

6.0

(xiv) SN

0

A

PD

5.0

4e-15

1.6e-14

8

Excitation Frequency, :

SN

P-3

SN

SN 4

SN

(xiii) SN

PD

0

Excitation Frequency, :

2.4e-14 SN

SN

SN

3.0e-5

0.0

8.0

143

0.0

1.5

8

P-3

SN

1.7

SN

SN 4

5.0

PD

6.0

7.0

8.0

Excitation Frequency, :

Excitation Frequency, :

(xv)

(xvi) Fig. 8 Continued.

the saddle-node bifurcation for period-6 motions. In Fig. 8 (ii), the harmonic amplitude A1/3 is presented. The bifurcation trees for two branches of period-3 motions are clearly observed. The quantity levels of such harmonic amplitudes are A1/3 ∼ 0.3 for the first branch and A1/3 ∼ 0.9 for the second branch of period-3 motions. In Fig. 8 (iii), the harmonic amplitude A2/3 is presented. The bifurcation trees of A2/3 for two branches of (3)

period-3 motions are similar to a0 . The quantity levels of the harmonic amplitudes are A2/3 ∼ 0.06 for the first branch and A2/3 ∼ 0.18 for the second branch of period-3 motions. To avoid abundant illustrations, harmonic amplitudes A j/3 (mod( j, 3) = 0) will not be presented. To show quantity levels and bifurcation variation of harmonic bifurcation trees, the harmonic amplitudes A j ( j = 2l − 1, l = 1, 2, . . . , 5) are presented in Fig. 8 (iv), (vi), (viii), (x) and (xii). The bifurcation trees of these harmonic amplitudes are similar to the harmonic amplitude A1/3 . The quantity levels of A1 are A1 ∈ (0.75, 0.88) for the first branch and A1 ∈ (0.15, 0.55) for the second branch. For the harmonic amplitude A3 , we have A3 ∈ (0.15, 0.5) for the first branch and A3 ∈ (0.0, 0.005) for the second branch. For the harmonic amplitude A5 , A5 ∈ (0.02, 0.15) for the first branch but A5 ∈ (0.0, 5 × 10−5 ) for the second branch. For the higher-order harmonics, the second branch, the quantity levels of harmonic amplitudes drop very fast. A7 ∈ (0.0, 5 × 10−7 ) and A9 ∈ (0.0, 6 × 10−9 ) for the second branch. However, for the first branch, A7 ∈ (0.01, 0.05) and A9 ∈ (103 , 2 × 10−2 ), which decrease very slowly. The harmonic amplitudes A j ( j = 2l, l = 1, 2, . . . , 5) are presented in Fig. 8 (v), (vii), (ix), and (xi). The bifurcation trees of these harmonic

144

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150 (3)

amplitudes are similar to constant term a0 and the harmonic amplitude A2/3 . The quantity levels of the harmonic amplitude A2 are A2 < 0.024 for the first branch and A2 < 0.012 for the second branch. For the harmonic amplitude A4 , A4 < 0.012 for the first branch and A4 < 3.0 × 10−4 for the second branch. For the harmonic amplitude A6 , A6 < 0.012 for the first branch but A6 < 4 × 10−6 for the second branch. For the higher-order harmonics, the second branch, the quantity levels of harmonic amplitudes also drop very fast. A8 < 6 × 10−8 for the second branch. However, for the first branch, A8 < 7.0 × 10−3 . For the first branch, the quantity levels are still high. To look into the effects of the higher-order harmonic amplitudes, the harmonic amplitudes A20 , A61/3 , A60 , A181/3 are presented in Fig. 8 (xiii)-(xvi), respectively. The bifurcation trees of the harmonic (3)

amplitude A20 and A60 are similar to the constant term a0 and the harmonic amplitude A2/3 . A20 < 2.5 × 10−5 and A60 < 2.5 × 10−14 for the first branch, but for the second branch, A20 < 10−16 and A60 < 10−16 . In fact, for the second branch, the quantity level should be much smaller because our computational algorithm cannot achieve more accurate results. The bifurcation trees of the harmonic amplitude A61/3 and A181/3 are similar to the the harmonic amplitude A1/3 . A61/3 < 5 × 10−5 and A181/3 < 3.5 × 10−14 for the first branch, but for the second branch, A161/3 < 10−16 and A181/3 < 10−16 . Once again, for the second branch, the quantity level should be much smaller owing to the computational accuracy of the discrete algorithm and time step. 5 Numerical simulations In this section, numerical illustrations are given from the semi-analytical solutions and numerical integration schemes. The initial conditions in numerical simulation are obtained from analytical prediction of periodic solutions. In all plots for illustration, circular symbols give analytical predictions, and solid curves give numerical simulation results. Acronym “IC” represents initial conditions. The initial points and the corresponding periodic points are depicted by the large circular symbols. In Fig. 9, consider excitation frequency Ω = 1.05 to demonstrate period-1 motion. Other parameters is presented in Eq. (53). The analytical prediction gives the initial conditions (x0 , y0 ) ≈ (0.888313, −2.694745). The displacement, velocity, trajectory and harmonic spectrum will be presented in Fig. 9 (a)-(d), respectively. In Fig. 9 (a) and (b), the time-histories of displacement and velocity are not the simple-sinusoidal alike periodic motion. One period (1T ) is labeled for the period-1 motion. The trajectory of period-1 motion is presented in Fig. 9 (c). The period-1 motion is like a period-3 motion. The analytical prediction results match very well with numerical simulation results. The harmonic amplitude spectrum is presented in Fig. 9 (d). The constant term is a0 ≈ 0.1235. The main harmonic amplitudes are A1 ≈ 0.9413, A2 ≈ 0.0949, A3 ≈ 0.1620, A4 ≈ 0.0924, A5 ≈ 0.2699, A6 ≈ 0.0978, A7 ≈ 0.1035, A8 ≈ 0.0277, A9 ≈ 0.0195, A10 ≈ 0.0166, A11 ≈ 0.0153, A12 ≈ 0.0134. The other harmonic amplitudes are A j ∈ (10−9 , 10−2 )( j = 13, 14, . . . , 50) and A50 ≈ 3.6100e-9. The harmonic amplitudes decrease very slowly with harmonic order. For this period-1 motion, we cannot use one harmonic term to approximate the periodic solutions. From the harmonic amplitudes, at lease 12 harmonic terms plus constant term should be included to obtain the rough estimate of periodic motion. Other harmonic amplitudes still can be presented. However, the quantity level is very small, and they will not be presented. In Fig. 10, consider excitation frequency Ω = 1.686 to demonstrate a complex period-3 motion. Other parameters is also presented in Eq. (53). Thee initial condition is (x0 , y0 ) ≈ (0.651260, 3.947260). The displacement, velocity, trajectory and harmonic spectrum for the period-3 motion will be presented in Fig. 10 (a)-(d), respectively. The time-histories of displacement and velocity are presented in F.10(a) and (b) , and three periods (3T ) is labeled for the period-1 motion. The trajectory of period-1 motions is presented in Fig. 10 (c). The period-3 motion is complex. The initial points with the corresponding periodic points are depicted through the large circular symbols. After three periods, the period-3 motion returns back to the initial condition. The harmonic (3) amplitude spectrum is presented in Fig. 10 (d). The constant term is a0 ≈ 7.2200e-3 The main harmonic amplitudes are A1/3 ≈ 0.2725, A2/3 ≈ 0.0318, A1 ≈ 0.8515, A4/3 ≈ 9.0457e-3, A5/3 ≈ 0.1088, A2 ≈ 0.0170, A7/3 ≈ 0.3172, A8/3 ≈ 0.0721, A3 ≈ 0.2823, A10/3 ≈ 0.0127, A11/3 ≈ 0.0733, A4 ≈ 6.1598e-3, A13/3 ≈ 0.1184, A14/3 ≈

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

2.0

5.0

T

T 2.5 Velocity, y

Displacement, x

IC 1.0

145

0.0

-1.0

0.0

-2.5 IC

-2.0 0.0

3.0

6.0

9.0

12.0

-5.0 0.0

15.0

3.0

6.0

Time, t

(a)

12.0

15.0

(b)

4.0

1e+1

1e-2

Harmonic Amplitude, Ak

0.0

-2.0

A20

A1

1e+0

2.0 Velocity, y

9.0 Time, t

A2

a0

A3

A5 A4

A50

1e-6

A6 A 7 A8

A9

1e-10 20

1e-2

40

A14

A16

A18 A20

IC -4.0 -1.6

-0.8

0.0 Displacement, x

(c)

0.8

1.6

1e-4

0

4

8

12

16

20

Harmonic Order, k

(d)

Fig. 9 Period-1 motions (Ω = 1.05): (a) displacement, (b) velocity, (c) trajectory, (d) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.888313, −2.694745). Parameters (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

0.0253, A5 ≈ 0.0530, A16/3 ≈ 5.5773e-3, A17/3 ≈ 0.0160, A6 ≈ 9.8557e-3, A19/3 ≈ 0.0440, A20/3 ≈ 7.1797e-3, A7 ≈ 7.2147e-3, A22/3 ≈ 3.9140e-3, A23/3 ≈ 6.4677e-3, A8 ≈ 5.0847e-3, A25/3 ≈ 0.0128. The other harmonic amplitudes are A j/3 ∈ (10−9 , 10−2 )( j = 26, 27, . . . , 90), and A30 ≈ 4.6072e-8. The harmonic amplitudes decrease very slowly with harmonic order. For the period-3 motion, we at lease 25 harmonic terms plus constant term should be included to obtain the rough estimate of periodic motion. The harmonic amplitudes decease non-uniformly. For the harmonic amplitudes, the primary harmonic terms of A1 ≈ 0.8515 plays an important role in the period-3 motion. In traditional analysis, such a period-3 motion cannot be called the super-harmonic or sub-harmonic motion. To avoid too many illustrations, only trajectories and harmonic amplitude spectrum are presented for periodic motions on the bifurcation tree of period-1 to period-4 motion. Consider an excitation frequency of Ω = 8.0 for period-1 motion. Because the Duffing oscillator possesses the twin-potential well, the two asymmetric solutions will be associated with the two potential well. Thus, two initial conditions for the two asymmetric period-1 motions are (x0 , y0 ) ≈ (0.301097, 0.195246), and (−0.655147, 0.215682). The trajectories of the two asymmetric period-1 motion are presented in Fig. 11 (a). The two asymmetric motions are skew symmetric. After one period, the periodic motion returns back to the initial point. The corresponding harmonic amplitudes are presented in Fig. 11 (b). The constant terms are aL0 = −aR0 , other harmonic amplitudes are the same for two periodic motions. However, the phases are different with ϕ Lj = mod(ϕ Rj + ( j + 1)π , 2π )( j = 1, 2, . . .). The

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6.0

2.0

3.0

1.0 IC

Velocity, y

Displacement, x

3T

3T

0.0

0.0

-3.0

-1.0

-2.0 0.0

5.0

10.0

-6.0 0.0

15.0

5.0

10.0

(a)

(b) 3T

10

IC

Velocity, y

2.5 1T

0.0 2T -2.5

Harmonic Amplitude, Ak/3

5.0

1

-0.8

0.0

0.8

1.6

1e-2

A1 A7/3

A1/3 A4/3

A10 A30

1e-5

A3 A11/3

0.1

A13/3

1e-8 10

A5

A19/3

A17/3 0.01 a0

-5.0 -1.6

15.0

Time, t

Time, t

20

A25/3 A7 A 23/3

(3)

0.001 0.0

2.0

4.0

6.0

8.0

30

A29/3

10.0

Harmonic Order, k/3

Displacement, x

(c)

(d)

Fig. 10 Period-3 motions (Ω = 1.686): (a) displacement, (b) velocity, (c) trajectory, (d) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.651260, 3.947260). Parameters (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

constant term is a0 = aR0 ≈ 0.9528. The main harmonic amplitudes are A1 ≈ 0.1788, and A2 ≈ 1.8480e-3. The other harmonic amplitudes are A j ∈ (10−9 , 10−3 )( j = 3, 4, . . . , 10), and A10 ≈ 1.9170e-16. For this case, only one harmonic term plus the constant term can provide a good approximation of the period-1 motion. Consider the excitation frequency of Ω = 6.9 for period-2 motions which is near the asymmetric period-1 motion of Ω = 8.0. The initial conditions for the two period-2 motions are (x0 , y0 ) ≈ (0.214375, 0.524608), and (−0.555016, 0.138223). The trajectories and harmonic amplitudes for such period-2 motions are presented in Fig. 11 (c) and (d), respectively. The trajectory of the period-2 motion is more complex than the period-1 motion. After two periods, the period-2 motion returns back to the initial point, and the point at one period (2)L (2)R is depicted with a large circular symbol, The constant terms are a0 = −a0 , and other harmonic amplitude are the same. However, the phases are different with ϕ Lj/2 = mod(ϕ Rj/2 + ( j/2 + 1)π , 2π )( j = 1, 2, . . .). (2)

(2)R

The constant term for the period-2 motion on the right hand side is a0 = a0 ≈ 0.8246. The main harmonic amplitudes are A1/2 ≈ 0.0985, A1 ≈ 0.2393, A3/2 ≈ 6.6685e-3, and A2 ≈ 4.1152e-3. The other harmonic amplitudes are A j/2 ∈ (10−9 , 10−3 )( j = 5, 6, . . . , 20) and A10 ≈ 3.5305e-14. For this period-2 motion, two harmonic terms plus the constant term can provide a good approximation. Consider the excitation frequency of Ω = 6.52 for period-4 motions. The corresponding trajectories and harmonic amplitudes are presented in Fig. 11 (e) and (f), respectively. The trajectory of the period-4 motion is much more complex

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

1e+1 a0

2.0

IC 1T

0.0

Harmonic Amplitude, Ak

Velocity, y

1.0 IC 1T

-1.0

A1

147

A2 A3 A4

1e-6

A5 A6

A7 A8

1e-12

A9 A10

-2.0 -0.8

-0.4

0.0

0.4

1e-19 0.0

0.8

2.0

4.0

(a) (2)

Harmonic Amplitude, Ak/2

1.2 Velocity, y

10.0

1e+1 a0

1T IC 2T

0.0

1T

IC 2T

-1.2

-2.4 -0.8

-0.4

0.0

0.4

A5

1e-8

A3/2

1e-2

A2

1e-14

A5/2

A10

5

10

A3 A7/2

1e-5

A4 A7/2 A5

1e-8

0.8

A1

A1/2

0.0

1.0

2.0

3.0

4.0

5.0

Harmonic Order, k/2

Displacement, x

(c)

(d)

2.4

1e+1 a0(4)

1T 3T

Harmonic Amplitude, Ak/4

1.2 Velocity, y

8.0

(b)

2.4

0.0

6.0

Harmonic Order, k

Displacement, x

1T IC

4T

2T

3T

IC 4T

2T

-1.2

A1

A1/2 A1/4

1e-2

A3/4

A5/4

A5

1e-8

A3/2

A2 A5/2

A7/4

A9/4

1e-5

1e-14

A3

5

A10

10

A7/2 A11/4 A13/4

A4

A15/4

A9/2 A5 A17/4 A19/4

-2.4 -0.8

-0.4

0.0 Displacement, x

(e)

0.4

0.8

1e-8

0.0

1.0

2.0

3.0

4.0

5.0

Harmonic Order, k/4

(f)

Fig. 11 Period-1 motions (Ω = 8.0): (a) trajectory, (b) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.301097, 0.195246), (−0.655147, 0.215682); Period-2 motions (Ω = 6.9): (c) trajectory, (d) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.214375, 0.524608), (−0.555016, 0.138223); Period-4 motions (Ω = 6.52): (e) trajectory, (f) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.062918, −0.051147), (−0.485633, 0.129886). Parameters (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

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than the period-1 motion. After four periods, the period-4 motion returns back to the initial point, and the points at one, two and three periods are depicted with large circular symbols. The constant terms are also (4)L (4)R a0 = −a0 , and other harmonic amplitude are still the same. However, the phases are different with ϕ Lj/4 = mod(ϕ Rj/4 + ( j/4 + 1)π , 2π )( j = 1, 2, . . .). The constant term for the period-4 motion on the right (4)

(4)R

hand side is a0 = a0 ≈ 0.7471. The main harmonic amplitudes are A1/4 ≈ 0.0262, A1/2 ≈ 0.1210, A3/4 ≈ 3.0916e-3, A1 ≈ 0.2632, A5/4 ≈ 2.1204e-3, A3/2 ≈ 9.3470e-3, A7/4 ≈ 8.5130e-5, A2 ≈ 5.2201e-3, The other harmonic amplitudes are A j/4 ∈ (10−9 , 10−3 )( j = 9, 10, . . . , 40) and A10 ≈ 1.8765e-13. Eight harmonic terms plus constant term can give a good approximation for period-4 motions. Periodic motions in the bifurcation tree of period-3 motion will be illustrated for demonstration of motion complexity. Consider an excitation frequency of Ω = 6.52 for period-3 motion. The period-3 motion crosses the separatrix of the non-damped Duffing oscillator. The initial condition for such an asymmetric period-3 motions is (x0 , y0 ) ≈ (−0.728926, 2.046610). The trajectories of the asymmetric period-3 motion are presented in Fig. 12 (a). After three periods, the period-3 motion returns back to the initial point. The corresponding harmonic (3) amplitudes are presented in Fig. 12 (b). The constant term is a0 ≈ 0.0660. The main harmonic amplitudes are A1/3 ≈ 0.8156, A2/3 ≈ 0.0843, A1 ≈ 0.2827, A4/3 ≈ 5.0215e-3, A5/3 ≈ 0.0222, A2 ≈ 4.0123e-3, and A7/3 ≈

1e+1

4.0

Velocity, y

2.0

Harmonic Amplitude, Ak/3

A1/3 1T 2T 0.0

-2.0

IC

3T

a0

(3)

A10

A2/3

A5/3 A4/3

1e-2

A5

1e-6

A1

A2 A7/3

1e-12

A8/3

A3

5

10

A10/3 A

11/3

1e-4

A4 A13/3 A14/3

-4.0 -1.2

-0.6

0.0

0.6

1e-6

1.2

0.0

1.0

2.0

(a)

Harmonic Amplitude, Ak/6

Velocity, y

4T 1T 2T

5T

0.0 IC

-4.0 -1.2

5.0

1e+1 A1/3

-2.0

4.0

(b)

4.0

2.0

3.0

Harmonic Order, k/3

Displacement, x

6T

3T

-0.6

0.0 Displacement, x

(c)

0.6

1.2

a0(6)

A2/3 A1/6

A10 A4/3

1e-2

A5

1e-6

A1 A5/3

A2 A7/3

A5/6

1e-12

A8/3 A3

5

10

A10/3 A

11/3

1e-4

1e-6

A4

0.0

1.0

2.0

3.0

4.0

A13/3 A14/3

5.0

Harmonic Order, k/6

(d)

Fig. 12 Period-3 motions (Ω = 6.52): (a) trajectory, (b) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (−0.728926, 2.046610); Period-6 motions (Ω = 6.37): (c) trajectory, (d) harmonic amplitude. Initial condition (x0 , y0 ) ≈ (0.448070, 1.286008). Parameters (α = 5.5, β = 20.0, δ = 1.0, Q0 = 10).

Albert C.J. Luo, Yu Guo / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 121–150

149

4.8011e-3. The other harmonic amplitudes are A j/3 ∈ (10−9 , 10−3 )( j = 8, 9, . . . , 30), and A10 ≈ 2.1566e-12. For the period-3 motion, only seven harmonic term plus the constant term can provide a good approximation of the period-3 motion. For the harmonic amplitudes, A1/3 ≈ 0.8156 plays an important role in the period-3 motion. In traditional analysis, such a period-3 motion is called the sub-harmonic motion. The center of such a period-3 motion is on the right hand side. However, there is another asymmetric period-3 motion possessing (3)L (3)R the center on the left hand side. The constant terms are a0 = −a0 , other harmonic amplitude are the L R same. However, the phases are different with ϕ j/2l m = mod(ϕ j/2l m + ( j/2l + 1)π , 2π )(m = 3; j = 1, 2, . . . ; l = (3)

(3)R

0, 1, 2, . . . ; ). The constant term is a0 = a0 ≈ 0.0660. Consider the excitation frequency of Ω = 6.37 for period-6 motions which is given by the analytical prediction. The initial condition for the period-6 motions are (x0 , y0 ) ≈ (−0.732997 2.006567). The trajectory and harmonic amplitudes for such a period-6 motion are presented in Fig. 11 (c) and (d), respectively. After six periods, the period-6 motion returns back to the initial point, and the periodic points for six periods are presented by the large circular symbols, The constant term for (6) (6)R the period-6 motion on the right hand side is a0 = a0 ≈ 0.0832. The main harmonic amplitudes are A1/6 ≈ 0.0166, A1/3 ≈ 0.7886, A1/2 ≈ 0.0272, A2/3 ≈ 0.1059, A5/6 ≈ 4.7814e-4, A1 ≈ 0.3043, A7/6 ≈ 2.4425e-3, A4/3 ≈ 4.9541e-3, A3/2 ≈ 1.0541e-3, A5/3 ≈ 0.0228, A11/6 ≈ 1.6912e-3, A2 ≈ 5.5408e-3, A13/6 ≈ 1.4457e-4, and A7/3 ≈ 5.7796e-3. The other harmonic amplitudes are A j/6 ∈ (10−12 , 10−3 )( j = 15, 16, . . . , 60) and A10 ≈ 4.9640e-12. For this period-6 motion, 14 harmonic terms plus the constant term can provide a good approximation. The (6)L (6)R constant terms are a0 = −a0 , and other harmonic amplitude are the same. However, the phases are different with ϕ Lj/2l m = mod(ϕ Rj/2l m + ( j/2l + 1)π , 2π )(m = 3, j = 1, 2, . . . , l = 1). 6 Conclusions In this paper, the periodic motions in the Duffing oscillator were predicted analytically through the mapping structures of discrete implicit maps. The discrete implicit maps were developed from differential equation of the Duffing oscillator. From mapping structures, period-m motions in such a Duffing oscillator were predicted analytically, and the bifurcation trees of periodic motions were developed for period-1 motion to chaos and for period-3 motion to chaos. Furthermore, the corresponding stability and bifurcations of periodic motions in the bifurcation trees were analyzed through the eigenvalue analysis. From the analytical prediction of periodic motions, the bifurcation trees of periodic motions were presented through the harmonic amplitudes determined by the discrete Fourier series of periodic motions. Finally, the periodic motions based on analytical prediction were presented with comparison of numerical simulation results. The two solutions are much very well. The harmonic amplitude spectrums were also presented, which show how many harmonic terms for each periodic motion can provide a good approximation, and the corresponding analytical expression of periodic motions can be given approximately. References [1] Lagrange, J. L. (1788), Mecanique Analytique (2 vol.) edition, Albert Balnchard, Paris, 1965. [2] Poincare, H. (1899), Methodes Nouvelles de la Mecanique Celeste, Vol.3, Gauthier-Villars, Paris. [3] van der Pol, B. (1920), A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, 701–710, 754–762. [4] Fatou, P. (1928), Sur le mouvement d’un systeme soumis ‘a des forces a courte periode, Bull.. Soc. Math. 56, 98–139 [5] Krylov, N.M. and Bogolyubov, N.N. (1935), Methodes approchees de la mecanique non-lineaire dans leurs application a l’Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s’y rapportant, Academie des Sciences d’Ukraine:Kiev. (in French). [6] Hayashi, C. (1964), Nonlinear oscillations in Physical Systems, McGraw-Hill Book Company, New York. [7] Barkham, P.G.D. and Soudack, A.C. (1969), An extension to the method of Krylov and Bogoliubov, International Journal of Control, 10, 377–392.

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[8] Barkham, P.G.D. and Soudack, A.C. (1970), Approximate solutions of nonlinear, non-autonomous second-order differential equations, International Journal of Control, 11, 763–767. [9] Rand, R.H. and Armbruster, D. (1987), Perturbation Methods, Bifurcation Theory, and Computer Algebra, Applied Mathematical Sciences, 65, Springer-Verlag, New York. [10] Garcia-Margallo, J. and & Bejarano, J.D. (1987), A generalization of the method of harmonic balance, Journal of Sound and Vibration, 116, 591–595. [11] Yuste, S.B. and Bejarano, J.D. (1989), Extension and improvement to the Krylov-Bogoliubov method that use elliptic functions, International Journal of Control, 49, 1127–1141. [12] Coppola, V.T. and Rand, R.H. (1990), Averaging using elliptic functions: Approximation of limit cycle, Acta Mechanica, 81, 125–142. [13] Luo, A.C.J. (2012) Continuous Dynamical Systems, HEP/L&H Scientific, Beijing/Glen Carbon. [14] Luo, A.C.J. and Huang, J.Z. (2012) Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. Journal of Vibration and Control, 18, 1661–1871. [15] Luo, A.C.J. and Huang, J.Z. (2012) Analytical dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No. 1250093 (29 pages). [16] Luo, A.C.J. and Huang, J.Z. (2012) Analytical routines of period-1 motions to chaos in a periodically forced Duffing oscillator with twin-well potential. Journal of Applied Nonlinear Dynamics, 1, 73–108. [17] Luo A.C.J. and Huang, J.Z. (2012) Unstable and stable period-m motions in a twin-well potential Duffing oscillator. Discontinuity, Nonlinearity and Complexity, 1, 113–145. [18] Luo, A.C.J. (2005), The mapping dynamics of periodic motions for a three-piecewise linear system under a periodic excitation, Journal of Sound and Vibration, 283, 723–748. [19] Luo, A.C.J. (2005), A theory for non-smooth dynamic systems on the connectable domains, Communications in Nonlinear Science and Numerical Simulation, 10, 1–55. [20] Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer, New York. [21] Luo, A.C.J. (2009), Discontinuous Dynamical Systems on Time-varying Domains, Higher Education Press/Springer, Beijing/Heidelberg. [22] Luo, A.C.J. (2015), Periodic flows in nonlinear dynamical systems based on discrete implicit maps, International Journal of Bifurcation and Chaos, 25(3), Article No. 1550044 (62 pages).

Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 151–162

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

Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters A.V. Milovanov1,2,4 and A. Iomin3,4† 1 ENEA

National Laboratory, Centro Ricerche Frascati, I-00044 Frascati, Rome, Italy Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia 3 Department of Physics and Solid State Institute, Technion, Haifa 32000, Israel 4 Max-Planck-Institut f¨ ur Physik komplexer Systeme, 01187 Dresden, Germany 2 Space

Abstract

Submission Info

This study is concerned with destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr¨odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.

Communicated by Xavier Leoncini Received 11 September 2014 Accepted 1 December 2014 Available online 1 July 2015 Keywords Subdiffusion Cayley tree Cantor set Connectivity index

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

1 Introduction We consider the problem of dynamical localization of waves in a nonlinear Schr¨odinger model with random potential on a lattice and arbitrary power nonlinearity,

∂ ψn = Hˆ L ψn + β |ψn |2s ψn , ∂t

(1)

Hˆ L ψn = εn ψn +V (ψn+1 + ψn−1 )

(2)

i where s (s ≥ 1) is a real number; † Corresponding author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.003

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is the Hamiltonian of a linear problem in the tight binding approximation; β (β > 0) characterizes the strength of nonlinearity; on-site energies εn are randomly distributed with zero mean across a finite energy range; V is hopping matrix element; and the total probability is normalized to ∑n |ψn |2 = 1. For β → 0, the model in Eqs. (1) and (2) reduces to the original Anderson model in Ref. [1]. In the absence of randomness, the nonlinear Schr¨odinger equation (NLSE) in Eq. (1) is completely integrable. Experimentally, Anderson localization has been reported for electron gases [2], acoustic waves [3], light waves [4,5], and matter waves in a controlled disorder [6]. It is generally agreed that the phenomena of Anderson localization are based on interference between multiple scattering paths, leading to localized wave functions with exponentially decaying profiles and dense eigenspectrum [1, 7]. Theoretically, nonlinear Schr¨odinger models offer a mean-field approximation, where the nonlinear term containing |ψn |2s absorbs the interactions between the components of the wave field. It has been discussed by a few authors [8–10] that NLSE with quadratic nonlinearity (i.e., s = 1) observes a localization-delocalization transition above a certain critical strength of nonlinear interaction. That means that the localized state is destroyed, and the nonlinear field can spread across the lattice despite the underlying disorder, provided just that the β value exceeds a maximal allowed value. Below the delocalization border, the field is dynamically localized similarly to the linear case. A generalization of this result to super-quadratic nonlinearity, with s > 1, is far from trivial. In a recent investigation of NLSE with disorder, we have shown [11] that the critical strength destroying localization is only preserved through dynamics, if s = 1. For s > 1 (and similarly for 0 < s < 1, a regime not considered here), the critical strength is dynamic in that it involves a dependence on the number of already excited modes (the latter are the exponentially localized modes of the linear disordered lattice). If the field is spread across Δn states, then the conservation of the probability implies that |ψn |2 ∝ 1/Δn. As the number of already excited modes is proportional to Δn, the distance between the frequencies obeys δ ω ∝ 1/Δn; whereas the nonlinear frequency shift varies as ΔωNL ∝ 1/(Δn)s . Hence δ ω /ΔωNL ∝ (Δn)s−1 is only independent of Δn, if the nonlinearity is quadratic, i.e., s = 1. The implication is that the effect of quadratic nonlinearity does not depend on the range of field distribution; but the effect of super-quadratic (as well as sub-quadratic, with 0 < s < 1) power nonlinearity does. Hence, if initial behavior is chaotic, say, chaos remains while spreading only for s = 1. For s > 1, a transition to regularity occurs, which blocks spreading in vicinity of the criticality beyond a certain limiting number of excited modes Δnmax (Δnmax  1). One sees that quadratic nonlinearity, characterized by s = 1, plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading of the wave field already at the delocalization border. This localization-delocalization transition bears signatures, enabling to associate it with a percolation transition on the infinite Cayley tree (Bethe lattice) [10, 11]. The main idea here is that delocalization occurs through infinite clusters of chaotic states on a Bethe lattice, with occupancy probabilities decided by the strength of nonlinear interaction. Then the percolation transition threshold can be translated into a critical value of the nonlinearity control parameter, such that above this value the field spreads to infinity, and is dynamically localized in spite of these nonlinearities otherwise. This critical value when account is taken for hierarchical geometry of the Cayley tree is found to be βc = 1/ ln 2 ≈ 1.4427 [10, 11], a fancy number representing the topology of nonlinear interaction posed by the quadratic power term. It was argued based on a random walk approach that in vicinity of the criticality the spreading of the wave field is subdiffusive in the limit t → +∞, and that the second moments grow with time as a power law (3) (Δn)2 (t) ∝ t α , t → +∞, with α = 1/3 exactly. This critical regime is modeled as a next-neighbor random walk at the onset of percolation on a Cayley tree. The phenomena of critical spreading find their significance in some connection with the general problem [12] of transport along separatrices of dynamical systems with many degrees of freedom and are mathematically related to a description [13–16] in terms of Hamiltonian pseudochaos (random non-chaotic dynamics with zero Lyapunov exponents) [17, 18] and time-fractional diffusion equations.

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For s > 1, the phenomena of field spreading are limited to finite clusters at the onset of delocalization [11]. Mathematically, this regime of field spreading is complicated by the fact that finiteness of clusters on which the transport processes concentrate conflicts with the assumptions of threshold percolation, breaking the universal scaling laws [19–22] which pertain to the infinite clusters. It is not clear, therefore, how to predict and explain transport on finite clusters, avoiding as the conceptual key element the use of percolation, and what the ensuing transport laws would be. The goal of the present study is to present a general solution to this problem. The approach, which we advocate, is based on topological methods and in a sketchy form comprises three basic steps explored in Sec. II: Step 1: Enabling an equivalent reduced dynamical model of field-spreading based on backbone map. Step 2: Projecting dynamical equations on a Cayley tree with appropriately large coordination number which accommodates the power nonlinearity s (s ≥ 1). Step 3: Calculating the index of anomalous diffusion based on combinatorial arguments, using a triangulation procedure in the mapping space and the notion of one-bond-connected (OBC) polyhedron. It is shown in Sec. III that the transport on finite clusters is subdiffusive with a power law memory kernel (for time scales for which the dynamics concentrate on a self-similar geometry) and pertains to a class of nonMarkovian transport processes described by generalized diffusion equations with the fractional derivative in time. We summarize our findings in Sec. IV. 2 The three-step topological approach Expanding ψn over a basis of linearly localized modes, the eigenfunctions of the linear problem, {φn,m }, m = 1, 2, . . . , we write, with time depending complex coefficients σm (t),

ψn = ∑ σm (t)φn,m .

(4)

m

We consider ψn , ψn ∈ {ψn }, as a vector in functional space whose basis vectors φn,m are the Anderson eigenstates. For strong disorder, dimensionality of this space is infinite (countable). It is convenient to think of each node n as comprising a countable number of “compactified” dimensions representing the components of the wave field. So these hidden dimensions when account is taken for Eq. (4) are “expanded” via a topological mapping procedure to form the functional space {ψn }. We consider this space as providing the embedding space for dynamics. Further, given any two vectors ψn ∈ {ψn } and φn ∈ {ψn }, we define the inner product, ψn ◦ ϕn , ψn ◦ ϕn  = ∑ ψn∗ ϕn ,

(5)

n

where star denotes complex conjugate. To this end, the functional space {ψn } becomes a Hilbert space, permitting the notions of length, angle, and orthogonality by standard methods [23]. With these implications in mind, we consider the functions φn,m as “orthogonal” basis vectors obeying ∗ φn,k = δm,k , ∑ φn,m

(6)

n

where δm,k is Kronecker’s delta. Then the total probability being equal to 1 implies ψn ◦ ψn  = ∑ ψn∗ ψn = ∑ σm∗ (t)σm (t) = 1. n

2.1

(7)

m

Step 1: The backbone map

We define the power 2s (2s ≥ 2) of the modulus of the wave field as the power s of the probability density, i.e., |ψn |2s ≡ [ψn ψn∗ ]s . Then in the basis of linear localized modes we can write, with the use of ψn = ∑m σm φn,m , |ψn |2s = [



m1 ,m2

∗ σm1 σm∗ 2 φn,m1 φn,m ]s . 2

(8)

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It is convenient to consider the expression on the right-hand side as a functional map Fˆ s : {φn,m } → [



m1 ,m2

∗ σm1 σm∗ 2 φn,m1 φn,m ]s 2

(9)

from the vector field {φn,m } into the scalar field |ψn |2s . It is noticed that the map in Eq. (9) is positive definite, and that it contains a self-similarity character in it, such that by stretching the basis vectors (by a stretch factor λ ) the value of Fˆ s is just renormalized (multiplied by |λ |2s ). We have, accordingly, Fˆ s {λ φn,m } = |λ |2s Fˆ s {φn,m }.

(10)

Consider expanding the powerlaw on the right-hand side of Eq. (8). If s is a positive integer, then a regular expansion can be obtained as a sum over s pairs of indices (m1,1 , m1,2 ) . . . (ms,1 , ms,2 ). The result is a homogeneous polynomial, an s-quadratic form [24]. In contrast, for fractional s, a simple procedure does not exist. Even so, with the aid of Eq. (10), one might circumvent the problem by proposing that the expansion goes as a homogeneous polynomial whose nonzero terms all have the same degree 2s. “Homogeneous” means that every term in the series is in some sense representative of the whole. Then one does not really need to obtain a complete expansion of Fˆ s in order to predict dynamical laws for the transport, since it will be sufficient to consider a certain collection of terms which by themselves completely characterize the algebraic structure of Fˆ s as a consequence of the homogeneity property. We dub this collection of terms the backbone, and we define it through the homogeneous map s φ ∗s . (11) Fˆ s : {φn,m } → ∑ σms 1 σm∗s2 φn,m 1 n,m2 m1 ,m2

In what follows, we consider the backbone as representing the algebraic structure of Fˆ s in the sense of Eq. (10). So, for fractional s, our analysis will be based on a reduced model which is obtained by replacing the original map Fˆ s by the backbone map Fˆ s . The claim is that the reduction Fˆ s → Fˆ s does not really alter the scaling exponents behind the wave-spreading, since the algebraic structure of the original map is there anyway. Note that Fˆ s and Fˆ s both have the same degree 2s, which is the sum of the exponents of the variables that appear in their terms. Note, also, that the original map coincides with its backbone in the limit s → 1. This property illustrates the significance of the quadratic nonlinearity vs. arbitrary power nonlinearity. Turning to NLSE (1), if we now substitute the original power nonlinearity with the backbone map, in the orthogonal basis of the Anderson eigenstates we find iσ˙ k − ωk σk = β where



m1 ,m2 ,m3

Vk,m1 ,m2 ,m3 σms 1 σm∗s2 σm3 ,

∗ s φn,m φ ∗s φ Vk,m1 ,m2 ,m3 = ∑ φn,k 1 n,m2 n,m3

(12)

(13)

n

are complex coefficients characterizing the overlap structure of the nonlinear field, and we have reintroduced the eigenvalues of the linear problem, ωk , satisfying Hˆ L φn,k = ωk φn,k . Although obvious, it should be emphasized that the use of the backbone map Fˆ s in place of the original map Fˆ s preserves the Hamiltonian character of the dynamics, but with a different interaction Hamiltonian, Hˆ int ,

β Vk,m1 ,m2 ,m3 σk∗ σms 1 σm∗s2 σm3 . Hˆ int = 1 + s k,m1∑ ,m2 ,m3

(14)

Note that Hˆ int includes self-interactions through the diagonal elements Vk,k,k,k . Another important point worth noting is that the strength of the interaction vanishes in the limit s → ∞ (as ∼ 1/s). Therefore, keeping the β parameter finite, and letting s → ∞, one generates a regime where the nonlinear field is asymptotically localized. One sees that high-power nonlinearities act as to reinstall the Anderson localization. This is confirmed in Step

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3 below by the direct calculation of respective transport exponents. Equations (12) define a system of coupled nonlinear oscillators with a parametric dependence on s. Similarly to the NLSE model with a quadratic power nonlinearity, each nonlinear oscillator with the Hamiltonian

and the equation of motion

β hˆ k = ωk σk∗ σk + Vk,k,k,k σk∗ σks σk∗s σk 1+s

(15)

iσ˙ k − ωk σk − β Vk,k,k,k σks σk∗s σk = 0

(16)

represents one nonlinear eigenstate in the system − identified by its wave number k, unperturbed frequency ωk , and nonlinear frequency shift Δωk = β Vk,k,k,k σks σk∗s . We reiterate that non-diagonal elements Vk,m1 ,m2 ,m3 characterize couplings between each four eigenstates with wave numbers k, m1 , m2 , and m3 . The comprehension of Hamiltonian character of the dynamics paves the way for a consistency analysis of the various transport scenarios behind the Anderson localization problem (with the topology of resonance overlap taken into account) [10, 11]. To this end, the transport problem for the wave function becomes essentially a topological problem in phase space. 2.2

Step 2: Mapping on a Cayley tree

The “edge” character of onset transport corresponds to infinite chains of next-neighbor interactions with a minimized number of links at every step. For the reasons of symmetry, when summing on the right-hand side of Eq. (12), the only combinations of terms to be taken into account, apart from the self-interaction term σks σk∗s σk , s σ ∗s σ s ∗s are, essentially, σk−1 k k+1 and σk+1 σk σk−1 . These terms will come with respective interaction amplitudes Vk,k,k,k , Vk,k−1,k,k+1 , and Vk,k+1,k,k−1 , which we shall denote simply by Vk , Vk− , and Vk+ . Then on the right-hand side (r.h.s.) of Eq. (12) we have s σk∗s σk∓1 . r.h.s. = β Vk σks σk∗s σk + β ∑ Vk± σk±1

(17)

±

The interaction Hamiltonian in Eq. (14) becomes

β s [Vk σk∗ σks σk∗s σk + ∑ Vk± σk∗ σk±1 σk∗s σk∓1 ] Hˆ int = 1+s ∑ ± k

(18)

representing the effective reduced Hˆ int for arbitrary real power s ≥ 1. Assuming that the exponent s is confined between two integer numbers, i.e., j ≤ s < j + 1, in the next-neighbor interaction term we can write

β j s− j ∗s− j = σk∗ j σk∓1 ]σk±1 σk , Hˆ int ∑ Vk±[σk∗ σk±1 1+s ∑ k ±

(19)

where the prime symbol indicates that we have extracted the self-interactions. When drawn on a graph in wavenumber space, the terms raised to the power s − j will correspond to disconnected bonds, thought as Cantor sets with the fractal dimensionality 0 ≤ s − j < 1. Hence, they will not contribute to field-spreading. These terms, therefore, can be cut off from the interaction Hamiltonian, suggesting that only those terms raised to the integer power, j, should be considered. We have, accordingly,

β j → σk∗ j σk∓1 . Hˆ int ∑ Vk± σk∗σk±1 1+s ∑ k ±

(20)

This is the desired result. Equation (20) defines the effective reduced interaction Hamiltonian in the parameter range of onset spreading for j ≤ s < j + 1. Focusing on the transport problem for the wave field, because the interactions are next-neighbor-like, it is convenient to project the system of coupled dynamical equations (17) on a Cayley tree, such that each node

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Fig. 1 Mapping Eqs. (12) on a Cayley tree for s = 1. Each node represents a nonlinear eigenstate, or nonlinear oscillator with the equation of motion iσ˙ k − ωk σk − β Vk,k,k,k σks σk∗s σk = 0. Blue circles represent oscillators in a chaotic (“dephased”) state. Black circles represent oscillators in regular state. One ingoing and two outgoing bonds on node k (k = 1, 2, ...) represent, respectively, the complex amplitudes σk∗ , σk−1 and σk+1 . (Adapted from Ref. [11])

with the coordinate k represents a nonlinear eigenstate, or nonlinear oscillator with the equation of motion (16); the outgoing bonds represent the complex amplitudes σk±1 and σk∓1 ; and the ingoing bonds, which involve complex conjugation, represent the complex amplitudes σk∗ . To make it with the amplitudes σk∗ when raised to the algebraic power s one needs for each node a fractional number s of the ingoing bonds. Confining the s value between two nearest integer numbers, j ≤ s < j + 1, we carry on with j connected bonds, which we charge to receive the interactions, and one disconnected bond, which corresponds to a Cantor set with the fractal dimensionality s − j, and which cannot transmit the waves. At this point we cut this bond off the tree. A similar procedure applied to the amplitudes σk±1 , coming up in the algebraic power s, generates j outgoing bonds, leaving one disconnected bond behind. Lastly, the remaining amplitude σk∓1 , which does not involve a nonlinear power, contributes with one outgoing bond for each combination of the indexes. One sees that the mapping requires a Cayley tree with the coordination number z = 2 j + 1. The mapping of dynamical equations (12) with the next-neighbor interaction term in Eq. (17) is illustrated in Fig. 1 for quadratic power nonlinearity, i.e., s = 1, generating the familiar Cayley tree with the coordination number z = 3. 2.3

Step 3: Obtaining the connectivity index

If the interactions are next-neighbor-like, and if the number of excited modes after t time steps is Δn(t), then self-similarity will imply that (21) (Δn)2 (t) ∝ t 2/(2+θ ) , t → +∞, where θ is the connectivity exponent of the structure on which the spreading processes occur. This exponent accounts for the deviation from the usual Fickian diffusion in a self-similar geometry [19–21, 25] and observes remarkable invariance properties under homeomorphic maps of fractals [22, 26]. The scaling law in Eq. (21) has been discussed by Gefen et al. [25] for anomalous diffusion on percolation clusters. The crucial assumption behind this scaling, however, is the assumption of self-similarity (and not of percolation) extending the range of validity of Eq. (21) to any self-similar fractal. Here we apply the scaling law in Eq. (21) to Cayley trees by appropriately choosing the θ value. We note in passing that self-similarity of the Cayley trees is not necessarily manifest in their folding in the embedding space, but is inherent in their connectedness and topology [27]. Indeed a Cayley tree is a graph without loops, where each node hosts the same number of branches (known as the coordination number). Therefore, one might expect from the outset that the value of θ will be a function of

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Fig. 2 Determination of one-bond-connections for a tetrahedron in R3 .

the coordination number, given that the dynamics occur on a Cayley tree. The coordination number, in its turn, will depend on the power s, thus paving the way for a theoretical prediction of onset spreading in association with the topology of interaction between the components of the wave field. More so, the power nonlinearity in Eq. (1) suggests that the connectivity value is a multiplicative function of s. Also one might expect this function to naturally reproduce the known value θ = 4 [19, 20, 28] for meanfield percolation on Bethe lattices in the limit s → 1. Then the obvious dependence satisfying these criteria is θ = 4s, where s ≥ 1. For the integer and half-integer s, this dependence can also be derived using the standard renormalization-group procedure [29] for self-similar clusters in Hilbert space. So restricting ourselves to the short times for which the dynamics concentrate on a self-similar geometry, we write, with Δnmax  1, (Δn)2 (t) ∝ t 1/(2s+1) , 1  t  (Δnmax )2(2s+1) ,

(22)

from which the scaling dependence in Eq. (3) can be deduced for quadratic nonlinearity, i.e., α = 1/3. Numerically, the field-spreading on finite clusters has been already discussed [30–33] based on computer simulation results, using one-dimensional disordered Klein-Gordon chains with tunable nonlinearity [32, 33]. It is noticed that the exponent of the power law, α = 1/(2s + 1), vanishes in the limit s → ∞, conformally with the previous considerations. To illustrate the determination of θ and to address the origin of subdiffusion in the regime of next-neighbor communication rule, we look directly into the connectivity properties of finite clusters. For this, we need a simple procedure by which calculations can be done exactly. We formulate such a procedure for integer and half-integer values of s using topological triangulation [34, 35] of the Cayley tree. For the purpose of formal analysis, it is essential to choose a node on a Cayley tree and a reference system of z next-neighbor connecting bonds (for a Cayley tree with the coordination number z = 2s + 1). Next we dispose the selected node of the Cayley tree and immerse it into a z-dimensional Euclidean space R2s+1 . The latter space is built on 2s + 1 orthonormal basis vectors. Note that there is a one-to-one correspondence between the basis vectors in R2s+1 and the reference bonds on the Cayley tree. Connecting the ending points of the basis vectors generates a polyhedron in R2s+1 , which reflects the connectivity of the original Cayley tree and the hierarchical composition of this. For the standard Cayley tree with z = 3 the associated geometric construction is illustrated in Fig. 2. More so, we apply the above procedure to all nodes of the original Cayley tree, such that the nodes which communicate via a next-neighbor rule on the tree go to the ending points of the corresponding basis vectors in R2s+1 . One sees that this procedure generates an infinite chain of mutually overlapping polyhedrons. The number of internal one-bond-connections (OBC’s) is obtained as the minimal number of bonds belonging to the same polyhedron and enabling an infinite connected mesh. We distinguish between “nodes” which compose a polyhedron, that is analyzed, and “node-vertexes” which are nodes pertaining to neighboring polyhedrons. In what follows, we identify the nodes with numbers, and node-vertexes with letters. For instance, the only node having a full family of nearest neighbors in Figs. 3 and 4 is the node marked as 1. The connectivity index θ is obtained as the number of paths (routes without self-crossings) connecting node 1 to node 2s + 2 via any nodes of the same polyhedron.

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a

2

s=1

1 b

4 c

3

a 2

3

1

d

s = 3/2

b 4

5 c

a 2

e

3 b

1 6

s=2

4 5

d

c

Fig. 3 Schematic representation of polyhedrons for s = 1, 3/2, and 2. Nodes belonging to the same polyhedron are marked by numbers: 1, 2, 3, etc. Nodes-vertexes pertaining to other polyhedrons are marked by letters: a, b, c, etc. One-bond-connections are shown by solid lines; virtual connections between the nodes, by broken lines.

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2

159

3 1

4

s = 5/2

7 5

6

2

9

3

4

1

8

s = 7/2

5 7

6

Fig. 4 Schematic representation of polyhedrons for the half-integer s = 5/2 and s = 7/2.

To illustrate, consider quadratic nonlinearity first, with s = 1 (see Fig. 2). Clearly, there are just three OBC’s defined by a tetrahedron with nodes 1, 2, 3 and 4. So one identifies these OBC’s with the bonds (1 − 2), (1 − 3), and (1 − 4). It is noticed that the connection between nodes 3 and 4 occurs via the node-vertex c; the connection between nodes 2 and 3 occurs via the node-vertex b; and the connection between nodes 2 and 4, via the nodevertex a. The connectivity index θ is the number of paths connecting node 1 to node 2s + 2 = 4. These paths are just four, namely, (1 − 4), (1 − 2 − 4), (1 − 3 − 4), and (1 − 2 − 3 − 4). Hence, θ = 4. This result is to be expected, as it also characterizes mean-field transport on lattice animals [19, 20] and trees [28]. With the aid of Eqs. (21) and (22) one also obtains (Δn)2 (t) ∝ t 1/3 for t  (Δnmax )6 consistently with the result of Ref. [11]. Let us now calculate the connectivity index θ for half-integer s = 3/2. Here one constructs a pentahedron in 4 R , with nodes marked 1, 2, 3, 4 and 5. The OBC’s are the bonds (1 − 2), (1 − 3), (1 − 4), and (1 − 5) (see Fig. 2). There are exactly six paths connecting node 1 to node 5, that is, (1 − 5), (1 − 2 − 5), (1 − 4 − 5), (1 − 3 − 2 − 5), (1 − 3 − 4 − 5), and (1 − 2 − 3 − 4 − 5). Thus, θ = 6, leading to a subdiffusive scaling of second moments (Δn)2 (t) ∝ t 1/4 for t  (Δnmax )8 . The same triangulation procedure applied to a hexahedron in R5 generates for s = 2 the following eight paths (see Fig. 2): (1−6), (1−2−6), (1−5−6), (1−3−2−6), (1−4−5−6), (1−4−3−2−6), (1−3−4−5−6), and (1 − 2 − 3 − 4 − 5 − 6), leading to θ = 8. The scaling of second moments is given by (Δn)2 (t) ∝ t 1/5 for t  (Δnmax )10 .

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In Fig. 4 we also present for reader’s convenience respective geometric constructions corresponding to halfinteger s = 5/2 and s = 7/2, facilitating the calculation of the paths and of respective connectivity values. By mathematical induction the connectivity index for integer and half-integer s is given by θ = 4s, yielding for the transport exponent α = 2/(2 + θ ) = 1/(2s + 1) consistently with the subdiffusive scaling in Eq. (22). Eliminating s with the aid of coordination number, z = 2s + 1, we also get α = 1/z. One sees that the transport is slowed down by complexity elements of clusters, contained in the z value. All in all, one sees that higher-order nonlinearities (s > 1) have a progressively weakening effect over the transport rates, with the fastest transport obtained for quadratic power nonlinearity. 3 Non-Markovian diffusion equation The next-neighbor communication rule which we associate with the phenomena of onset spreading must have implications for anomalous diffusion on the short times for which the dynamics concentrate on a self-similar geometry of finite clusters. This equation has been already discussed [13–16] and has been shown to be a non-Markovian variant of the diffusion equation with power law memory kernel: ˆ 1 ∂ t ∂ dt ∂2 f (t, Δn) = [Wθ f (t , Δn)], (23) ∂t Γ(α ) ∂ t 0 (t − t )1−α ∂ (Δn)2 where 1 − α = θ /(2 + θ ); θ is the connectivity exponent; Γ(α ) is the Euler gamma function; Wθ absorbs the parameters of the transport model; and we have chosen t = 0 as the beginning of the system’s time evolution. The integral term on the right-hand side has the analytical structure of fractional time the so-called RiemannLiouville fractional derivative [36]. In a compact form,

∂ ∂ 1−α ∂ 2 f (t, Δn) = 1−α [Wθ f (t, Δn)]. ∂t ∂t ∂ (Δn)2

(24)

The fractional order of time differentiation in Eq. (24) is determined by the connectivity value through 1 − α = θ /(2 + θ ) and is exactly zero for θ = 0. Then the fractional derivative of the zero order is a unity operator, implying that no fractional properties come into play for homogeneous spaces. Also in writing Eq. (23) we have adopted results of Refs. [13–15] to diffusion processes on a single cluster. Equations (23) and (24) when account is taken for the initial value problem can be rephrased [13] in terms of the Caputo fractional derivative [36]. When properly derived, the fractional diffusion equation can equally be written in terms of the RiemannLiouville or Caputo fractional derivatives. However, the latter is more appreciable in the applications, as it shows a better behavior under the Laplace transform. One sees that the dispersion law in Eq. (21) can be obtained as a second moment of the fractional diffusion equation (23), with α = 2/(2 + θ ). Using for the connectivity exponent θ = 4s, one also finds the fractional order 1 − α = 2s/(2s + 1) in the entire parameter range s ≥ 1, showing that ordinary differentiation is reinstalled on the right-hand side of Eq. (24) in the limit s → ∞. For s = 1, one gets 1 − α = 2/3, implying that the diffusion process is essentially non-Markovian with power-law correlations in the regime of quadratic nonlinearity. We associate this non-Markovian character of field-spreading with the effect of complexity elements of the Cayley tree, contained in the z = 2s + 1 value. It is noticed that the fractional diffusion equation in Eq. (24) is “born” within the exact mathematical framework of nonlinear Schr¨odinger equation with usual time differentiation. Indeed, no ad hoc introduction of fractional time differentiation in the dynamic Eq. (1) has been assumed to obtain this subdiffusion. It is, in fact, the interplay between nonlinearity and randomness, which leads to a non-Markovian transport of the wave function at criticality, and to a time-fractional kinetic equation in the end. This observation also emphasizes the different physics implications behind the fractional kinetic vs. dynamical equations [16, 37, 38]. Equation (24) shows that the onset spreading is a matter of fractional, or “strange,” kinetics [39–41] consistently with the implication of critical behavior [13, 16, 18].

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The fundamental solution or Green’s function of the fractional Eq. (24) is evidenced in Table 1 of Ref. [42]. 4 Conclusions This study is concerned with destruction of Anderson localization by a nonlinearity of the power-law type. It has been proposed using an NLSE with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localizationdelocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to a diffusion process on finite clusters. We have suggested an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinear Anderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. Also we predict that the transport of waves at the border of delocalization is subdiffusive, with the exponent α which is inversely proportional with the power nonlinearity increased by one. For quadratic nonlinearity we have (Δn)2 (t) ∝ t 1/3 for t → +∞ consistently with the previous investigations [10, 11]. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations. Note, that this dynamics, encoded in Eqs. (23) and (24), is nonergodic in the sense that individual realizations of the associated processes show pronounced amplitude fluctuations (see the recent review in Ref. [43]). Acknowledgement A.V.M. and A.I. thank the Max-Planck-Institute for the Physics of Complex Systems for hospitality and financial support. This work was supported in part by the Israel Science Foundation (ISF) and by the ISSI project “SelfOrganized Criticality and Turbulence” (Bern, Switzerland). References [1] Anderson, P.W. (1958), Absence of diffusion in certain random lattices, Physical Review, 109, 1492–1506. [2] Akkermans, E. and Montambaux, G. (2006), Mesoscopic Physics of Electrons and Photons, Cambridge University Press, Cambridge. [3] Weaver, R.L. (1990), Anderson localization of ultrasound, Wave Motion, 12, 129–142. [4] St¨orzer, M., Gross, P, Aegerter, C.M., and Maret, G. (2006), Observation of the critical regime near Anderson localization of light, Physical Review Letters, 96, 063904. [5] Schwartz, T., Bartal, G., Fishman, S., and Segev, M. (2007), Transport and Anderson localization in disordered twodimensional photonic lattices, Nature (London), 446, 52–55. [6] Billy, J., Josse, V., Zuo, Z., Bernard, A., Hambrecht, B., Lugan, P., Cl´ement, D., Sanchez-Palencia, L., Bouyer, P., and Aspect, A. (2008), Direct observation of Anderson localization of matter waves in a controlled disorder, Nature (London), 453, 891–894. [7] Abou-Chacra, R., Anderson, P.W., and Thouless, D.J. (1973), A selfconsistent theory of localization, Journal of Physics C (Solid State Physics), 6, 1734–1752. [8] Shepelyansky, D.L. (1993), Delocalization of quantum chaos by weak nonlinearity, Physical Review Letters, 70, 1787– 1791. [9] Pikovsky, A.S. and Shepelyansky, D.L. (2008), Destruction of Anderson localization by a weak nonlinearity, Physical Review Letters, 100, 094101. [10] Milovanov, A.V. and Iomin, A. (2012), Localization-delocalization transition on a separatrix system of nonlinear Schr¨odinger equation with disorder, Europhysics Letters, 100, 10006. [11] Milovanov, A.V. and Iomin, A. (2014), Topological approximation of the nonlinear Anderson model, Physical Review E, 89, 062921. [12] Chirikov, B.V. and Vecheslavov, V.V. (1997) Arnold diffusion in large systems, Journal of Experimental and Theoretical Physics, 112, 616–624.

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Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

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

A Method for Solving Nonlinear Differential Equations: An Application to λ φ 4 Model Danilo V. Ruy † Instituto de F´ısica Te´orica-UNESP, Rua Dr Bento Teobaldo Ferraz 271, Bloco II S˜ao Paulo, 01140-070, Brazil Submission Info Communicated by Xavier Leoncini Received 27 August 2014 Accepted 27 January 2015 Available online 1 July 2015

Abstract Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad´e approximants for solving nonlinear partial differential equations without requiring a one-dimensional reduction. This method is applied to the λ φ 4 model in 4 dimensions and new solutions are obtained.

Keywords Integrable equations in physics Integrable field theory Pad´e approximants λ φ 4 model ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction For many years, nonlinear differential equations have been an important topic of study in many branches of knowledge. This interest has led to the development of many techniques through the last few years in order to obtain exact solutions without requiring further properties of the differential equation (for example [1–18]). In [13], it was proposed the multiple Exp-function method for finding exact solutions of partial differential equation (PDE) without requiring a one-dimensional reduction. Although this method is very powerful, there are a large number of parameters to be determined. Here, we present a simpler algorithm based on Pad´e approximants and apply it to the classical equation of the λ φ 4 model with m = 0. There are several papers generalizing Pad´e approximants for multivariate functions [19–25]. Here, we focus on the homogeneous Pad´e approximant, introduced in [21] (see also [24]). The λ φ 4 model is one of the simplest example of a renormalizable scalar field theory and it is defined by the Lagrangian density m2 λ 1 (1) L = ∂μ φ ∂ μ φ − φ 2 − φ 4 , 2 2 4 where we use the metric η = (− + ++) and the notation of repeated indices summed. The Euler-Lagrange † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.004

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D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

equation, i. e

∂L ∂L − = 0, ∂ ∂μ φ ∂φ

∂μ of (1) yields the classical equation of motion

−∂t2 φ + ∇2 φ + m2 φ + λ φ 3 = 0.

(2)

Static solutions of the two-dimensional λ φ 4 model was already presented in [27] and [26]. Here, we construct some solutions of equation (2) representing travelling waves and the scattering of two travelling waves. The paper is organized in two main sections. In section 2, I show a practical introduction to the homogeneous multivariate Pad´e approximant and present a new approach for solving nonlinear PDEs. The section 3 is devoted to the λ φ 4 model. 2 The method 2.1

Homogeneous multivariate Pad´e approximants

Consider a function f (z) = f (z1 , . . . , zD ) regular at origin and with Taylor expansion around z = 0 given by f (z) =



∑ cJzJ ≡

J=0





∑ ∑

j1 =0 j2 =0

...





jD =0

D

c j1 , j2 ,..., jD ∏ zdjd . d=1

The homogeneous multivariate Pad´e approximant consist in rearranging the coefficients such that we can use the Pad´e approximant in one dimension. Through the map z → ξ z, the Taylor Expansion can be rearranged as f (ξ z) =

L+M

∑ an (z)ξ n + O(ξ L+M+1)

n=0

where an (z) =

n n− j1

n−∑D−2 r=1 jr

j1 =0 j2 =0

jD−1 =0

∑ ∑ ... ∑

D−1

n−∑D−1 r=1 jr

c j1 , j2 ,..., jD−1 ,n−∑D−1 jr ( ∏ zdjd )zD r=1

.

d=1

This rearrangement allows us to compute the univariate Pad´e approximant of f (ξ z) on ξ , i. e. ∑Lj=0 p j (z)ξ j Pz,L (ξ ) = , [L/M]z (ξ ) ≡ Qz,M (ξ ) 1 + ∑Mj=1 q j (z)ξ j where the coefficients p j (z) and q j (z) are determined such that the Pad´e approximant agrees with f (ξ z) up to the degree L + M, i. e. f (ξ z) = [L/M]z (ξ ) + O(ξ L+M+1 ). Thus we need to solve the following system of equations j = 0, 1, . . . , L + M. ∑ ar (z)qs (z) − p j (z) = 0, r+s= j

This system can be easily solved by a symbolic computation software. Concluding, the homogeneous Pad´e approximant for f(z) is obtained by setting f (z) = f (ξ z)|ξ =1 . 2.2

The functional ansatz

Consider now a system of Ne equations with Ne fields in D dimensions, i. e. Ek (xμ , φi , ∂μ φi , . . . ; S0 ) = 0,

k = 1, . . . , Ne ,

D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

165

where S0 is the space formed by the Cartesian product of the set of parameters, μ = 1, . . . , D and i = 1, . . . , Ne . Now, let us suppose at least one solution for this system can be expressed as functionals of a set of functions ρ = (ρ1 , . . . , ρNρ ), i.e. φi (xμ ) = φˆi (ρ1 , . . . , ρNρ ), i = 1, . . . , Ne , where ρk = ρk (xμ ) for k = 1, . . . , Nρ . Moreover, suppose the first derivative of all ρk are known in terms of ρ , i. e. ∂μ ρk = Fμ ,k (ρ1 , . . . , ρNρ ; S1 ), μ = 1, . . . , D, k = 1, . . . , Nρ , (3) where S1 is the space formed by the Cartesian product of the sets of parameters introduced by ρ . The choice of ρ is the first ansatz of the algorithm and it yields the transformation Ek (xμ , φi , ∂μ φi , . . . ; S0 ) = Eˆk (ρk , φˆi , ∂k φˆi , . . . ; S0 × S1 ) = 0,

k = 1, . . . , Ne ,

(4)

where Eˆ k is polynomial or rational in ρk , φˆi and its derivatives. Now, the system (4) can be worked out as a system in Nρ dimensions. If at least one particular solution for the set of fields φˆi are regular at origin, we can consider a multivariate Taylor expansion at ρ = 0, i. e.

φˆi (ρ ) =





∑ ∑

j1 =0 j2 =0

...





jNρ =0

ci; j1 , j2 ,..., jNρ



∏ ρdj ,

i = 1, . . . , Ne ,

d

(5)

d=1

where ci; j1 , j2 ,..., jNρ = ci; j1 , j2 ,..., jNρ (S0 × S1 ). Observe that the expansion (5) can be drastically changed due to the particular combination of the parameters in the space formed by S0 × S1 . Therefore, for obtaining particular solutions, we can consider a set of constraints ψi = ψi (S0 × S1 ) = 0 acting on ρ and Eˆk (ρk , φˆi , ∂k φˆi , . . . ; S0 × S1 ) = 0. When a constraint ψ is considered, the notation ρ¯ ≡ ρ |ψ =0 will be employed. Let us call S2 the space formed by the Cartesian product of all undetermined ci; j1 , j2 ,..., jNρ and define S = S0 × S1 × S2 . Now, we can use the homogeneous multivariate Pad´e approximant in φˆi (ρ ). Mapping ρ → ξ ρ , we can rearrange the expansion (5) as

φˆi (ξ ρ ) =

Li +Mi



ai;n (ρ )ξ n + O(ξ Li +Mi +1 ),

i = 1, . . . , Ne ,

n=0

ai;n (ρ ) =

n n− j1

Nρ −2

n−∑r=1

∑ ∑ ... ∑

j1 =0 j2 =0

Nρ −1

jr

jNρ −1 =0

ci; j

Nρ −1 1 , j2 ,..., jNρ −1 ,n−∑r=1 jr

(

d

d=1

Nρ −1

n−∑r=1

∏ ρdj )ρN

ρ

jr

.

Thus, we can apply the univariate Pad´e approximant on ξ in order to obtain an approximation of the solution, i.e. Pρ ,Li (ξ ; S ) + O(ξ Li +Mi +1 ). φˆi (ξ ρ ) = (6) Qρ ,Mi (ξ ; S ) Finally, we can apply the second ansatz. Let us assume that there is a particular subset Sˆ ⊂ S , such that expression (6) yields an exact solution when ξ = 1, i. e.  Pρ ,Li (ξ ; Sˆ )  ˆ φi (ρ ) = . (7) Qρ ,Mi (ξ ; Sˆ ) ξ =1 This idea was used in [18] for the simpler case when D = Ne = Nρ = 1 and ρ1 = z (where z was the onedimensional variable). In order to determine Sˆ , let us substitute (7) in (4). This yields ˆ ∑Λ Eˆk;n (S) = 0, Eˆ k (ρk , φˆi , ∂k φˆi , . . . ; S0 × S1 ) = n=0 ˆ Dk (S)

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D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

ˆ = Eˆ k;n (S)

Nρ −2

n−∑r=1

n n− j1

∑ ∑ ... ∑

j1 =0 j2 =0

jr

jNρ −1 =0

Nρ −1

E˜k; j

Nρ −1 1 , j2 ,..., jNρ −1 ,n−∑r=1 jr

(

Nρ −1

n−∑r=1

∏ ρdj )ρN d

ρ

d=1

jr

.

(8)

where Λ is determined by the choices of Li , Mi , the set ρ and the differential equation under consideration. Hence, in order to determine all elements of Sˆ , we need to solve the following algebraic system: ˆ = 0, k = 1, . . . , Ne , E˜ k; j1 , j2 ,..., jNρ (S)

l−1

jl = 0, . . . , n − ∑ jr ,

n = 0, . . . , Λ,

(9a)

r=1

ˆ = 0, k = 1, . . . , Ne . Dk (S)

(9b)

Step (9a) may require a huge computational power for some models, but it yields a system of algebraic equations smaller than the one we should solve by using the multiple Exp-function method [13]. 3 An application to the λ φ 4 theory in 4 dimensions Here, we apply the algorithm presented in section 2.2 for the λ φ 4 theory, i. e. −∂t2 φ + ∇2 φ + m2 φ + λ φ 3 = 0,

(10)

by using two different functional ansatz: (i) φ (xμ ) = φˆ (ρ1 ), (ii) φ (xμ ) = φˆ (ρ1 , ρ2 ),

ρ1 = ei(k1,0 t+k1,1 x+k1,2 y+k1,3 z) , ρ1 = ei(k1,0 t+k1,1 x+k1,2 y+k1,3 z) ,

(11) i(k2,0 t+k2,1 x+k2,2 y+k2,3 z)

ρ2 = e

.

(12)

For simplicity, let us use the notation kj = (k j,1 , k j,2 , k j,3 ), where ki .kj = ki,1 k j,1 + ki,2 k j,2 + ki,3 k j,3 . 3.1

Ansatz (i)

First, consider the ansatz (i). Obviously, this set for ρ satisfies condition (3), namely

∂t ρ1 = ik1,0 ρ1 ,

∂x ρ1 = ik1,1 ρ1 ,

∂y ρ1 = ik1,2 ρ1 ,

∂z ρ1 = ik1,3 ρ1 ,

and yields the equation 2 ˆ ρ1 , φˆ , ∂ρ1 φˆ , ∂ρ2 φˆ ; S0 × S1 ) ≡ (k1,0 − k1 2 )(ρ12 ∂ρ21 φˆ + ρ1 ∂ρ1 φˆ ) + m2 φˆ + λ φˆ 3 = 0. E( 1

The first element of the Taylor expansion of φ can be c0 = 0 or c0 = any constraint ψ , this expansion yields two trivial solutions, namely,

φˆ = 0

and

iμ m φˆ = √ , λ

i√ μm λ

(13)

where μ = ±1. Without imposing

μ = ±1.

(14)

However, a convenient choice for ψ leads us to a more interesting expansion. If we impose 2 ψ = −k1,0 + k1 2 − m2 = 0

(15)

on equation (13) and the set ρ by eliminating k1,0 , we get

where ρ¯1 = ei(ν form



−m2 (ρ¯12 ∂ρ2¯1 φˆ + ρ¯1 ∂ρ¯1 φˆ ) + m2 φˆ + λ φˆ 3 = 0,

(k1 2 −m2 )t+k1,1 x+k1,2 y+k1,3 z)

(16)

and ν = ±1. The expansion of φˆ starting with c0 = 0 then has the

c3 λ c5 λ 2 c7 λ 3 φˆ = (c1 ρ¯1 )ξ + ( 1 2 ρ¯3 )ξ 3 + ( 1 4 ρ¯5 )ξ 5 + ( 1 6 ρ¯7 )ξ 7 + · · · , 8m 64m 512m

m = 0

(17)

D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

167

while the expansion starting with c0 = i√μ m truncate at the constant term. By employing the Pad´e approximant λ [1/1](ρ1 ) (ξ )|ξ =1 of expansion (17), ansatz (7) yields

φˆ = c1 ρ¯1 ,

(18)

whose substitution into equation (20) leads us to the conditions: ˆ = c31 λ = 0, E˜ 1;3 (S) ˆ = 1 = 0. D1 (S) Here, we see that the condition for (18) to be a solution of (20) is c1 = 0 or λ = 0, which represents the vacuum solution and the Klein-Gordon limit respectively. Now, let us consider the Pad´e approximant [2/2](ρ1 ) (ξ )|ξ =1 . In this case, the ansatz (7) yields

φˆ =

8c1 m2 ρ¯1 . 8m2 − c21 λ ρ¯12

(19)

By substituting (19) into (20), we can check that expression (19) already is an exact solution without requiring any further conditions. We also obtain (19) if we use the ansatz [3/3](ρ1 ) (ξ )|ξ =1 , [4/4](ρ1 ) (ξ )|ξ =1 or [5/5](ρ1 ) (ξ )|ξ =1 . Now, consider the constraint 2 ψ = −k1,0 + k1 2 + 2m2 = 0. By eliminating k1,0 , equation (13) yields 2m2 (ρ¯12 ∂ρ2¯1 φˆ + ρ¯1 ∂ρ¯1 φˆ ) + m2 φˆ + λ φˆ 3 = 0,

(20)

√ 2 2 where ρ¯1 = ei(ν (k1 +2m )t+k1,1 x+k1,2 y+k1,3 z) for ν = ±1. Using this constraint, the expansion of φˆ starting with c0 = 0 truncates at the constant term, while the expansion starting with c0 = i√μ m has the form λ



c3 λ iμ m iμ c2 λ 2 2 iμ c4 λ 3/2 φˆ = √ + (c1 ρ¯1 )ξ − ( 1 ρ¯1 )ξ − ( 1 2 ρ¯13 )ξ 3 + ( 1 3 ρ¯14 )ξ 4 + · · · 2m 4m 8m λ

m = 0.

(21)

By employing the Pad´e approximant [1/1](ρ1 ) (ξ )|ξ =1 and [2/2](ρ1 ) (ξ )|ξ =1 of expansion (21), ansatz (7) yields √ m(c λ ρ¯1 + 2iμ m) 1 √ , (22) φˆ = [1/1](ρ1 ) (ξ )|ξ =1 = √ λ (2m + iμ c1 λ ρ¯1 ) √ 4c1 λ m2 ρ¯1 + iμ (4m3 − c21 λ mρ¯12 ) ˆ √ . (23) φ = [2/2](ρ1 ) (ξ )|ξ =1 = λ (4m2 + c21 λ ρ¯12 ) Both (22) and (23) are exact solutions without requiring further conditions on the space of constants S . The solution (23) is also obtained if we consider the ansatz [3/3](ρ1 ) (ξ )|ξ =1 , [4/4](ρ1 ) (ξ )|ξ =1 or [5/5](ρ1 ) (ξ )|ξ =1 . 3.2

Ansatz (ii)

Now, consider the ansatz (ii). Clearly, this set for ρ also satisfies condition (3), namely

∂t ρ1 = ik1,0 ρ1 , ∂t ρ2 = ik2,0 ρ2 ,

∂x ρ1 = ik1,1 ρ1 , ∂x ρ2 = ik2,1 ρ2 ,

∂y ρ1 = ik1,2 ρ1 , ∂y ρ2 = ik2,2 ρ2 ,

∂z ρ1 = ik1,3 ρ1 , ∂z ρ2 = ik2,3 ρ2 ,

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D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

and yields the equation 2 2 ˆ ρ1 , ρ2 , φˆ , , . . . ; S0 × S1 ) ≡ (k1,0 E( − k1 2 )(ρ12 ∂ρ21 φˆ + ρ1 ∂ρ1 φˆ ) + (k2,0 − k2 2 )(ρ22 ∂ρ22 φˆ +ρ2 ∂ρ2 φˆ ) + 2(k1,0 k2,0 − k1 .k2 )ρ1 ρ2 ∂ρ1 ∂ρ2 φˆ + m2 φˆ

(24)

+λ φˆ 3 = 0.

Without imposing any constraint φ , this functional ansatz yields solution (14). In addition, if we impose the constraint 2 ψ1 = −k1,0 + k1 2 − m2 = 0, (25) or

2 ψ2 = −k1,0 + k2 2 − m2 = 0,

(26)

we obtain solution (19) up to a redefinition of an arbitrary constant. However, if we employ constraints (25) and (26) and eliminate k1,0 and k2,0 , equation (24) yields −m2 (ρ12 ∂ρ21 φˆ + ρ1 ∂ρ1 φˆ + ρ22 ∂ρ22 φˆ + ρ2 ∂ρ2 φˆ ) + 2(k1,0 k2,0 − k1 .k2 )ρ1 ρ2 ∂ρ1 ∂ρ2 φˆ + m2 φˆ + λ φˆ 3 = 0.

(27)

Observe that we did not eliminate the linear terms of k1,0 and k2,0 at this stage in order to avoid mistakes with the sign of the root square. The multivariate Taylor expansion of (27) yields

φˆ = (c1,0 ρ¯1 + c0,1 ρ¯2 )ξ + (

λ (c31,0 ρ¯13 + c30,1 ρ¯23 ) 3λ (c21,0 c0,1 ρ¯12 ρ¯2 + c1,0 c20,1 ρ¯1 ρ¯22 ) 3 )ξ + · · · , − 8m2 4(k1,0 k2,0 − k1 .k2 − m2 )

(28)

for m = 0 and k1,0 k2,0 − k1 .k2 − m2 = 0. The Pad´e approximant [1/1](ρ1 ,ρ2 ) (ξ )|ξ =1 of expansion (28) yield the ansatz

φˆ = c1,0 ρ¯1 + c0,1 ρ¯2 ,

(29)

such that the conditions for (29) to become an exact solution are ˆ = c31,0 λ = 0, E˜ 1;3,0 (S) ˆ = 3c21,0 c0,1 λ = 0, E˜ 1;2,1 (S) ˆ = 3c1,0 c20,1 λ = 0, E˜ 1;1,2 (S) ˆ = c30,1 λ = 0, E˜ 1;0,3 (S) ˆ = 1 = 0. D1 (S) Hence, the only possibility for (29) to be a solution of (24) is in the Klein-Gordon limit, i. e. λ = 0. Employing the ansatz [2/2](ρ1 ,ρ2 ) (ξ )|ξ =1 , we obtain

φˆ =

8m2 (−k1,0 k2,0 + k1 .k2 + m2 )(c1,0 ρ¯1 + c0,1 ρ¯2 ) . (k1,0 k2,0 − k1 .k2 − m2 )(λ (c21,0 ρ¯12 − c1,0 c0,1 ρ¯1 ρ¯2 + c20,1 ρ¯22 ) − 8m2 ) − 6m2 c1,0 c0,1 λ ρ¯1 ρ¯2

By substituting ansatz (30) into (24), we obtain the conditions ˆ = 64c41,0 c0,1 λ 2 m2 (k1,0 k2,0 − k1 .k2 − m2 )2 (k1,0 k2,0 − k1 .k2 + m2 ) E˜ 1;4,1 (S) (k1,0 k2,0 − k1 .k2 + 2m2 ) = 0, ˆ = −96c31,0 c20,1 λ 2 m2 (k1,0 k2,0 − k1 .k2 − m2 )3 (k1,0 k2,0 − k1 .k2 + m2 ) = 0, E˜ 1;3,2 (S) ˆ = −96c21,0 c30,1 λ 2 m2 (k1,0 k2,0 − k1 .k2 − m2 )3 (k1,0 k2,0 − k1 .k2 + m2 ) = 0, E˜ 1;2,3 (S) ˆ = 64c1,0 c40,1 λ 2 m2 (k1,0 k2,0 − k1 .k2 − m2 )2 (k1,0 k2,0 − k1 .k2 + m2 ) E˜ 1;1,4 (S) (k1,0 k2,0 − k1 .k2 + 2m2 ) = 0, ˆ = [(k1,0 k2,0 − k1 .k2 − m2 )(c21,0 λ ρ¯12 − c1,0 c0,1 λ ρ¯1 ρ¯2 + c20,1 λ ρ¯22 − 8m2 ) D1 (S) −6m2 c1,0 c0,1 λ ρ¯1 ρ¯2 ]3 = 0.

(30)

D. V. Ruy / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 163–171

169

The system of conditions above has four possibilities for solutions, namely c1,0 = 0, c0,1 = 0, λ = 0 or k1,0 k2,0 − k1 .k2 + m2 = 0.

(31)

It is easy to see that conditions c1,0 = 0 and c0,1 = 0 yield the same solution (19) up to a redefinition of the arbitrary constants, while the condition λ = 0 yields the solution (29). We can also use condition (31) and the constraints (25) and (26) for eliminating three constants, for example

k1,0 =

ν1 k2,1

k2,0 = ν2 k1,1 =





m2 ∑3j=2 (k1, j − k2, j )2 − (k1,3 k2,2 − k1,2 k2,3 )2 + ν2



k2 2 − m2 (∑3j=2 k1, j k2, j − m2 )

2 − m2 ∑3j=2 k2, j

k2 2 − m2 ,

k2,1 (∑3j=2 k1, j k2, j − m2 ) + ν1 ν2



,

(32) (33)

(k2 2 − m2 )[m2 ∑3j=2 (k1, j − k2, j )2 − (k1,3 k2,2 − k1,2 k2,3 )2 ] 2 − m2 ∑3j=2 k2, j

,

(34)

where ν1 = ±1 and ν2 = ±1. Therefore, expression (30) is a solution of (10) with ρ¯1 = ei(k1,0 t+k1,1 x+k1,2 y+k1,3 z) , ρ¯2 = ei(k2,0t+k2,1 x+k2,2 y+k2,3 z) and the constants (32), (33) and (34). Now, let us consider equation (24) and the set ρ with the constraints 2 ψ1 = −k1,0 + k1 2 + 2m2 = 0,

(35)

2 −k2,0 + k2 2 + 2m2

(36)

ψ2 =

= 0.

If we consider only one of these constants, we will obtain the solution (22) again, up to a redefinition of the arbitrary constants. However, if both constraints are considered we can eliminate k1,0 , k1,0 , such that (24) yields 2m2 (ρ12 ∂ρ21 φˆ + ρ1 ∂ρ1 φˆ + ρ22 ∂ρ22 φˆ + ρ2 ∂ρ2 φˆ ) + 2(k1,0 k2,0 − k1 .k2 )ρ1 ρ2 ∂ρ1 ∂ρ2 φˆ + m2 φˆ + λ φˆ 3 = 0.

(37)

Observe that we did not substitute linear terms of k1,0 and k2,0 again. The multivariate Taylor expansion of (37) yields √ iμ λ (c21,0 ρ¯12 + c20,1 ρ¯22 ) 3iμ mλ c1,0 c0,1 ρ¯1 ρ¯2 i μ m + )ξ 2 + · · · , φˆ = √ + (c1,0 ρ¯1 + c0,1 ρ¯2 )ξ − ( 2m (k1,0 k2,0 − k1 .k2 + m2 ) λ

(38)

for m = 0, λ = 0 and k1,0 k2,0 − k1 .k2 + m2 = 0. The Pad´e approximant [1/1](ρ1 ,ρ2 ) (ξ )|ξ =1 of (38) yields the ansatz √ φˆ = m((k1,0 k2,0 − k1 .k2 + m2 )( λ (c21,0 ρ¯12 + c20,1 ρ¯22 + 4c1,0 c0,1 ρ¯1 ρ¯2 ) + 2imμ (c1,0 ρ¯1 + c0,1 ρ¯2 )) √ √ −6m2 λ c1,0 c0,1 ρ¯1 ρ¯2 )/((k1,0 k2,0 − k1 .k2 + m2 )(iμλ (c21,0 ρ¯12 + c20,1 ρ¯22 ) + 2m λ (c1,0 ρ¯1 +c0,1 ρ¯2 )) + 6iμλ m2 c1,0 c0,1 ρ¯1 ρ¯2 ),

(39)

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and the conditions for (39) to become an exact solution are ˆ = 8c51,0 c0,1 λ 3/2 m(k1,0 k2,0 − k1 .k2 − m2 )(k1,0 k2,0 − k1 .k2 + m2 )2 E˜ 1;5,1 (S) (k1,0 k2,0 − k1 .k2 − 2m2 ) = 0, ˆ = −96c41,0 c20,1 λ 3/2 m5 (k1,0 k2,0 − k1 .k2 + m2 )(k1,0 k2,0 − k1 .k2 − 2m2 ) = 0, E˜ 1;4,2 (S) ˆ = 16c31,0 c30,1 λ 3/2 m(k1,0 k2,0 − k1 .k2 + m2 )(k1,0 k2,0 − k1 .k2 − 2m2 )[8m2 (k1,0 k2,0 , E˜ 1;3,3 (S) −k1 .k2 ) − 3((k1,0 k2,0 − k1 .k2 ))2 + 5m4 ] = 0, ˆ = −96c21,0 c40,1 λ 3/2 m5 (k1,0 k2,0 − k1 .k2 + m2 )(k1,0 k2,0 − k1 .k2 − 2m2 ), E˜ 1;2,4 (S) ˆ = 8c1,0 c50,1 λ 3/2 m(k1,0 k2,0 − k1 .k2 − m2 )(k1,0 k2,0 − k1 .k2 + m2 )2 E˜ 1;1,5 (S) (k1,0 k2,0 − k1 .k2 − 2m2 ) = 0,

√ ˆ = [(k1,0 k2,0 − k1 .k2 + m2 )(iμλ (c21,0 λ ρ¯12 + c20,1 λ ρ¯22 ) + 2m λ (c1,0 ρ¯1 + c0,1 ρ¯2 ) D1 (S) +6iμλ m2 c1,0 c0,1 ρ¯1 ρ¯2 )] = 0. The system of conditions above has three possibilities of solutions, namely c1,0 = 0, c0,1 = 0, or k1,0 k2,0 − k1 .k2 − 2m2 = 0.

(40)

The conditions c1,0 = 0 and c0,1 = 0 yield the solution (22) up to a redefinition of the arbitrary constants, while condition (40) and the constraints (35) and (36) eliminate three constants, for example,   ν1 k2,1 −2m2 ∑3j=2 (k1, j − k2, j )2 − (k1,3 k2,2 − k1,2 k2,3 )2 + ν2 k2 2 + 2m2 (∑3j=2 k1, j k2, j + 2m2 ) , k1,0 = 2 + 2m2 ∑3j=2 k2, j  k2,0 = ν2 k1,1 =

(41) k2 2 + 2m2 ,

 k2,1 (∑3j=2 k1, j k2, j + 2m2 ) + ν1 ν2 (k2 2 + 2m2 )[−2m2 ∑3j=2 (k1, j − k2, j )2 − (k1,3 k2,2 − k1,2 k2,3 )2 ] 2 + 2m2 ∑3j=2 k2, j

(42) , (43)

where ν1 = ±1 and ν2 = ±1. Therefore, the expression (39) is a solution of the λ φ 4 model with ρ¯1 = ei(k1,0 t+k1,1 x+k1,2 y+k1,3 z) , ρ¯2 = ei(k2,0 t+k2,1 x+k2,2 y+k2,3 z) and the constants (41), (42) and (43). The algorithm presented here could also be used with Pad´e approximants of higher degree or different functional ansatz. However, we will not consider other ansatz due computational limitations. 4 Conclusions In this paper, it was shown an algorithm for solving nonlinear partial differential equations based on Pad´e approximants. The algorithm was applied to the λ φ 4 model in 4 dimensions by using two funcional ansatzes and it lead us to new solutions for the model. There are many recent papers proposing methods for solving differential equations [1–18] and the approach presented here could be an easier algorithm for applying to more complicated model. Acknowledgments I am thankful to H. Aratyn, J. F. Gomes and A. H. Zimerman for discussions. The author also thanks FAPESP (2010/18110-9) for financial support.

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171

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Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 173–185

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

Synchronization of Micro-Electro-Mechanical-Systems in Finite Time Hadi Delavari1†, Ayyob Asadbeigi2, and Omid Heydarnia3 1 Department

of Electrical Engineering, Hamedan University of Technology, Hamedan, 65155, Iran Department, Institute for Advanced Studies in Basic Sciences (IASBS) Gava zang Zanjan 3 Department of Robotic Engineering, Hamedan University of Technology, Hamedan, 65155, Iran 2 Mathematics

Submission Info Communicated by Lev Osctrovsky Received 17 June 2014 Accepted 4 February 2015 Available online 1 July 2015 Keywords Chaotic dynamical systems Micro-Electro-Mechanical-Systems Synchronization Dead-zone input

Abstract Finite time synchronization of chaotic Micro-Electro-Mechanical Sys-tems (MEMS) is considered. In particular, a Lyapunov-based adaptive controller is developed such that convergence of synchronization error is guaranteed globally in the presence unknown perturbations. The system under consideration suffers from bounded parametric uncertainties, additive external disturbances as well as dead zone input nonlinearities. We establish the controller on being resistance against hard nonlinearities by a novel scheme which can be developed to general chaotic systems even. We provide rigorous stability analysis to come up with sufficient conditions that guarantee finite time error convergence of perturbed system. Several simulation scenarios are carried out to verify the effectiveness of obtained theoretical results. ©2015 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Chaotic dynamic systems belong to a very complex non-linear class of systems that present doubtful, erratic behavior. A chaotic system has some special characteristics such as extreme sensitivity to initial conditions and system parameter variations, broad spectra of Fourier transform fractal properties of the motion in phase space, and strange attractors. Synchronization of autonomous chaotic systems is one of the outstanding studying issues and it has received considerable attention among researchers [1–5]. Existence and stability of periodic solutions of a canonical mass-spring model of electrostatically actuated Micro-Electro-Mechanical Systems (MEMS) are studied in [6]. In [7], the MEMS are modeled as kicked damped oscillator; dynamics of iterative maps is considered numerically. Micro-electro-mechanical systems (MEMS) refer to device that have a characteristic length of less than 1mm but more than 1 that combine electrical and mechanical component that are fabricated using integrated circuits batch-processing technologies [8]. Chaotic behavior of MEMS devices is shown in [9–12]. Synchronization of chaos can be utilized in various field such as biology, fluid dynamics, secure communications, nonlinear optics, and electronics [13]. The Most common applications of MEMS resonator are mentioned in [14] such as radiofrequency filters, scanned probe microscopy, and Nano scale positioning. Nonlinearity is † Corresponding

author. Email address: [email protected], [email protected],[email protected],[email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.005

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an impartible issue in MEMS dynamics; many researches have studied on nonlinear dynamics of MEMS like nonlinear springs and damping mechanism [15], nonlinear resistive, inductive and capacitive circuit elements [16] and nonlinear surface, fluid, electric and magnetic forces [17]. There are various approaches to model nonlinear dynamics of MEMS, in [18] a nonlinear model is proposed to obtain observed behavior of the system, based on experimental studies. Dynamics of MEMS resonators under superharmonic and subharmonic excitations analyzed and simulated in [19], and new nonlinear dynamic properties of MEMS also observed under super harmonic excitations in [10]. Bistability and existence of a strange attractor in the MEMS device is exhibited and synchronized chaos is applied on secure communications [20]. In [12] the chaotic motion of MEMS resonant systems in the vicinity of specific resonant separatrix is investigated based on corresponding resonant condition. A chaotic motion on certain frequency band of a (MEMS) with variable capacitor has been obtained in [21]. Chaotic motion of a micro-electro-mechanical cantilever beam under both open and close loop control has also been reported [11]. The chaotic behavior of particular micro-electro-mechanical system had been analyzed and the irregular oscillatory movement of the non-linear system reduced, by using an optimal linear control technique [24]. Linearization method of C-V response improved and maximum tenability increased for high nonlinearity and structural instability in electrostatically actuated MEMS capacitors [22]. An adaptive feedback system designed to detect resonant frequencies for a wide range of MEMS resonators [23]. A phase portrait, maximum Lyapunov exponent and bifurcation diagram had been used to find chaotic dynamics of the micro-electro-mechanical system and a robust fuzzy sliding mode control had been proposed to suppress chaotic motion in [22, 25], Studying fuzzy controllers is another procedure, A robust fuzzy sliding mode control recommended for nonlinear and chaotic systems in [26, 27], and a second-order sliding mode control with PID sliding surface used to develop tracking performance of a 2-degree-of-freedom (2D) torsional MEMS Micro mirror with sidewall electrodes and reduce chattering phenomena [28]. Fuzzy PID type controller with the set of control rules and fuzzy inference had been discussed [24, 25]. Moreover fuzzy sliding mode control with PID surface had been studied in [29]. In many industrial situations dead zone input is an unavoidable pat of the system. In this paper, a finite time adaptive controller is presented for a chaotic MEMS resonator with parameter uncertainties and external disturbances. A new nonlinear dead zone is considered on input of the system. The proposed controller is then applied to the control of the system with and without dead zone. Simulation results are presented to validate the analysis. 2 Mathematical Descriptions of Micro-Electro-Mechanical-Systems (MEMS) 2.1

Mathematical modeling of Micro-Electro-Mechanical-Systems

In this paper we considered a two order equation of motion for the dynamics of the MEMS resonator system equation, which can be obtained as: mz + bz + k1 z + k3 z3 = Fact ,

(1)

where m, b, k, r, are respectively effective lumped mass, damping coefficient, linear and cubic mechanical stiffness parameters of the system, Fact shows the net actuation force and is considered accordance to [18]. The electrostatically actuated micro-beam is shown in Fig. 1, where d is initial width of the gap and z is the vertical displacement of the beam. An external driving force on the resonator is applied via an electrical driving voltage that causes electrostatic excitation with DC-bias voltage between electrodes and the resonator: Vi = Vb +VAC sin(ω t) where Vb is the bias voltage, VAC and Ω are AC amplitude and frequency, respectively. The amplitude of the AC driving voltage is assumed to be much lower than the bias voltage, yielding nondimensional equation of motion [9, 25]. The MEMS cantilever is modeled as a nonlinear single mass-spring-damper with electrostatic actuation [11].

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175

Fig. 1 A schematic image of the electrostatically micromechanical resonator.

Fig. 2 A simplified mechanical model of the MEMS with a time-varying capacitor.

A mechanical model for MEMS, with time-varying capacitor shown in Fig. 2. [9, 12, 21]. x¨ + μ x˙ + α x + β x3 = γ (

1 1 A − )+ sin(ω t), 2 2 (1 − x) (1 + x) (1 − x)2

(2)

where dimensionless variables defined as: x = z/d,

ω = Ω/ω0 ,

A = 2γ ,

h(x) = γ (

1 1 − ). (1 − x)2 (1 + x)2

(3)

In (3), ω0 is the purely elastic natural frequency. The MEMS (2) exhibits chaotic and complex behavior and has been studied by Haghighi and Markazi [9] for values of VAC in the range 0 < VAC < 0.47 and constant values α = 1, β = 12, μ = 0.338,Vb = 0.01, ω = 0.5, Chaotic behavior and irregular motion of MEMS at VAC = 0.2 with zero initial condition for state variable x1 has been presented in Figs. 3, 4. 2.2

Problem statement

The non-autonomous chaotic MEMS (2) with model uncertainties, external disturbances and unknown parameter can be rewritten in the following form: x˙1 = x2 , x˙2 = −α x1 − β x31 − μ x2 + h(x1 ) +

A sin(ω t). (1 − x1 )2

(4)

In (4), x = [x1 , x2 ] shows the state vector of the system, Δ fi and di , are unknown bounded time- varying model uncertainties, and disturbance perturbations, for i = 1, 2, respectively.

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Fig. 3 Behavior of first state variable.

Fig. 4 Phase Plan trajectory.

By regarding the chaotic MEMS (4) as the drive systems, the non-autonomous chaotic response MEMS with control inputs is defined as follows: y˙1 = y2 + Δg1 (y1 , y2 ) + u1 (t) + d1 (t), y˙2 = −α y1 − β y31 + Δg2 (y1 , y2 ) + μ y2 + h(y1 ) + [

A ] sin(ω t) + u2 (t) + d2 (t), (1 − y1 )2

(5)

where y = [y1 , y2 ] is the state vector of the systems, Δgi are unknown bounded time-varying model uncertainties, and di be disturbance perturbations, for i = 1, 2. e˙1 = e2 + Δg1 (y1 , y2 ) − Δ f1 (x1 , x2 ) + d1 (t) − d1 (t) + u1 (t), e˙2 = −α e1 − β (y31 − x31 ) + Δg2 (y1 , y2 ) − Δ f2 (x1 , x2 ) + h(y1 ) − h(x1 ) A A − ] sin(ω t) + d2 (t) − d2 (t) + u2 (t). +[ (1 − y1 )2 (1 − x1 )2

(6)

Our aim is to enforce error tend to zero by designing control input signals. 3 Preliminaries and assumptions Throughout this paper, we consider the following definitions, lemmas and assumptions: Lemma 1. Suppose that a continuous, positive definite function V (t) satisfies the following deferential inequality: V˙ (t) ≤ −σ V τ (t), for all t ≥ t0 ,V (t) ≥ 0 where σ > 0, 0 < τ < 1 are two constants. For any given t0 ,V (t) satisfies the following inequality: V 1−τ (t) ≤ V 1−τ (t0 ) − σ (1 − τ )(t − t0 ),

t0 ≤ t ≤ t1 .

(7)

And V (t) ≡ 0 for all t ≥ t1 with t1 is given by: t1 =

V 1−τ (t0 ) + t0 . σ (1 − τ )

(8)

Proof. See [34]. Lemma 2. For π1 , π2 , . . . , πn belong to R, the following inequality holds:  π1 + π2 + · · · + πn ≥ π12 + π22 + · · · + πn2 .

(9)

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177

Proof. See [33]. Assumption 1. The trajectories of the chaotic systems always considered bounded [32]; we assume that the uncertainties and disturbance perturbations di (t) satisfy the following conditions: |Δ fi (t) − Δgi (t)| ≤ ki ,

(10)

|di (t) − di (t)| ≤ δi . For i = 1, 2, ki and δi are positive constants.

Assumption 2. Suppose ν = [ν1 , ν2 , ν3 , ν4 ]T = [α , β , μ , A]T as the vector of the unknown parameters of the synchronization error system. Subsequently, in order to guarantee the existence of the chaos for the drive and response MEMS, the following assumption is made. Assumption 3. We consider the unknown vector parameter ν is above norm bounded by a positive constant in the Euclidean norm space. Definition 1. Consider the master and slave chaotic non-autonomous MEMS described by Eqs. (4) and (5), respectively. Suppose that there exists a constant T = T (e(0)) > 0 , such that lim e(t) = 0 and for all t ≥ T , in t→T

such conditions we say that the MEMS of equations (4) and (5) will be synchronized in finite time. Definition 2. Uncertain Dead-zone nonlinear function is described by the following relations: ⎧ ⎪ βi (u(t) − u+i (t))2 ≤ (u(t) − u+i (t))ϕi (ui (t)) ≤ βi (ui (t) − u+i (t))2 , ui (t) < u−i (t), ⎪ ⎪ ⎨ ϕi (ui ) = 0, if u−i (t) ≤ ui (t) ≤ u+i (t), ⎪ ⎪ ⎪ ⎩ β (u(t) + u (t))2 ≤ (u(t) + u (t))ϕ (u (t)) ≤ β (u(t) + u (t))2 , u (t) > u (t). i −i −i i i i −i i +i (11) In addition, βi are the gain reduction tolerances for i = 1, 2, and 0 < β1 < β2 , ϕ (u) shows the uncertain dead-zone input. The following figure shows a general nonlinear Dead-zone;

Fig. 5 Nonlinear Dead-Zone.

4 Controller design and synchronization results 4.1

Synchronization of two MEMS with linear input

In this part, we deal with the problem of designing the finite time controllers for the linear input; which is essential for the synchronization results of the systems.

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The control laws for the system can be considered as follow: 

u1 = −e2 − λ sgn(e1 )[G(υ ) + υ ] − (k1 + c1 )sgn(e1 ), 







u2 = −ν 1 e2 − ν 2 (x31 − y31 ) − ν 3 e2 − ν 4 [

A A − ] sin(ω t) (1 − y1 )2 (1 − x1 )2

(12)



−λ sgn(e2 )[G(υ ) + υ ] − (k2 + c2 )sgn(e2 ). Here, an estimation for the unknown parameter ν is given by ν = [ν1 , ν2 , ν3 , ν4 ]T ; and λ = min{ci }; For i = 1, 2,  and ci > 0 , are constant gains, G(ν ) = inf ν − ν  where sign() is the sign function. The following update laws can be considered, to deal with the unknown parameters: ·





ν 1 (t) = e1 e2 , ·



and ν 2 (t) = (y31 − x31 )e2 , ·





and ν 3 (t) = e22 ,



ν 1 (0) = υ 10 , 



ν 2 (0) = υ 20 ,

(13)



υ 3 (0) = υ 30 , · A A  − ] sin(ω t)e2 , and ν 4 (t) = −[ 2 (1 − y1 ) (1 − x1 )2











υ 4 (0) = υ 40 ,



where υ 10 , υ 20 and υ 30 , υ 40 are the initial values of the update parameters which are mentioned above. Theorem 1. Suppose the synchronization error system (6) with linear control inputs. If this system is controlled by the control laws (12) with update laws (13), then the system trajectories will converge to zero in the finite time T1 , determined by: √  2 1  (e(0)2 + υ (0) − υ (0)2 ). (14) T1 = ρ 2 Proof. Consider the following Lyaponuv function:

Its derivative is:

1  V (t) = (e2 + υ (t) − υ )2 . 2

(15)

V˙ (t) = e˙1 e2 + e1 e˙2 + (ν˙ (t) − ν )T ν˙ (t).

(16)

After substituting from the error system: ·

V ≤ e1 (e2 + u1 (t)) + |Δg1 (y,t) − Δ f1 (x,t)||e1 | + |d1 (t) − d1 (t)||e1 | A A − ] sin(ω t) + u2 (t))e2 +(−α e1 − β (y31 − x31 ) − μ e2 + [ (1 − y1 )2 (1 − x1 )2 +|Δg2 (y,t) − Δ f2 (x,t)||e2 | + e2 u2 (t) + |h(y1 ) − h(x1 )||e2 | + |d2 (t) − d2 (t)||e2 |

(17)

·

+(υˆ (t) − ν )T υˆ (t), By considering: −α e1 e2 − β (y31 − x31 ) − μ e22 + (h(y1 ) − h(x1 ))e2 + [

·

T  A A − ] sin( ω t)e = υ (t) υ (t), 2 (1 − y1 )2 (1 − x1 )2

(18)

We will get to: V˙ (t) ≤ e1 (e2 + u1 (t)) + |Δg1 (y,t) − Δ f1 (y,t)||e1 | + |d1 (t) − d1 (t)||e1 | T

·



+|Δg2 (y,t) − Δ f2 (y,t)||e2 | + e2 u2 (t) + υ (t) υ (t).

(19)

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179

By entering the controllers, we will have:  V˙ (t) ≤ e1 (e2 + [−e2 − λ sgn(e1 )[G(υ ) + υ ] − (k1 + c1 )sgn(e1 )]) + k1 |e1 | + δ1 |e1 | A A     − ] sin(ω t) +e2 (−ν 1 e2 − ν 2 (y31 − x31 ) − ν 3 e2 + ν 4 [ 2 (1 − y1 ) (1 − x1 )2 

(20)

·





−λ sgn(e2 )[G(υ ) + υ ] − (k2 + c2 )sgn(e2 )) + k2 |e2 | + δ2 |e2 | + (υ (t))T υ (t). By taking: 







ν 1 e22 + ν 2 (y31 − x31 )e2 + ν 3 e22 + ν 4 [ We will obtain:

·

A A   − ] sin(ω t) = (υ (t))T υ (t). 2 2 (1 − y1 ) (1 − x1 )

 V˙ (t) ≤ −λ1 |e1 | − λ2 |e2 | − λ (G(υ ) + υ ).

(21) (22)

In which λi = ki + ci , ρ = min{λi , λ } for i = 1, 2, so the last relation can be written as:  V˙ (t) ≤ −ρ |e1 | − ρ |e2 | − ρ (G(υ ) + υ ).

(23)

And from this we will achieve to:

√  2  (24) V˙ (t) ≤ −ρ (e + υ − υ ) ≤ − √ ρ (e2 + υ − υ 2 ). 2 √   Now, Lemma 1 implies that; in finite time T1 = ρ2 12 (e(0)2 + υ (0) − υ (0)2 ) the error system e(t) tends to zero. So in the finite time the state trajectories of the non-autonomous uncertain response MEMS (4), will approach to the state trajectories of the non-autonomous uncertain drive MEMS (3). 

4.2

Synchronization of two MEMS with dead-zone input

Now we give the main result of this paper which is Synchronization of two MEMS with dead-zone input, by finite-time controller: Consider the drive MEMS (4) with uncertain dead-zone inputs as the response system let us introduce the following MEMS: y˙1 = y2 + Δg1 (y1 , y2 ) + d1 (t) + ϕ1 (u1 (t)), y˙2 = −α y1 − β y31 + Δg2 (y1 , y2 ) + μ y2 + h(y1 ) + [

A ] sin(ω t) + d2 (t) + ϕ2 (u2 (t)), (1 − y1 )2

(25)

where, ϕi (ui ) are as the definition of uncertain dead-zone (25). The error system of the drive MEMS (4) and the response MEMS (25) is obtained as follows: e˙1 = e2 + Δg1 (y1 , y2 ) − Δ f1 (x1 , x2 ) + d1 (t) − d1 (t) + ϕ1 (u1 (t)), e˙2 = −α e1 − β (y31 − x31 ) + Δg2 (x2 , y2 ) − Δ f2 (x1 , x2 ) + μ (y2 − x2 ) + (h(y1 ) − h(x1 )) A A − ] sin(ω t) + d2 (t) − d2 (t) + ϕ2 (u2 (t)). +[ (1 − y1 )2 (1 − x1 )2

(26)

Adaptation laws for the nonlinear case with dead zone are as follows: ·



ν 1 (t) = α e1 e2 , 





ν 1 (0) = ν 10 







ν 2 (t) = β (y31 − x31 )e2 , ν 2 (0) = ν20 , ν 3 (t) = −μ e2 , ν 3 (0) = ν 30 A A    ˙ − ) sin(ω t) + h(y1 ) − h(x1 )]e2 , ν 4 (0) = ν 40 and ν 4 (t) = −[( 2 2 (1 − y1 ) (1 − x1 )

(27)

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Hadi Delavari, Ayyob Asadbeigi, Omid Heydarnia / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 173–185

For studying finite-time synchronization of the uncertain chaotic non-autonomous MEMS of Eqs (4) and (25) with input dead zone nonlinearities, the following control laws are proposed: ⎧ ⎨ −pqi sgn(ei (t)) − |u−i |, ei (t) > 0, ui (t) = 0, (28) ei (t) = 0, ⎩ pqi sgn(ei (t)) + |u+i |, ei (t) < 0, where 1 > p > min βi for i = 1, 2 , p and η is an arbitrary positive constant, now we define q1 and q2 as: 1  q1 = (−|e2 | + η [G(υ ) + υ ] + k1 + s1 ) β1

⎛ ⎞ A A     3 3 − sin(ω t) ⎠ 1 ⎝ −|ν 1 | + |ν 2 (y1 − x1 )| − ν 3 + h(y1 ) − h(x1 ) + ν 4 q2 = (1 − y1 )2 (1 − x1 )2  β2 +|d2 − d2 | + η [G(υ ) + υ ] + k2 + s2

(29)

Lemma 2. For i = 1, 2 we have ei ϕi (ui ) ≤ −βi pqi |ei |. Proof. Suppose that ei > 0. From definition of nonlinear dead-zone (11) and after substituting in the corresponding formulas we will have:

βi (ui (t) − u+i (t)) ≤ (ui (t) − u+i (t))ϕi (ui (t)).

(30)

−pqi |ei |ei ϕi (ui ) ≥ βi pq2i |ei |2 .

(31)

βi (ui (t) + u+i (t)) ≤ (ui (t) + u+i (t))ϕi (ui (t)).

(32)

ei ϕi (ui ) ≤ −βi pqi |ei |.

(33)

So If ei < 0 then So we have The proof is completed.

Theorem 2. Consider the synchronization error system (25) with input nonlinearities. If this system is controlled by the control then the system trajectories will converge to zero in the finite time T2 determined by: √  2 2  (e(0)2 + ∑ (υ i (0) − υi (0))2 ). (34) T2 = Ω i=1 Proof. We introduce the Lyaponuv candidate as: 2 1  V (t) = (e2 + ∑ (υ i (t) − υi )2 ). 2 i=1

Its derivative is:

2

  ˙ V˙ (t) = e˙1 e1 + e2 e˙2 + ∑ (υ i (t) − υi )T υ i (t).

(35)

(36)

i=1

After substituting from the error system: 2

˙

∑ (υ i (t) − υi)T υ i (t) 



i=1

V˙ (t) ≤ e1 (e2 + Δg1 (y1 , y2 ) − Δ f1 (x1 , x2 ) + d1 (t) − d1 (t) + ϕ1 (u1 (t))) ⎞ ⎛ −α e1 − β (y31 − x31 ) + Δg2 (x2 , y2 ) − Δ f2 (x1 , x2 ) + μ (y2 − x2 ) + (h(y1 ) − h(x1 )) ⎟ ⎜

+e2 ⎝ ⎠ A A  − sin(ω t) + d2 (t) − d2 (t) + ϕ2 (u2 (t)) + 2 2 (1 − y1 ) (1 − x1 ) 2

  ˙ + ∑ (υ i (t) − υi )T υ i (t).

i=1

(37)

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181

By substituting the following relations and from Lemma 3 and relation (29) and considering: 2

˙

∑ (υi )T υ i(t) = 

i=1

e2 e1 − α e1 e2 − β (y31 − x31 )e2 + (h(y1 ) − h(x1 ))e2 +[

We will have:

(38)

A A − ] sin(ω t)e2 . 2 (1 − y1 ) (1 − x1 )2

2

T

·

  V˙ (t) ≤ ∑ −|ei |(βi pqi ) + ai |ei | + ν i (t)ν i (t),

(39)

i=1

where a1 = k1 + δ1 and a2 = k2 + δ2 + μ e∞ . Now by regarding the following relations: ·

T



T



ν 1 (t)ν 1 (t) = |e1 ||e2 | + k1 |e1 |, ·







υ 2 (t)υ 2 (t) = |ν 1 e2 | + |ν 2 (y31 − x31 )e2 | + |ν 3 e2 | + |d2 − d2 ||e2 | + k2 |e2 | A A − ] sin(ω t) + h(y1 ) − h(x1 )||e2 |. |[ 2 (1 − y1 ) (1 − x1 )2 We will have:

2

 V˙ ≤ −p ∑ |ei |η ([G(υ ) + υ ] + si ).

(40)

(41)

i=1

We can rewrite the previous inequality as:   V˙ (t) ≤ −B|e1 | − B|e2 | − B|e1 |(G(υ ) + υ ) − B|e2 |(G(υ ) + υ ) 

≤ −Be − Be(G(υ ) + υ ).

(42)

In which B = min{pη , si }. From (42) and considering Ω = min{Be, B}, We obtain the following: √  2   ˙ V (t) ≤ −Ω(e + υ − υ ) ≤ − √ Ω( e2 + υ − υ 2 ). 2  Now Lemma 1 implies that e(t) approaches to zero, in T2 =

√ 2 Ω

2

(43)



(e(0)2 + ∑ (υ i (0) − υi (0))2 ) And this i=1

completes the proof. 5 Numerical illustrations In this section, the efficiency and usefulness of the proposed finite-time controller, the synchronization of the non-autonomous response MEMS drive (4) with unknown parameters, model uncertainties and external disturbances, is validated. The initial conditions are considered as; ν10 = 0.01, ν20 = 0.01, ν30 = 0.01, ν40 = 0.01. Initial states in master and slave systems are; x10 = 0, x20 = 0 and y10 = 0.06, y20 = 0.04. Two numerical simulations are presented. The simulations are carried out using the MATLAB software, and the following model uncertainties and external disturbances are applied: Master π Δ f1 = 0.01 sin( x1 ), d1 = 0.01, 10 (44) π Δ f2 = 0.015 sin( x2 ), d2 = 0.15. 10

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Slave

π y1 ) + 0.03 sin(10t), d1 = 0.03, 10 π Δg2 = 0.03 sin( y2 ) + 0.04 sin(20t), d2 = 0.45. 10 ⎧ (0.07 + 0.02 sin ui )(ui + 0.4) if ui < −0.4; ⎪ ⎪ ⎨ ϕi (ui ) = 0 if − 0.4 ≤ ui (t) ≤ 0.4, ⎪ ⎪ ⎩ (0.07 + 0.02 sin u )(u − 0.4), if u > −0.4. Δg1 = 0.03 sin(

Dead Zone

i

i

(45)

(46)

i

Example 1. In this example, the efficiency and usefulness of the proposed finite-time controller for synchronization of the non-autonomous drive MEMS (4) and response MEMS (5) with unknown parameters, model uncertainties and external disturbances is validated. The simulation, in the following model uncertainties and external disturbances are applied: u1 = −e2 − 0.65sgn(e1 ) − 0.03G(υ )sgn(e1 ), A A − ] sin(ω t) − 0.075sgn(e2 ). u2 = −υˆ1 e1 − υˆ 2 (y3 − x3 ) − υˆ 3 e2 − υˆ 4 [ (1 − y1 )2 (1 − x1 )2

(47)



In which, the parameters are as follows: G(υ ) = υ + 0.1, k1 = 0.035 and k2 = 0.025, c1 = 0.030 and c2 = 0.05. The state trajectories, and error of the uncertain chaotic system in Example 1 respectively are illustrated in Figs. 6 and 8, where the control inputs are applied at t = 100 s. It is observed that the system trajectories converge to zero quickly, after that the controller is turned on. Example 2. After that, the efficiency of the controllers for dead-zone zone inputs and the controllers as:  0.55q1 sgn(e1 (t)) − 0.4, if u1 (t) = −0.55q1 sgn(e2 (t)) − 0.4, if  0.55q2 sgn(e1 (t)) − 0.4, if u2 (t) = −0.55q2 sgn(e2 (t)) − 0.4, if

input is shown: In the case of the dead e1 (t) ≥ 0, e2 (t) < 0, e1 (t) ≥ 0,

(48)

e2 (t) < 0.

Results for Example 2 are given in Figs. 7 and 9, respectively. The control inputs are considered at t = 100 s, and consequently It is observed that the system trajectories approach to zero.

Fig. 6 State trajectories of the master and slave systems without dead zone input.

Hadi Delavari, Ayyob Asadbeigi, Omid Heydarnia / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 173–185

183

where q1 and q2 are considered as follow: 1 (−|e2 | + 0.065 G(v) + 0.15), 0.05 A A 1     − ] sin(ω t) + 0.065 G(v) + 0.5180 ). ( −|ν 1 | + |ν 2 (y31 − x31 )| + ν 3 − ν 4 [ q2 = (1 − y1 )2 (1 − x1 )2 0.09

q1 =

(49)



And the other data on the system are given by: G(v) = υ  + 0.4, β1 = 0.05, δ1 = 0.025, k1 = 0.06, s1 = 0.065, p = 0.55, η = 0.065, β2 = 0.09 and δ2 = 0.035, k2 = 0.07, s2 = 0.075, p = 0.55, η = 0.065. 6 Conclusions The problem of finite-time synchronization of Micro-electro-mechanical-systems is investigated in this paper. We had considered the dead-zone input, and linear inputs. The parameters of the Micro-electro-mechanical systems are fully unknown in advance and the system is perturbed by unknown model uncertainties and external disturbances. The finite-time stability and convergence of the proposed controller are shown.

Fig. 7 State trajectories of the master and slave systems with dead zone input.

Fig. 8 Error for the system without dead-zone input.

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Fig. 9 Error for the system with dead-zone input.

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Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 187–197

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

Scaling Modeling of the Emitted Substance Dispersion Transported by Advection Caused by Non-homogeneous Wind Field and by Isotropic and Anisotropic Diffusion in Vicinity of Obstacles Ranis N. Ibragimov†, Andrew Barnes, Peter Spaeth, Radislav Potyrailo, and Majid Nayeri GE Global Research, 1 Research Circle, Niskayuna, NY 12309, USA Submission Info Communicated by Xavier Leoncini Received 1 December 2014 Accepted 1 March 2015 Available online 1 July 2015 Keywords Source localization Advection - diffusion flows Scaling analysis

Abstract A simple two-dimensional mathematical approach for source localization of contaminants in the vicinity of individual simple two-dimensional obstacles is proposed. The approach consists of scaling analysis of advectiondiffusion potential flows that can be used in the vicinity of two-dimensional cylindrical obstacles. Three different modeling scenarios are developed in order to simulate the effects of wind. Particularly, the model incorporates the cases of anisotropic diffusion and spatially and temporary inhomogeneous airflow speeds.

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

1 Introduction If a region is filled non-uniformly by a gas, then diffusion takes place from points of higher concentration to points of lower concentration, as shown schematically in Figure 1. This phenomenon also takes place in solutions, if the concentration of the solute is not constant throughout the volume. The effects of wind in dispersing pollution from sources above level ground is a problem of pressing importance to architects, planners, and public health officials for obvious reasons [1, 2]. The dispersion of matter from a point or line source can be studied by calculating the displacement caused by fluid or atmospheric motions and molecular motion (i.e. diffusion) of the ensemble of all particles which have passed through the source. In recent years, problems related to air pollution have become of particular importance as humanity has reached a peak in the growth and development of industrial potential and motor vehicles. Development of new approaches to the modeling of harmful pollutants in the atmosphere are vital to determine the optimal mathematical modeling and its integration into mathematical application packages that are needed to provide information on the state of air quality, they also needed to pollution environmental services. Odor recognition techniques very well developed and applied in many kinds of industry areas, such as dairy industry and beverage industry [3, 4]. On the other hand, source localization is usually performed by electronic noses (Enose) built-in † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.006

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robotics. This approach could successfully solve this problem even in natural environments (see e.g. [5, 6]). But it will face difficulty to accomplish this task in some situations, for example if the source is instantaneous; or the landscape is very hard for a robotic vehicle to reach the source position. Therefore, a requirement of mathematical modeling capable to locate the source position is raised. Usually, the concentration of a contaminant released into the air may be described by the diffusion equation, which is a second-order partial differential equation of parabolic type [7,8], which in a simplest one-dimensional case is

∂ ∂q ∂q (K ) = ±c , ∂x ∂x ∂t

(1)

in which q (x,t) the concentration at section x at time t, K is a coefficient of diffusion (in general, K might not be a constant) and c (x) is the coefficient of porosity (the coefficient of porosity is the ratio of the volume of the pores to the total volume Sdx). We remark that Eq. (1) is completely analogous to the equation of heat conduction. In deducing this equation describing the process of diffusion in a hollow tube, or in a tube filled with a porous medium (assuming that at any moment of time the concentration q of the gas in a section of the tube is the same), we assumed that there were no sources of matter in the tube and that diffusion across the surface of the tube was absent. If the effects of advection are added, e.g. by means of airflow with spatially and time dependent components of the wind velocity u and v, then, in a linear approximation, in the case of constant diffusion coefficient K and when no sources of matter in the tube and that diffusion across the surface of the tube are absent, Eq (1) is modified to the form ∂q ∂ 2q ∂q ∂q + K 2 + u (t, x, y) + v (t, x, y) = 0. ∂t ∂x ∂x ∂y

1.4

q0

1.2

q0 tot max q0

Gas concentration

1 0.8 0.6 0.4 0.2 0 −0.2 −1 10

0

10

Log x [m]

1

10

Fig. 1 Diffusion takes place from points of higher concentration to points of lower concentration.

The main objective of this paper is to reduce the model to its simplest form, which can be integrated in mathematical packages designed to visualize the state of air pollution and to localize the source by spatial and temporary distribution of advection and diffusion. To reduce the dimension of the system, we assume that the

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189

source and the electronic nose network can be placed on the impermeable surface z = 0. Thus the mathematical model could be considered in a two dimensional xy−plane. 2 Dispersal model In terms of mathematical modeling, the dispersal behavior of an emission is characterized by diffusion and advection (if wind is present). In real-world problems, the wind speed and direction are functions of space and time, which we use in our modeling, in order to obtain an analytic solution for the dispersal time, which can be used for source localization (see e.g. [9] ). We consider the problem of determining the concentration at time t0 from a point source located at (x0 , y0 ) . The intensity is considered to be a given value. The linearized two-dimensional behavior of the concentration q(x, y,t) caused by a point source with a step-function source rate q(t) = q∗ (t − t0 ) is described by the inhomogeneous diffusion-advection equation ( [10])

∂q ∂q ∂q ∂ 2q ∂ 2q + u (t, x, y) + v (t, x, y) + Kx 2 + Ky 2 = λ (t) q(t) + 2q∗ (t − t0 ) δ (x − x0 ) δ (y − y0 ) , ∂t ∂x ∂y ∂x ∂y

(2)

→ where the source position is denoted by − x 0 = (x0 , y0 ), and the source start time is t0 , so that the initial condition of the model is: → (3) q (− x ,t0 ) = Q0 , where Q0 is a given constant, corresponding to max {q} , which stands for the maximum concentration, as (x,y,r)

shown schematically in Figure 1. Additionally, in Eq. (2), δ −is a function of a point source (usual  Dirac  delta function), the values Kx and Ky represent the constant components of the diffusion coefficient K m2 /s , u and v are the  spatially and time dependent components of the wind velocity, q is the average impurity concentration g/m3 , and λ is the loss rate of the impurity (m/s) . Thus a sensor network system for localization and quantifation of gas or methane leaks along gas pipelines can be tested by means of analyzing the maximum values of explicit analytic solutions corresponding to the two-dimensional version of Eq. (1) describing molecular diffusion-advection modeling with a small diffusion coefficient K (for example, as shown in [9], typically K = 7.8 · 10−6 m2 /s for toluene in air at 20o C). In terms of laboratory experiments (see e.g. [11, 12]), for the source localization, it is assumed that the dispersal parameters K and the spatially homogeneous and steady wind velocity are identified prior to experiments. It is also possible to measure the wind velocity with aneometers. The direct measurement of the diffusion coefficient is not possible. Theoretically, the dependency of K from the constant wind velocity can be used to estimate K based on the anemometer measurements, but practically, it is hard to fund a such functional dependence. In terms of previous analytic studies, the model (2) has been considered under simplifying assumption that either the wind components are spatially homogeneous (i.e. u = u(t) and v = v(t) only) or that the diffusion is isotropic, i.e. Kx = Ky . Particularly, the latter assumption on the spatial wind homogeneity does not allow to include and analyze the effects of obstacles. Particularly, the dispersal model (2) has been considered by Matthes in [9] under simplifying condition of one dimensional constant airflow speed Vx in the x-direction. In terms of statistical modeling, turbulent dispersion from sources near two-dimensional obstacles was studied in [2] and [13]. In addition to this molecular diffusion-advection, turbulence in the air leads to so-called turbulent diffusion. The turbulence is caused by thermal effects, moving of objects, wind, etc. The turbulent diffusion is very complex for analytic modeling. However, the diffusion-advection model (2) is a reasonable approximation in many cases (see e.g. [14] ), especially if some averaging of the measured concentration is applied citeMatt. In this work, we shall use different scaling scenarios to simplify the analytic form of Eq. (2), but, unlike the previous analytic studies, our mathematical model will incorporate the cases of spatially and temporally inhomogeneous airflow speeds u (x, y,t) , v (x, y,t) , and anisotropic diffusion.

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3 Three scenarios of scaling analysis Without loss of generality, the term λ (t) q (t) can be eliminated from the model (2), i.e., it can be eliminated by means of transformation (see also [15]) ´t − λ (ξ )d ξ

q (x, y,t) = e

t0

q ,

(4)

where q is a new dependent variable. Thus, dropping the prime, we further study Eq. (2) written in the form

∂q ∂q ∂q ∂ 2q ∂ 2q + u (t, x, y) + v (t, x, y) + Kx 2 + Ky 2 = 2q∗ (t − t0 ) δ (x − x0 ) δ (y − y0 ) . ∂t ∂x ∂y ∂x ∂y

(5)

We consider three scaling scenarios. Scenario 1. Isotropic diffusion. Here we consider a simple reparameterization of the model (2) by introducing the following nondimensional parameters: x = L x, y = H y, u = U u, v =

L H  q∗ = U 2 q∗ , U v, t = U  t , Kx = Ky = HU K, L U

(6)

where Kx and Ky are constants. We next introduce the parameter ε = H/L > 1 to indicate that the time evolution is mostly in y−direction. Using the homogeneity property of the delta function, δ (α x) = δ (x) / |α | , we write the model (5) in the new variables (6) as

ε

∂q ∂q ∂q ∂ 2q ∂ 2q + ε (u + v ) + K(ε 2 2 + 2 ) = 2q∗ (t − t0 ) δ (x − x0 ) δ (y − y0 ) . ∂t ∂x ∂y ∂x ∂y

We look for a series solution



q = ∑ ε i qi .

(7)

(8)

i=1

Substitution of the presentation (8) into Eq. (7) and comparison of the terms with the same order of ε in this equation gives a recurrent system of ODEs for determination of all functions qi , i.e., 2 ∂ 2 q0 = q∗ (t − t0 ) δ (x − x0 ) δ (y − y0 ) , 2 ∂y K

(9)

1 ∂ q0 ∂ 2 q1 ∂ q0 ∂ q0 +u +v ), = ( 2 ∂y K ∂t ∂x ∂y

(10)

..... 1 ∂ qk−1 ∂ 2 qk ∂ qk−1 ∂ qk−1 ∂ 2 qk−2 ( + u + v ) − = , (11) ∂ y2 K ∂t ∂x ∂y ∂ x2 → and so on. So we represent q (− x ,t) as the solution of the Cauchy problem with the given initial condition (3) for q0 and zero initial conditions for qi (i > 0) .

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191

For the purpose of visualization, we shall use the Lorentz’s approximation of the Dirac delta function, i.e.

δ (y) ≈

1 γ , γ 0. Let us take the 38 configurations with M < 0, after one step of the Q2R algorithm, the 38 initial states end as following: 8 of them remain in the same partition with M < 0, 22 of them pass to M = 0 and 8 of them get a positive magnetization. Therefore the first column of the Wˆ matrix is (8/38, 22/38, 8/38), naturally their sum is the unity. In a similar way one can build systematically all the other casesh . The Wˆ -matrices read for distinct energies (we shall omit here the cases with E = ±8 which are not mixing cases): ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 4/5 1/2 0 4/19 11/38 4/19 0 1/2 4/5 Wˆ E=−4 = ⎝ 1/5 0 1/5 ⎠ , Wˆ E=0 = ⎝ 11/19 8/19 11/19 ⎠ , Wˆ E=4 = ⎝ 1/5 0 1/5 ⎠ . 0 1/2 4/5 4/19 11/38 4/19 4/5 1/2 0 As a first sight we observe a symmetry property between the cases Wˆ E=±4 . We shall discuss this fact later. The eigenvalues and the invariant probability distributions (the corresponding Eigenvectors associated to the unique unitary Eigenvalue) of these matrices are: ⎧ ⎧ {1, 4/5, −1/5} , E = −4 (5/12, 1/6, 5/12) , E = −4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ λ= {1, −3/19, 0} , E = 0 and f eq = (1/4, 1/2, 1/4) , E = 0 (7) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ {1, −4/5, −1/5} , E = 4 (5/12, 1/6, 5/12) , E = 4 5.1.2

Finest grained partition

The finest grained partition consider the exact values of magnetization M = {−4, −2, 0, 2, 4}, then the 5 × 5 matrices are: ⎛

⎞ 0 1/4 0 0 0 ⎜ 1 1/2 1/2 0 0 ⎟ ⎜ ⎟ ⎟, ˆ 0 1/4 0 1/4 0 WE=−4 = ⎜ ⎜ ⎟ ⎝ 0 0 1/2 1/2 1 ⎠ 0 0 0 1/4 0



⎞ 0 0 3/38 0 0 ⎜ 0 1/4 4/19 1/4 0 ⎟ ⎜ ⎟ ⎟, ˆ WE=0 = ⎜ 1 1/2 8/19 1/2 1 ⎜ ⎟ ⎝ 0 1/4 4/19 1/4 0 ⎠ 0 0 3/38 0 0



⎞ 0 0 0 1/4 0 ⎜ 0 0 1/2 1/2 1 ⎟ ⎜ ⎟ ⎟. ˆ WE=4 = ⎜ 0 1/4 0 1/4 0 ⎜ ⎟ ⎝ 1 1/2 1/2 0 0 ⎠ 0 1/4 0 0 0

The corresponding Eigenvalues and invariant probability distributions are:  ⎧  √  1 1 √   1 ⎪ 1, 5 , − , 5 ,0 , E = −4 1 + 1 − ⎪ 4 2 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨    √  1 √ 1 λ= 465 − 3 , 0, 0 , E = 0 3 + 465 , 76 1, − 76 ⎪ ⎪ ⎪ ⎪ ⎪    √  ⎪ √  ⎪ ⎩ 5− 1 ,0 , E =4 1, − 14 1 + 5 , − 12 , 14 h The

present calculation is recovered in the first column of Wˆ E=0 .

⎧ (1/12, 1/3, 1/6, 1/3, 1/12) , E = −4 ⎪ ⎪ ⎪ ⎪ ⎨ f eq = (3/76, 4/19, 1/2, 4/19, 3/76) , E = 0 (8) . ⎪ ⎪ ⎪ ⎪ ⎩ (1/12, 1/3, 1/6, 1/3, 1/12) E = 4

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Despite the evident differences among the coarse and the fine graining partitions, one notices that both partitions predicts at least qualitatively the same behavior of the equilibrium distribution. Finally, we emphasize the following remarks: 1. Both partitions are symmetric in the sign of M, further we observe that the equilibrium probability are symmetric under the transformation M → −M. 2. The equilibrium probability are identically for the cases E = ±4, recovering a hidden symmetry of the system. However the non-equilibrium behavior is different because the corresponding Eigenvalues have distinct signs. Notice, however, that this “hidden symmetry” is apparently not observed in numerical simulations of the Q2R model (see Fig. 1). A more careful inspection of the dynamics indicates that in the cases of the initial conditions R2 and R4 (Fig. 1), the magnetization is swapping constantly in time, for instance, if the sequence of values of magnetization for R1 is {M0 , M1 , M2 , M3 , . . . }, thus, the sequence for R3 would be {M0 , −M1 , M2 , −M3 , . . . }. Therefore, the temporal average of the magnetization, as computed in Fig. 1, would be zero for the cases of R3 and R4. Moreover, taking an average but each every two steps one recovers the Ising bifurcation for positive values of energies. Therefore, the symmetry among positive and negative energies is recovered in the phase diagram. 3. It is noticed, that there is qualitative difference for distinct energies: for E = −4 the equilibrium distribution has a maximum for M = 0, while its maximum is located at M = 0 for the case E = 0, this is the precursor of the Ising transition, as observed in Fig. 1. 5.2

Sampling for a 256 × 256 system size.

We shall consider now a very large system in a lattice with 256 × 256 sites, for this case it is not possible to perform all possible configuration to build a probability transfer matrix, therefore we consider a reduced sampling. In practice for a given p, we use a sample of 104 states, but among them, only a fraction of these states have exactly the same energy. Then, these states maybe expanded by a factor two by taking changing {x, y} → {−x, −y}. For instance, for an energy E/N = −1.8082 only 4882 states posses the same total energy. In this particular case, one notices that the distributions are well separated in two distinct cases with positive and negative M. Moreover, the dynamical rule does not allow any transfer of states being at −M into states at +M, hence the system is well separated in phase space. Mixing is possible only between close magnetization regions. It is tempted to write   10 ˆ W= . 01 But this partition does not consider all the possible values of M because the interval contained M = 0 is empty. One may cure, this singular behavior, adding a small number of configuration with zero magnetization, and using the partition: M > 0, M = 0 and M < 0. But the resulting the Wˆ matrix should be also close to the identity matrix, therefore any coarse grained distribution f eq is invariant. For a larger energy, the magnetization mixes among states having negative, positive, and null values of magnetization. Below we reproduces a probability transfer matrix for E/N = −0.0466: ⎛

⎞ 1/2 2/5 0 Wˆ = ⎝ 1/2 11/15 1/2 ⎠ . 0 2/15 1/2 The corresponding Eigenvalues and invariant probability distributions are:

λ = {1, 1/2, 7/30} ,

f eq = (0.174, 0.652, 0.174) .

(9)

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207

One may see, that qualitatively the cases of large and low energies display the same qualitative behavior of the previous sections. 6 Slow modes and transport coefficients The approach to equilibrium follows from the already discussed solution ft = Wˆ t f0 , which maybe expanded in terms of the eigenvectors of the Wˆ -matrix, getting ft = ∑Ki=1 αi λit χi . Therefore, the Eigenvalues near the unity behaves as slow modes. If one defines σi = − log λi one obtain the usual slow mode relaxation: f t = ∑Ki=1 αi e−σit χ i . Moreover, the eigenvalues closest to the unity, represents the transport coefficients, which we shall investigate in the following. We have consider the case of a 256 × 256 system size with an energy E/N = −1.445, which is closest to the energy of Ising transition therefore big fluctuations are expected. The magnetization runs over the interval M ∈ [32768, 52448]. We have performed a uniform partition with a ΔM = 24, getting a 820 × 820 matrix, which we shall not write for obvious reasons. Fig. 3 displays |λi |, ordered by decreasing absolute value, as a function of its order. As a first sight we have the impression that λi ≈ 1 − β i2 (for i < 15) characteristic of a diffusive mode, however for larger value of i one sees λi ≈ 1 − γ i . The Eigenmodes corresponding to i = 1, to 5 are also plotted showing the usual behavior of a confined Eigenvalue problem, which does not seem to agree with the diffusive mode.

Χ1

Λi

1.0    0.8      0.6     0.4  0.2 i 0.0 5 10 15 20 25 30 35 a) 0.2

Χ2

0.06 0.04 0.02 0.02 0.04 c) 0.06

200

400

600

800

k

0.08 0.06 0.04 0.02

b)

Χ3

400

600

800

200

400

600

800

200

400

600

800

0.08 0.06 0.04 0.02 0.02 0.04 d) 0.06

Χ4

200

k

k

Χ5

0.05

0.05 200

400

600

800

k

0.05

0.05

e) 0.10

f) 0.10

k

Fig. 3 Slow modes of the case 256 × 256 for an energy E/N = −1.445 with a uniform partition in M inside the interval M ∈ [32768, 52448] such a that ΔM = 24 which gives a 820 × 820 matrix. a) The first 50 Eigenvalues as a function of the index i. b-e) The first 5 Eigenmodes of the Wˆ matrices. The Eigenmode χ1 corresponds to the invariant probability vector.

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A possible explanation of the behavior of the eigenvalues, λi ≈ 1 − γ i, is in agreement with a continuous limit approximation of the Master equation (5) leading a Fokker-Planck-type equation: ˆ ft lim ( f t+1 − f t ) = (Wˆ − 1)

ΔM→0



∂P ∂P ∂ = (β + γ MP). ∂t ∂M ∂M

(10)

The slow mode dynamics is provided by the Eigenvalue problem:

−σ ϕ = which has a solution

σi = γ i and

 ∂  β ϕ + γ Mϕ . ∂M

(11)

   γ M2 γ ϕi = Hi M e− 2β , 2β

where Hi (x) is the Hermite polynomial of degree i which is a nonnegative integer: i = 0, 1, 2 . . . . Though, the behavior of the Eigenvalues is not the good one for i < 15, the behavior of the Eigenmodes seems to be the adequate. This exploration deserves a more deep study. 7 Conclusions Though this article presents an overview of the method, we can see that if the partitions are well done, this coarse graining technique is a powerful tool to reduce the information of whole system in a tractable probability transfer matrix which simplify the original master equation. One central property of this matrix, is the existence of an invariant probability distribution vector (the eigenvector with unitary eigenvalue), which is the coarse grained equilibrium probability distribution of the system. The studied cases agrees, at least qualitatively, with the numerical simulations. This study may provide the non-equilibrium properties of the system as the slow mode behavior presented in Section 6 A deep study of the present overview seems to be necessary, which is in realization. F.U. acknowledges support from the Programma de Becas de Doctorado CONICYT and SR and ET acknowledge the FONDECYT grants N 1130709 and 1120329 respectively. References [1] Nicolis, G. and Nicolis, C. (1988). Master-equation approach to deterministic chaos, Physical Review A 38, 427–433 [2] Nicolis, G., Martinez, S. and Tirapegui, E. (1991). Finite coarse-graining and Chapman-Kolmogorov equation in conservative dynamical systems, Chaos, Solitons and Fractals, 1, 25–37. [3] Vichniac, G. (1984), Simulating Physics with Cellular Automata, Physica, D 10, 96-116. [4] Pomeau, Y. (1984), Invariant in cellular automata, Journal of Physics A: Mathematical and General , 17 L415–L418. [5] Herrmann , H. (1986). Fast algorithm for the simulation of Ising models, Journal of Statistical Physics 45, 145–151 [6] Takesue, S (1987). Reversible Cellular Automata and Statistical Mechanics, Physical Review Letters 59, 2499–4503. [7] Herrmann, H.J., Carmesin, H.O. and Stauffer, D. (1987). Periods and clusters in Ising cellular automata, Journal of Physics A: Mathematical and General , 20, 4939–4948. [8] Goles, E. and Rica, S. (2011), Irreversibility and spontaneous appearance of coherent behavior in reversible systems, The European Physical Journal B, D 62, 127–137. [9] Onsager, L. (1944). Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Physical Review 65, 117–149. [10] Yang, C.N. (1952). The Spontaneous Magnetization of a Two-Dimensional Ising Model, Physical Review 85, 808–816.

Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

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

On Selective Decay States of 2D Magnetohydrodynamic Flows Mei-Qin Zhan† Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL32224, USA Submission Info Communicated by Valentin Afraimovich Received 6 January 2015 Accepted 10 March 2015 Available online 1 July 2015 Keywords selective decay principle 2D magnetohydrodynamic vorticity-flux instability MHD

Abstract The selective decay phenomena has been observed by physicists for many dynamic flows such as Navier-Stoke flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. Rigorous mathematical works have been carried out for both Navier-Stoke and barotropic geophysical flows. In our previous work, we have rigorously showed the existence of selective states for 2D MHD flows. In this paper, we present a partial result on instability of the selective states

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

1 Introduction One of the interesting selection principles that characterize the large time asymptotic states of turbulent flows is the selective decay principle. This principle states that, due to the transfer of the dissipated quantities to short wavelengths where the dissipation coefficients become effective, one or more ideal invariants are dissipated more rapidly relative to another in the turbulent process. In other words, in the selective decay process associated with turbulence, nonlinearity causes a preferential migration of one or more invariants to higher wavenumber, and another to lower wavenumber. Many researchers have done a great amount of work on the selective principles range from physical observations, numerical analysis and simulations, to mathematic rigorous justification. We have list a few such works in the reference ( [1]– [20]). Inspired by the work of Majada and Wang [13], in M. Zhan [20], we gave the following definition of the selective decay principle: Definition 1. Selective Decay Principle: After a long time, the solutions of the the 2D MHD (Magnetohydrodynamic) approach those states which minimize the energy for a given mean square vector potential. The appeal of such principle is that it reduces the calculation of the asymptotic states of the MHD to a simpler problem in the calculus of variation. As discussed in Majada and Wang [14], the solutions of the MHD may † Corresponding

author. Email address: [email protected]

ISSN 2164 − 6376, eISSN 2164 − 6414/$-see front materials © 2015 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/DNC.2015.06.008

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not approach only a minimizing state, but rather some critical point of the energy. We include in the discussion all states rather than just the energy minimizers at constant mean square vector potential. For conciseness, we refer to such a critical point of the energy at constant mean square vector potential as a selective decay state (for detailed discussion and rationale, see Majada and Wang [14]). From theoretic point of view, to justify the selective decay principle, one needs to address the following issues (Majada and Wang [14]): • Invariance. This is the one that make the selective decay state meaningful. • Convergence. This is needed to justify the selective decay principle. • Stability. This is useful in interpreting the numerical results which indicate all flows converge to some ground states. Before we describe our results in detail, we first introduce the 2D MHD, the two dimensional MHD in the vorticity-flux form is (see for instance Longcope et al [11] and Singh, M., et al [18]), Δφt + J(φ , Δφ ) − J(ψ , Δψ ) − μ Δ2φ = 0,

(1)

ψt + J(φ , ψ ) = ν Δψ ,

(2)

where J( f , g) = fx gy − fy gx is the Jacobi operator, ω = Δφ is the vorticity, ψ is the magnetic flux, φ is the velocity flux (also called stream function), v = ⊥ ψ =< −∂y ψ , ∂x ψ > is the magnetic field,

u = ⊥ φ =< −∂y φ , ∂x φ >

is the velocity, and j = −Δψ is the current. The kinematic viscosity μ is a reciprocal Reynolds number and the magnetic diffusivity ν is a reciprocal of a magnetic Reynolds number. Both μ and ν are positive constants. We will supply the 2D MHD with the periodic boundary conditions for both ´ ´ equations on the region defined by T 2 = [0, π ]× [0, π ], and assume the zero average conditions T 2 φ dxdy = T 2 ψ dxdy = 0 for the initial values. We also introduce the following three quotients that are used in study of selective decay principle for 2D MHD (see [20] for details),   φ 2 +   ψ 2 E = C (t) = A ψ 2 be the generalized Dirichlet quotient, and b(t) =

  ψ 2   φ 2 , and a(t) = . ψ 2 ψ 2

In our previous work, we have shown that the quotients C (t) and a(t) converges to an eigenvalues of Laplace operator −Δ, and the correspondent eigenstates are tehthe selective states which are invariant for 2D MHD. In this short article we will show that the selective decay states are unstable under small perturbation. Similar result has been proved by Majda and Wang for the barotropic geophysical flows (Majda and Wang [14]) ) and was implied by the normal form derived by Foias and Saut for 2D Navier-Stokes Equations. For the barotropic geophysical flows and the 2D Navier-Stokes Equations, the instability are simple to prove as the Dirichlet quotients are monotonic decreasing in all time. The case for 2D MHD are slight more complex

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

211

as the Dirichlet quotients are not monotonic in general. We have to give a detail analysis to show that, under certain condition on the initial data, the generalized Dirichlet quotient is monotonically decreasing in time. The paper is organized in such a way that, in section 2, we will present some preliminary results that shows the timely exponential decay of the relevant quantities which will be used in proving the decaying of ψ to decayed states. In section 3 we will state and prove our main results. 2 The preliminaries In this section we present the exponential decay in time of various quantities that are needed in the proof of decaying of ψ and φ to the selective states. We also prove an elementary result concerning existence of limit of a given function at infinity. As pointed out in the introduction, we will supply the 2D MHD with the periodic boundary conditions for 2 both equations ´ on the region defined by T = [0, 2π ] × [0, 2π ], and assume that the zero average conditions ´ T 2 φ dxdy = T 2 ψ dxdy = 0. We begin our exposition with the of following decay results, in the following, C will be a constant independent of μ , ν and initial values, Proposition 2. (1) The total energy E(t) = u2 + v2 and the mean square vector potential A (t) = ψ 2 decay exponentially in time as t → ∞. In fact, we have A (t) ≤ ψ (0)2 e−2ν t , 2μ t

e ˆ



0

2

2ν t

φ  + e

(3)

2

2

2

ψ  ≤ φ (0) + ψ (0) ,

(4)

μ Δφ (t)2 + ν Δψ (t)2 dt ≤ φ (0)2 + ψ (0)2 ,

(5)

(2) let B(μ , ν , E(0)) = νC2 ( ν1 + μ1 )2 (E(0))2 , and δ = min{μ , ν }. The sum of the enstrophy E and the mean square current J decays exponentially as t → ∞. That is, there exists a constant C independent of the initial data, μ , and ν , such that, 3

e 2 δ t (Δψ (t)2 + Δφ (t)2 ) ≤ (Δψ (0)2 + Δφ (0)2 ) 1 1 C B(μ , ν , E(0)) ( + ) e2B(μ ,ν ,E(0)) (Δψ (0)2 + Δφ (0)2 )2 , + ν ν μ and 3 2

ˆ 0



(6)

(μ   Δφ 2 + ν   Δψ 2 ) ds

≤ (Δψ (0)2 + Δφ (0)2 )(1 + B(μ , ν , E(0))eB(μ ,ν ,E(0)) ). Proof. Proof of (1): It is easy to see the following identity, d ψ 2 + 2ν   ψ 2 = 0, dt d (  φ 2 +   ψ 2 ) + 2(μ Δφ 2 + ν Δψ 2 ) = 0. dt Then the Poincar´e inequalities ψ 2 ≤   ψ 2 ,

  ψ 2 ≤ Δψ 2 ,

φ 2 ≤   φ 2 , and

  φ 2 ≤ Δφ 2

(7)

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show the results. Proof of (2):First we have, ˆ |

ˆ 2J(ψ , Δψ )Δφ dxdy| ≤ 2

|  ψ ||  Δψ ||Δφ | dxdy

≤   Δψ  Δφ    ψ ∞ 3

1

≤   Δψ  2 Δφ    ψ  2 (Agmon inequality) C ν ≤   Δψ 2 + 3 Δφ 4   ψ 2 , 4 ν

(8)

and notice that,

∂x J(φ , ψ ) = J(∂x φ , ψ ) + J(φ , ∂x ψ ), ∂y J(φ , ψ ) = J(∂y φ , ψ ) + J(φ , ∂y ψ ), so

ˆ 2|

J(φ , ψ )Δ2 ψ dxdy| = 2|

(9) (10)

ˆ J(φ , ψ )  Δψ dxdy|

≤ 2(Δφ    ψ ∞ + Δψ    φ ∞ )  Δψ  1

3

1

1

≤ 2Δφ    ψ  2   Δψ  2 + 2Δψ    φ  2   Δφ  2   Δψ  C C μ ν Δψ 4   φ 2 ≤   Δφ 2 +   Δψ 2 + 3 Δφ 4   ψ 2 + 4 4 ν μν 2 μ ν ≤   Δφ 2 +   Δψ 2 4 4 C 1 1 + 2 ( + )(  ψ 2 +   φ 2 )(Δψ 2 + Δφ 2 )2 . ν ν μ

(11)

Since, d (Δψ 2 + Δφ 2 ) − 2 < J(ψ , Δψ ), Δφ > dt +2 < J(φ , ψ ), Δ2 ψ > +2(μ   Δφ 2 + ν   Δψ 2 ) = 0.

(12)

Put together, we get 3 d (Δψ 2 + Δφ 2 ) + (μ   Δφ 2 + ν   Δψ 2 ) dt 2 C 1 1 2 ≤ 2 ( + )(  ψ  +   φ 2 )(Δψ 2 + Δφ 2 )2 . ν ν μ

(13)

Let

C 1 1 ( + ) (  ψ (t)2 +   φ (t)2 ) (Δψ (t)2 + Δφ (t)2 ). ν2 ν μ From estimation (3) and (5) we have ˆ ∞ C 1 1 F(s) ds ≤ 2 ( + )2 (  ψ (0)2 +   φ (0)2 )2 = B(μ , ν , E(0)) < ∞. ν ν μ 0 F(t) =

So from (13), we have 3 d (Δψ 2 + Δφ 2 ) + (μ   Δφ 2 + ν   Δψ 2 ) dt 2 ≤ F(t)(Δψ 2 + Δφ 2 ),

(14)

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

213

elementary argument shows that Δψ (t)2 + Δφ (t)2 ≤ (Δψ (0)2 + Δφ (0)2 )eB(μ ,ν ,E(0)) is uniformly bounded for all t > 0. This, in turn, leads to the following exponential decay result for F(t), thanks to the estimates (3) and (5), C 1 1 ( + ) (  ψ (t)2 +   φ (t)2 ) (Δψ (t)2 + Δφ (t)2 ) ν2 ν μ C 1 1 ≤ 2 ( + ) (Δψ (0)2 + Δφ (0)2 )eB(μ ,ν ,E(0)) (  ψ (t)2 +   φ (t)2 ) ν ν μ C B(μ , ν , E(0)) (Δψ (0)2 + Δφ (0)2 )eB(μ ,ν ,E(0)) e−2δ t . ≤ ν F(t) =

Hence the inequality (14) can be written, 3 d (Δψ 2 + Δφ 2 ) + (μ   Δφ 2 + ν   Δψ 2 ) dt 2 2 ≤ (Δψ (0) + Δφ (0)2 )eB(μ ,ν ,E(0))F(t).

(15)

With the help of the Poincar´e inequalities Δψ 2 ≤   Δψ 2 and Δφ 2 ≤   Δφ 2 , we can deduce that, 3

e 2 δ t (Δψ (t)2 + Δφ (t)2 ) 2

2

2

2

B( μ ,ν ,E(0))

≤ (Δψ (0) + Δφ (0) )(1 + e

ˆ



3

F(s)e 2 δ s ds)

0

≤ (Δψ (0) + Δφ (0) ) 1 1 C B(μ , ν , E(0)) ( + ) e2B(μ ,ν ,E(0)) (Δψ (0)2 + Δφ (0)2 )2 + ν ν μ and 3 2

ˆ 0



2

2

2

2

B( μ ,ν ,E(0))

(μ   Δφ  + ν   Δψ  ) ds ≤ (Δψ (0) + Δφ (0) )(1 + e

ˆ



F(s) ds)

0

≤ (Δψ (0)2 + Δφ (0)2 )(1 + B(μ , ν , E(0))eB(μ ,ν ,E(0)) ds) That finishes the proof. With the proceeding results, we have, A = (Δψ (0)2 + Δφ (0)2 Proposition 3. φ (t)W 1,∞ ∈ L2 ([0, ∞)) ∩ L1 ([0, ∞)). Furthermore, there is a constant C which does not dependent on initial data and δ , such that φ (t)W 1,∞ + ψ (t)W 1,∞ ≤ C(Δψ (0)2 + Δφ (0)2 )e−δ t 1 1 1 1 [E(0) + B(μ , ν , E(0)) ( + ) e2B(μ ,ν ,E(0))] 2 ν ν μ

(16)

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Proof. Recall that the Agmon type inequality indicates that 1

1

1

1

φ (t)W 1,∞ + ψ (t)W 1,∞ ≤ Cφ (t)H2 1 φ (t)H2 2 + φ (t)H2 1 φ (t)H2 2 So the results of Proposition 2 give φ (t)W 1,∞ + ψ (t)W 1,∞ ≤ C(Δψ (0)2 + Δφ (0)2 )e−δ t 1 1 1 1 [E(0) + B(μ , ν , E(0)) ( + ) e2B(μ ,ν ,E(0)) ] 2 ν ν μ Next, we present the following elementary result concerning existence of limit of a given function at infinity. For completeness, we shall present a proof. Lemma 4. For t0 > 0, let g(t) ≥ 0, f (t) ≥ 0 be defined on [t0 , ∞). Suppose that limt→∞ g(t) = 1, and for any t1 > t0 , we have f (t) ≤ g(t1 ) f (t1 ), for all t > t1 , then limt→∞ f (t) exists. Proof. Choose t1 = 1 + t0 we have 0 ≤ f (t) ≤ g(1 + t1 ) f (1 + t1 ) for all t > t1 . Hence f (t) is bounded on [1 + t1 , ∞). Now suppose limt→∞ f (t) does not exist. Then we could find two sequences tn1 ,tn2 → ∞, as n1, n2 → ∞ such that lim f (tn1 ) = a1 > lim f (tn2 ) = a2 . n1→∞

n2→∞

But, for any fixed n2 and for all n1 large enough such that tn1 > tn2 we have, f (tn1 ) ≤ g(tn2 ) f (tn2 ), hence lim f (tn1 ) = a1 ≤ g(tn2 ) f (tn2 ).

n1→∞

Let tn2 → ∞ in above inequality, we get

a1 ≤ lim f (tn2 ) = a2 , n2→∞

a contradiction. That proves the lemma. 3 The Main Results In this section, we will state and prove our main results. First we recall that C (t) =

  φ (t)2 +   ψ (t)2 E = A ψ (t)2

be the generalized Dirichlet quotient, and b(t) =

  ψ (t)2   φ (t)2 , and a(t) = . 2 ψ (t) ψ (t)2

First we have the following estimate: | < J(ψ , φ ), Δψ > | =< J(ψ , φ ), Δψ + a(t)ψ > | ≤   φ (t)∞   ψ  Δψ + a(t)ψ  2 ≤   φ (t)2∞   ψ 2 + 2ν Δψ + a(t)ψ 2 . ν

(17)

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

215

With this at hand, we can have a better estimate for a(t) then what we have obtained in M. Zhan [20]. −2(ν Δψ 2 + < J(φ , ψ ), Δψ >) 2ν   ψ 4 d a(t) = + , dt ψ 2 ψ 4 for the term

Δψ 2 ψ 2 −ψ 4 , ψ 4

(18)

we have

1 Δψ 2 ψ 2 −   ψ 4 = (Δψ 2 − a(t)  ψ 2 ) 4 ψ  ψ 2 1 = Δψ + a(t)ψ 2 ≥ 0. ψ 2

(19)

Using the inequality (17) for j(t), we have 2ν < J(φ , ψ ), Δψ > d a(t) ≤ − Δψ + a(t)ψ 2 − 2 2 dt ψ  ψ 2 2 2   ψ 2 ≤   φ (t)2∞ =   φ (t)2∞ a(t). 2 ν ψ  ν

(20)

The Gronwall’s inequality shows that for any t > t1 > 0, a(t) ≤ a(t1 )e ≤ a(t1 )e

´ 2 t 2 ν t1 φ (s)∞ ds

´ 2 t ν t1 φ (s) Δφ (s) ds 2

≤ a(t1 )e ν

(

´t t1

4

≤ a(t1 )e 3ν 3

´t t1

1

Δφ (s)2 ds) 2 1

((φ (0)2 +ψ (0)2 ) (Δψ (0)2 +Δφ (0)2 ) (1+B( μ ,ν ,E(0))eB(μ ,ν ,E(0)) )) 2 e−ν t1

− ν t1

≤ a(t1 )eMe

1

φ (s)2 ds) 2 (

,

(21) 1 2

where M = 3ν4 3 ((φ (0)2 + ψ (0)2 ) (Δψ (0)2 + Δφ (0)2 ) (1+ B(μ , ν , E(0))eB(μ ,ν ,E(0)) )) . In particular we have a(t) ≤ a(0)eM for all t > 0

(22)

Me−ν t

, we have limt→∞ h(t) = 1. With the help of Lemma 4, we see that a∗ = limt→∞ a(t) exist. Let h(t) = e We have proven in (Zhan 2010) that a∗ is an eigenvalue of −Δ. That is, −Δψ ∗ = a∗ ψ ∗ . Let j(t) =

) , ψ 2

with the help of preliminary results, one can show that | j(t)| ≤ Cφ (t)W 2,∞ a(t).

(23)

Now look at b(t), taking derivative with respect to t and using the equations, we have −2(μ Δφ 2 + < J(ψ , Δψ ), φ >) 2ν   φ 2   ψ 2 d b(t) = + . dt ψ 2 ψ 4

(24)

With the help of (22) we have ψ 2 Δφ 2 − p  φ 2   ψ 2 d b(t) = −2μ − 2 j(t) dt ψ 4 ψ 2   φ 2 − p  φ 2   ψ 2 − 2 j(t) ≤ −2μ ψ 4   φ 2 − 2 j(t) ≤ −2μ (1 − pa(t)) ψ 2 = −μ (1 − pa(t))b(t) +Cφ (t)W 2,∞ a(t).

(25)

216

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

Therefore, if

λ = 1 − pa(0)eM > 0,

(26)

ˆ t b(t) ≤ Ce−λ t ( a(t)φ (t)W 2,∞ eλ s ds + 1) → 0 as t → ∞,

(27)

we have 1 − pa(t) ≥ λ > 0, ∀t > 0

0

and

ˆ

∞ 0

b(s) ds < ∞.

(28)

Remark 5. Please notice that λ = 1 − pa(0)eM approaches to 1 when   ψ (0) → 0. So when   ψ (0) is small, the value p can be large. From the previous discussions we arrive at our main result, Lemma 6. Under the condition 1 − pa(0)eM > 0, the generalized Dirichlet quotient C (t) is strictly decreasing in t and limt→∞ C (t) exists such that 1 ≤ a∗ = lim C (t) = lim a(t), t→∞

t→∞

(29)

where a∗ is an eigenvalue of the Laplacian Δ. Proof. Taking the derivative with respect to t, we have d C (t) dt −2(μ Δφ 2 + ν Δψ 2 ) 2ν (  φ 2 +   ψ 2 )(  ψ 2 ) + = ψ 2 ψ 4 2 2 2 2 μ ψ  Δφ  − ν   φ    ψ  Δψ 2 ψ 2 −   ψ 4 = −2 − 2 ν ψ 4 ψ 4 ≤ −2μ (1 − pa(t))b(t),

(30)

notice again ,

μ ψ 2 Δφ 2 − ν   φ 2   ψ 2 ≥ μ (1 − pa(t))b(t). ψ 4

(31)

Therefore the condition 1 − pa(0)eM > 0 lead to, d C (t) ≤ 0. dt So the limit limt→∞ C (t) exists. Since limt→∞ b(t) = 0, we have 2 ≤ a∗ = limt→∞ C (t) = limt→∞ a(t), With above preparations, we can prove the following result, the detail of proof can be found in M. Zhan [20]. Theorem 7. In the absence of external forces, for any sufficiently smooth initial data satisfying ψ (0) = 0, and 1 − pa(0)eM > 0, the 2D magnetohydrodynamic(MHD) flows possess the following selective decay principle: there exists an eigenvalue a∗ of the Laplacian such that lim C (t) = a∗ ,

t→∞

(32)

Such that for any increasing sequence of times {t j }∞j=1 such that t j → ∞, there exists a subsequence {t jk }∞ k=1 and a selective decay state ψ ∗ corresponding to the eigenvalue a∗ , so that  (t jk ) − ψ ∗  → 0 and   φ(t jk ) → 0 as k → ∞. ψ

(33)

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

217

The above result show existence of selective states and converges of solutions to the selective states dictated by the initial values and any imposed symmetries. However, in order to have a good mathematical understanding of why only the ground state (respecting imposed symmetries) is actually observed, we need to show that the flows approaching higher states are unstable under small perturbations from the lower eigenmodes. The following result shows that is true for the 2D MHD flow, Theorem 8. In the absence of external forces, for any sufficiently smooth initial data satisfying ψ (0) = 0, and 1 − pa(0)eM > 0, the selective states associated with eigenvalue a∗ of the Laplacian such that lim C (t) = a∗ ,

(34)

t→∞

is unstable if a∗ is eigenvalue Λk with k > 1. Proof. Since at selective state φ (t) = 0, we can choose φ (0) = 0 as initial value for φ . Start with a selective decay state ψk ∈ Ek , the eigenspace associated with eigenvalue Λk with k > 1, that satisfies

1−

ν   ψk 2 M e > 0, μ ψk 2

(35) 1

notice that at the choice of initial value for φ , M = 3ν4 3 (ψk 2 Δψk 2 )(1 + B(μ , ν , ψk )eB(μ ,ν ,ψk ) )) 2 , where B(μ , ν , ψk ) = νC2 ( ν1 + μ1 )2   ψk 2 . Let ψ j ∈ E j with j < k, ψk  = 1, and consider the flow ψk + εψ j , for small perturbation strength ε > 0 such that the inequality (35) still true. Due to orthogonality, we have ψk + εψ j  = ψk  + ε , and   (ψk + εψ j ) =   ψk  + ε 2 Λ j . Therefore, the generalize Dirichlet quotient   (ψk + εψ j )2 ψk + εψ j 2

(36)

Λk ψk 2 + ε 2 Λ j ψk 2 + ε 2

(37)

ε2 (Λ j − Λk ) < Λk ψk 2 + ε 2

(38)

C (ψk + εψ j ) =

= ≤ Λk +

Since, under the condition 1 − pa(0)eM > 0, C (t) is decreasing, we have a∗ (ψk + εψ j ) < Λk . This shows that, the asymptotic structure of perturbed flow, for small enough ε , will resemble selective decay states from a lower eigenspace. 4 Conclusion remark Self-organization is an important aspect of MHD turbulence. It is intimately connected with selective dissipation of the ideal invariants that leading to large-scale quasi-static magnetic structures. In our previous works, we have shown the existence of selective decay states for the 2D MHD flow. In this short remark, we discuss the unstability of selective decay states for 2D MHD flow. This unstability result gives a good mathematical understanding of why, in some configuration, the 2D MHD flow eventually evolves to the state with the mean square vector potential has crowded into the longest wavelengths permitted by the imposed

218

Mei-Qin Zhan / Discontinuity, Nonlinearity, and Complexity 4(2) (2015) 209–218

symmetries (periodic boundary conditions) and the kinetic energy has been dissipated, namely the 2D MHD flow organizes itself into a large-scale quasi-static magnetic structures. However, we do not analyze how the flow actually approaches the selective decay state. Intriguing dynamics can arise; numerical results show that before the flow reaches the final selective decay state, such selective decay state may already be satisfied locally in certain coherent regions, magnetic eddies of approximately circular shape generated during turbulence decay (Biskamp [1]). On the other hand, a flow might approach a certain “higher” selective decay state, hover there for a while, then cascade down to the lower decay states. Such a phenomenon is known as a coarsening transition. More discussion of the selective decay process will be carried out in our future researches.

References [1] Biskamp, D. ( 2003), Magnetohydrodynamic Turbulence, Cambridge University Press. [2] Brown, M.R. (1997), Experimental evidence of rapid relaxation to large-scale structures in turbulent fluids: selective decay and maximal entropy , Plasma Physics, 57, 203–227. [3] Constantin, P. and Foias, C. (1988), Navier-Stokes Equations, Chicago University Press, Chicago. [4] Embid, O. and Majda, A., (1998), Low Froude number limiting dynamics for stably stratified flow with small or finite Rossby numbers, Geophysical & Astrophysical Fluid Dynamics, 87, 1–50. [5] Foias, C., Manley, O.P., Rosa R., and Temam, R. (2001), Cascade of energy in turbulent flows , C.R. Acad. Sci. Paris S´er., 332, 1–6 [6] Foias, C. and Saut, (1984), Asymptotic behaviour, as t → ∞ of solutions of Navier-Stokes equations and non-linear spectral manifolds, Indiana University Mathematics Journal, 33, 459–477. [7] Foias, C., and Temam, R., (1989), Gevrey class regularity for the solutions of the Navier-Stokes equations, Journal of Functional Analysis, 87, 359-369. [8] Kraichnan, R.H. (1959), The structure of isotropic turbulence at very high Reynolds numberss,Journal of Fluid Mechanics, 5, 497–543. [9] Kraichnan, R.H. (1965), Lagrangian-history closure approximation for turbulence, Physics of Fluids, 8, 575–598. [10] Kraichnan, R.H. (1967), Inertial ranges in two-dimentional turbulence, Physics of Fluids, 10, 1417–1423. [11] Longcope, D.W. and Strauss, H.R. (1993), The coalescence instability and the development of current sheets in twodimensional magnetohydrodynamics,Physics of Fluids, 135, 1858-2869. [12] Majda, A., and Holen, M. (1998), Dissipation, topography, and statistical theories for large scale coherent structure, CPAM, L, 1183–1234. [13] Majda, A., and Wang, X. (2001), The selective decay principle for barotropic geophysical flows, MAA, textbf 8, 579– 594. [14] Majda, A., and Wang, X. (2006), Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press. [15] Matthaeus, W.H., and Montgomery, D. (1980), Selective decay hypothesis at high mechanical and magnetic Reynolds numbers, Annals New York Academy of Sciences, 203–222. [16] Matthaeus, W.H., Stribling, W.T., Martinez, D., Oughton, S., and Montgomery, D. (1991), Decaying two-dimensional Navier-Stokes turbulence at very long times, Physica D, 51, 531–538. [17] Montgomery, D., Shan, X., Matthaeus, W.H. (1993), Navier-Stokes relaxation to sinh-Poisson states at finite Reynolds numbers, Physics of Fluids A, 5(9), 2207–2216. [18] Singh, M., Khosla, H. K., and Malik, J.S.K. (1998), Nonlinear dispersive instabilities in Kelvin Helmholtz MHD flows Plasma Physics, textbf59, 27–37. [19] Temam, R. (1997), Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd Edition , Springer-Verlag, New York. [20] Zhan, M.(2010), Selective Decay Principle For 2D Magnetohydrodynamic Flow, Asymptotic Analysis, 67, 125–146.

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Aims and Scope The interdisciplinary journal publishes original and new results on recent developments, discoveries and progresses on Discontinuity, Nonlinearity and Complexity in physical and social sciences. The aim of the journal is to stimulate more research interest for exploration of discontinuity, complexity, nonlinearity and chaos in complex systems. The manuscripts in 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 discontinuity, complexity, nonlinearity and chaos in physical and social sciences. Topics of interest include but not limited to • • • • • • • • • • • • • •

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

Tassilo Küpper Mathematical Institute University of Cologne, Weyertal 86-90 D-50931 Cologne, Germany Fax: +49 221 470 5021 Email: [email protected]

Nikolai Rulkov BioCircuits Institute, University of California, San Diego, 9500 Gilman Drive #0328 La Jolla, CA 92093-0328, USA Fax: (858) 534-1892 Email: [email protected]

Didier Bénisti CEA, DAM, DIF 91297 Arpajon Cedex France Fax: +33 169 267 106 Email: [email protected]

Marc Leonetti IRPHE, Aix-Marseille Université UMR CNRS 6594, Technopôle de ChâteauGombert 13384 Marseilles Cedex 13 France Fax: + 33 4 13 55 20 01 Email: [email protected]

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

Alexandre N. Carvalho Departamento de Matem´atica, Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao Universidade de S˜ao Paulo - Campus de S˜ao Carlos, Caixa Postal 668, 13560-970 S˜ao Carlos SP, Brazil Email: [email protected]

Yuri Maistrenko Institute of Mathematics National Academy of Sciences of Ukraine Volodymyrska Str. 54, room 232 01030 Kiev, Ukraine E-mail: [email protected]

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119992 Moscow, Russia Fax: +7 495 939 03 97 Email: [email protected]

Maurice Courbage CNRS-UMR 7057 "Matière et Systèmes Complexes", 75205 Paris Cedex 13 France Email: [email protected]

Mikhail Malkin Department of Mathematics and Mechanics Nizhny Novgorod State University, Nizhny Novgorod, Russia Fax: +7 831 465 76 01 Email: [email protected]

Marco Thiel Institute for Mathematical Biology and Complex Systems University of Aberdeen AB243UE Aberdeen, Scotland, UK Fax: +44 1224 273105 Email: [email protected]

Michal Feckan Department of Mathematical Analysis and Numerical Mathematics, Comenius University Mlynska dolina 842 48 Bratislava, Slovakia Fax: +421 2 654 12 305 Email: [email protected]

Vladimir I. Nekorkin Institute of Applied Physics of RAS 46 Ul'yanov Street, 603950, Nizhny Novgorod, Russia Email: [email protected]

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Marie-Christine Firpo Laboratoire de Physique des Plasmas CNRS UMR 7648, Ecole Polytechnique 91128 Palaiseau cedex, France Tel: (00 33) 1 69 33 59 04 Fax: (00 33) 1 69 33 59 06 E-mail: [email protected]

Dmitry E. Pelinovsky Department of Mathematics & Statistics McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 Fax: +1 905 522 0935 Email: [email protected]

Edgardo Ugalde Instituto de Fisica Universidad Autonoma de San Luis Potosi Av. Manuel Nava 6, Zona Universitaria San Luis Potosi SLP CP 78290, Mexico Email: [email protected]

Stefano Galatolo Dipartimento di Matematica Applicata Via Buonattoti 1 56127 Pisa, Italy Email: [email protected]

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An Interdisciplinary Journal of Discontinuity, Nonlinearity, and Complexity Volume 4, Issue 2

June 2015

Contents Poincaré Recurrences in the Circle Map: Fibonacci Stairs V.S. Anishchenko, N.I. Semenova, and T.E. Vadivasova .........................................

111-119

A Semi-analytical Prediction of Periodic Motions in Duffing Oscillator through Mapping Structures Albert C. J. Luo and Yu Guo………...……...…………………….………………..

121-150

Topology of Delocalization in the Nonlinear Anderson Model and Anomalous Diffusion on Finite Clusters A.V. Milovanov and A. Iomin....................................................................................

151-162

A Method for Solving Nonlinear Differential Equations: An Application to  4 Model Danilo V. Ruy…………………………………………...……………..…………...

163-171

Synchronization of Micro-Electro-Mechanical-Systems in Finite Time Hadi Delavari, Ayyob Asadbeigi, and Omid Heydarnia……...…...…...…………..

173-185

Scaling Modeling of the Emitted Substance Dispersion Transported by Advection Caused by Non-homogeneous Wind Field and by Isotropic and Anisotropic Diffusion in Vicinity of Obstacles Ranis N. Ibragimov, Andrew Barnes, Peter Spaeth, Radislav Potyrailo, and Majid Nayeri…………………………………........…………..…………………...…..….

187-197

Coarse-Graining and Master Equation in a Reversible and Conservative System F. Urbina, S. Rica, and E. Tirapegui....…………………………..…………….....

199-208

On Selective Decay States of 2D Magnetohydrodynamic Flows Mei-Qin Zhan....………………………….………………………..…………….....

209-218

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

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