Synchronization on the accuracy of chaotic oscillators simulations

Numerical problems are considered on general synchronization of chaotic oscillators, through the evaluation of the Lower Bound Error index on two case studies: a Lorenz system unidirectionally coupled to a Duffing system and a Duffing system unidirec…

Authors: Gabriel H. A. Silva, Igor C. Silva, Wilson R. L. Junior

Synchronization on the accuracy of chaotic oscillators simulations
DINCON 2017 CONFER ˆ ENCIA BRASILEIRA DE DIN ˆ AMICA, CONTR OLE E APLICAC ¸ ˜ OES 30 de outubro a 01 de no v em bro de 2017 – S˜ ao Jos ´ e do Rio Preto/SP Sync hronization on the accuracy of c haotic oscillators sim ulations Gabriel Hugo ´ Alv ares Silv a 1 Igor Carlini Silv a 2 Wilson Ro c ha Lacerda Junior 3 Samir Angelo Milani Martins 4 M´ arcio F alc˜ ao San tos Barroso 5 Eriv elton Geraldo Nep om uceno 6 Con trol and Mo deling Group, UFSJ, S˜ ao Jo˜ ao del-Rei, MG, Brazil Departmen t of Electrical Engineering, UFSJ, S˜ ao Jo˜ ao del-Rei, MG, Brazil Abstract . Numerical problems are considered on general sync hronization of c haotic oscilla- tors, through the ev aluation of the Lo w er Bound Error index on tw o case studies: a Lorenz system unidirectionally coupled to a Duffing system and a Duffing system unidirectionally coupled to a Rossler system. It w as p ossible to observ e, in each case, that the b ehavior of the sla ve’s LBE curv e tends to follo w the b eha vior of the master’s as the v alue of the coupling constant is increased up to a certain v alue, and th us, that synchronization can affect n umerical calculations. Key-w ords . General synchronization, Chaotic oscillators, Low er Bound Error, Numerical Computation. 1 In tro duction Since the work of Lorenz in 1963 [3], chaos synchronization has b een extensively stud- ied by several researchers. Sync hronization of chaos is often understo o d as a regime in whic h tw o coupled chaotic systems exhibit iden tical, but still chaotic, oscillations [10]. 1 gabrielh ugo0701@gmail.com 2 igorufsj@y aho o.com.br 3 wilsonrljr@outlo ok.com 4 martins@ufsj.edu.br 5 barroso@ufsj.edu.br 6 nep om uceno@ufsj.edu.br 2 Chaos synchronization has b een applied in electrical [13], biological, chemical, and secure comm unication [5] problems. In this context, it is well kno wn that countless researc hers identifies a chaotic system b eha viour [9] b y analyzing n umerical solutions, obtained using p opular softw are, in which the reliability of results is not carefully v erified [4]. Nep om uceno [6] shows that a simple sequence of iterations of discrete logistic mo del may generate a steady state result that con v erges to the wrong answ er. It is w orth noting that inv estigation of propagation error is not a recent issue [2]. In fact, there are many w orks based on deterministic or sto c hastic to ols that provide some confidence in simulation of recursiv e functions. Based on the fact that although in terv al extensions are mathematically equiv alent, they ma y generate different computer sim ulation outcomes, Nep omuceno and Martins [7] in tro duced an approach to ev aluate a low er b ound error in recursive functions. The Low er Bound Error (LBE) index may conduce the understanding of the solutions generated by nonlinear dynamical systems. Nep om uceno and Mendes [8] show the existence of multiple pseudo-orbits for nonlinear dynamics systems when discretization schemes are used, when the step-size and the initial conditions are k ept unc hanged. Although there are man y studies regarding synchronization of chaotic oscillators and regarding numerical problems, as far as w e know, no study deals with numerical problems in the sync hronization of c haotic systems. This is the main scop e of this pap er. W e dev elop ed t w o cases studies in which w e coupled a Lorenz system to a Duffing system and a Duffing system to a Rossler system, in order to ev aluate the effects of general sync hronization on the rec k oning of Low er Bound Error. This paper is laid out as follo ws: In Section 2, the preliminary concepts are briefly review ed. The proposed method base d on LBE is presen ted in Section 3. The results as w ell as the discussion are presented in Section 4, while concluding remarks and persp ectiv es for future researc h are sho wn in Section 5. 2 Preliminary Concepts 2.1 Generalized synchronization The simplest concept of synchronization b et w een c haotic oscillators, called complete sync hronization, o ccurs when the distance b etw een the state v ariables of t w o dynamical systems con v erges to zero while evolving in time. Let ˙ x = F ( x ) and ˙ y = G ( y ) b e the represen tations of tw o chaotic systems, where x is a n -dimensional state vector and y is a m -dimensional state vector. F and G are vector fields, F : R n → R n , and G : R m → R m . Those systems are said to b e completely synchronized if l im t →∞ || x ( t ) − y ( t ) || = 0. In [12], Rulk ov et al presented a generalization of the definition of complete synchro- nization, where the state v ariables of the systems considered do not hav e to b e identical. F or this case, it is sufficient if they presen t a functional relation. Let ˙ x = F ( x ) and ˙ y = G ( y , h µ ( x )) be t w o unidirectionally coupled systems, where x is the n -dimensional state vector of the driver and y is the m -dimensional state vector of the resp onse. F and G are vector fields, F : R n → R n , and G : R m → R m . The vector field h µ ( x ) : R m → R m rule the couple b et w een resp onse and driver. When the parameter µ = 0, b oth systems 3 are chaotic, since there is no relation b etw een their ev olution. When µ = 0, the systems are considered generally synchronized if exists a transformation ψ : x → y which is able to map asymptotically the tra jectories of the driver attractor in to the ones of the resp onse [1]. 2.2 Lo w er Bound Error Before introducing LBE’s concept, we need to presen t the definitions of orbits and pseudo-orbits. An orbit is a sequence of v alues of a map, represen ted by x n = [ x 0 , x 1 , x 2 , ..., x n ] . A pseudo-orbit is an approximation of an orbit and is expressed as ˆ x i,n = [ ˆ x i, 0 , ˆ x i, 1 , ˆ x i, 2 , ..., ˆ x i,n ] . The Low er Bound Error (LBE) is a metho d presen ted by Nep omuceno and Martins in [7], which aims to ev aluate the error propagation due to round off in digital computers. The pro cedure to calculate the LBE is based on the comparison of tw o pseudo-orbits pro duced from tw o mathematical equiv alent mo dels, but differen t from the p oint of view of floating point represen tation. Therefore, LBE’s mathematical representation in given b y 2 δ α,n = | ˆ x a,n − ˆ x b,n | , where δ α,n represen ts the lo w er b ound error betw een tw o pseudo- orbits ˆ x a,n e ˆ x b,n . 3 Metho dology In order to verify the influence of synchronization on the rec koning of Lo w er Bound Error, tw o case studies were considered. In the first one, a Lorenz system (slav e) was unidirectionally coupled to a Duffing system (master). In the second case, the same pro cedure was follow ed for a Duffing system (slav e) and a Rossler system (master). 3.1 Duffing - Lorenz The Duffing and Lorenz systems are giv en b y Equations (1) and (2), resp ectiv ely: ˙ x 1 = x 2 ˙ x 2 = x 1 − x 3 1 − δ x 2 + γ cos ( ω t ) , (1) ˙ y 1 = − 10 y 1 + 10 y 2 ˙ y 2 = 28 y 1 − y 2 − y 1 y 3 + K x 1 (2) ˙ y 3 = y 1 y 2 − 8 3 y 3 . T o couple the systems, w e added the term K x 1 on the equation ˙ y 2 of the Lorenz sys- tem, where K is called coupling constan t and determines how strong the sync hronization b et ween the oscillators is [11]. The state v ariable x 1 w as chosen arbitrarily and could b e x 2 as w ell. The initial conditions used for the Duffing system w ere x 1 (0) = 3 and x 2 (0) = 4 and for the Lorenz system, y 1 (0) = y 2 (0) = y 3 (0) = 1. During the exp eriments, we increased the v alue of K from zero, when the systems are completely unsynchronized, up to a v alue where general synchronization is observ ed. In order to v erify if the systems are in fact sync hronized, we considered the metho d presen ted in [11]. The auxiliary equation used is given by ˙ y 0 2 = 28 y 0 1 − y 0 2 − y 0 1 y 0 3 + K x 1 . The initial conditions used to compute the auxiliary equation w ere y 1 (0) = y 2 (0) = y 3 (0) = 5. As Pyragas defines in [11], the tw o systems can be considered to be strongly synchronized in a general manner if complete sync hronization exists b et ween ˙ y 2 and ˙ y 0 2 . 4 Finally , LBE was calculated for tw o pseudo-orbits of the resp onse system for different v alues of K. The pseudo-orbits were obtained by a simple mathematical manipulation on ˙ y 1 . The equation obtained after the manipulation is given b y ˙ y 1 = 10( y 2 − y 1 ). Thereb y , w e calculated the LBE using the Equations giv en by ˙ y 2 of the t w o pseudo-orbits. 3.2 Rossler - Duffing W e follow ed the same pro cedures describ ed previously to study the b eha vior of a Duffing system being driven b y a Rossler system. The equations of Rossler circuit are giv en b y ˙ x 1 = − x 2 + x 3 , ˙ x 2 = x 1 + 0 . 2 x 2 and ˙ x 3 = 0 . 2 + x + 3( x 1 − 5 . 7). The Duffing system is the same used in the previous case of study (Equation (1)), with the addition of the coupling term to ˙ x 1 , thus, it b ecame ˙ x 1 = x 2 + K y 1 . W e used the same initial conditions as in Section 3.1. The auxiliary equation is ˙ x 0 1 = x 0 2 + K y 1 and, for this equation, the initial conditions are x 1 (0) = 5 and x 2 (0) = 6. The initial conditions used for the Rossler system are x 1 (0) = x 2 (0) = x 3 (0) = 1. T o calculate LBE, we p erformed a mathematical manipulation on Equation (1), th us it b ecame ˙ x 2 = x 1 − x 1 x 1 x 1 − δ x 2 + γ cos ( ω t ). Finally , we calculated LBE b et w een the Equations given b y ˙ x 1 of the t w o pseudo-orbits. 4 Results 4.1 Duffing - Lorenz As we increased the v alue of K, we applied Pyragas’ metho d to verify the synchroniza- tion b et w een the oscillators. W e observed that for v alues greater than 30, we could see a straigh t line on the phase p ortrait of y and y 0 . Therefore, we c hose K = 40 to represent in this pap er. Figure 1(a) sho ws the dynamics of the systems when they are unsync hronized, that is, K = 0. In figure 1(b), we represent the dynamics of the systems when there is general sync hronization b et ween them for k = 40. T o v alidate the results, Figure 2(a) shows the phase portrait for K = 0, where y and y 0 are clearly unsync hronized. Figure 2(b) represen t the phase p ortrait of the systems for K = 40, where a straigh t line is visible, whic h indicates complete synchron yzation b et ween y and y 0 , and, consequen tly , general sync hronization b et ween the Duffing and Lorenz systems. The results of the computation of LBE b et w een Equations ˙ x 2 of the tw o pseudo-orbits for different v alues of K ranging from 0 to 30 are shown in Figure 3. As one can see, the LBE’s curv es tend to tak e more iterations to increase to higher v alues as K increase. F or example, for K = 0, it takes 1266 iterations for the LBE to go to -0.3 and for K = 25, it takes 5100 iterations for the LBE to go to the same v alue. Apparently , the resp onse’s LBE curve is follo wing the b ehavior of the master’s. Ho w ev er, for K = 30, Duffing’s LBE curv e will take ab out 5000 iterations to start increasing, as can b e seen in Figure 5 for K = 0 and, in Figure 3, for K = 30, Lorenz’s takes ab out 15000 iterations to start. This observ ation could mean that sync hronization is also dela ying the error propagation. It is also imp ortan t to mention that, for v alues greater than 30, like 40, w e were not able to 5 (a) Duffing and Lorenz dynamics for K = 0. (b) Duffing and Lorenz dynamics for K = 40. Figure 1: Dynamics of the systems when they are not sync hronized (K=0) and when generalized sync hronization exists (K=40). (a) Phase p ortrait for K = 0. (b) Phase p ortrait for K = 40. Figure 2: Synchronization b etw een the systems using the auxiliary equation y’. represen t LBE curv e. Since we are using logarithmic notation to represen t the results, this fact may imply that the Low er Bound Error go es to zero when the systems are strongly coupled. Figure 3: Evolution of LBE according to v alue of K for Lorenz system. 4.2 Rossler - Duffing In order to analyze the behavior of Duffing system b een driven by a Rossler, w e follow ed the same pro cedure adopted in Section 4.1. In this case, we observed the o ccurrence of sync hronization for v alues of K greater than 300. W e chose 400 to represent. Figures 4(a) 6 and 4(b) sho w the dynamics of the systems and the phase p ortrait for k = 400. (a) Rossler and Duffing dynamics for K = 400 (b) Phase p ortrait for K = 400. Figure 4: Dynamics of the systems and phase p ortrait b et w een y and y 0 . The results of the computation of LBE b et ween Equations ˙ x 2 of the tw o pseudo- orbits for different v alues of k ranging from 0 to 300 are shown in Figure 5. In this case, LBE’s curv e for k = 0 takes 9698 iterations to go -0.3, while for k = 100 and k = 200, it tak es ab out 3500 and 6500 iterations, resp ectiv ely . Therefore, when the systems are w eakly coupled, Duffing’s curve will follow Rossler’s. How ever, as K go as high as 300, for example, the curve will tak e ab out 14300 to get to -0.3. W e w ere not able to represent LBE curv e for v alues greater than 300. The same observ ations from Section 4.1 can b e p oin ted in this case. Figure 5: Evolution of LBE according to v alue of K for Duffing system. 5 Conclusions The results presen ted in Sec tion 4 show that when a chaotic oscillator is synchronized to another, the LBE of t w o pseudo-orbits of the resp onse system tend to follow the b eha vior of the driver un til a certain v alue of K is reac hed and, consequently , we demonstrated that synchronization can affect n umerical calculations. W e could observ e that, in the t w o case studies in vestigated, after that v alue of K, the LBE index take more iterations to raise, what could mean that the LBE b etw een the pseudo-orbits of the resp onse system w as decreasing. Therefore, sync hronization can b e further in v estigated to b e used as a 7 to ol to reduce the error on numerical simulations. Also, in future w orks, other types of sync hronization can b e inv estigated, as complete, phase and lag synchronization. 6 Ac knowledgemen t The authors are thankful to the Brazilian agencies CNPq, F APEMIG, and to UFSJ. References [1] S. Boccaletti, J. Kurths, G. Osip o v, D. L. V alladares and C. S. Zhou, The syncroniza- tion of c haotic systems, Ph ysics Rep orts, vol. 366(1), 1-101 , Elsevier, (2002). [2] S. M. Hammel, J. A. Y orke and C. Greb ogi, Do n umerical orbits of c haotic dynamical pro cesses represent true orbits?, Journal of Complexity , vol. 3(2), 136-145, (1987). [3] E. N. Lorenz, Deterministic nonperio dic flo w, Journal of the atmospheric sciences, v ol. 20(2), 130–141, (1963). [4] R. Lozi, Can w e trust in numerical computations of chaotic solutions of dynamical systems?, W orld Scien tific Series in Nonlinear Science Series A, v ol. 84, (2013). [5] B. Naderi and H. 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