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Spatially periodic and temporally chaotic pattern in coupled nonidentical chaotic systems
Chaos, Solitons and Fractals 24 (2005) 767–774 www.elsevier.com/locate/chaos
Spatially periodic and temporally chaotic pattern in coupled nonidentical chaotic systems Meng Zhan a, Xingang Wang a, Xiaofeng Gong a, C.-H. Lai a b
b,*
Temasek Laboratories, National University of Singapore, 117508 Singapore Department of Physics, National University of Singapore, 117542 Singapore Accepted 14 September 2004
Abstract A particular spatio-temporal pattern, the spatially periodic and temporally chaotic pattern (SPTCP), can be observed in coupled one-way ring and linear array systems. This is a state chaotic in time while periodic in space in a strict sense. In this work, a driven system, of coupled nonidentical chaotic elements, which supports this structure is studied. We find that the appearance of the pattern is closely connected with the cascade of generalized synchronization in the ring. In particular, the establishment of the spatial periodicity of the SPTCP is determined by the condition that all of the coupled sites in the ring stay in the generalized synchronous state. Ó 2004 Elsevier Ltd. All rights reserved.
For several decades pattern formation [1,2] has become a hot research topic in the studies of several disciplines of science, including physics, chemistry, biology, hydrodynamics, materials science, and pattern recognition. Usually the formation of a specific pattern, or the establishment of coherent behavior in a spatio-temporal system, determines the systemÕs global features. The studies of pattern formation are mainly concerned with the conditions (the system structure, the proper system parameters, the external environment, etc.) under which a particular pattern can be realized, the relationship between the patterns, and the possibility for realistic applications. It is also of significance to exploit the emergence of interesting patterns to study the underlying dynamics. An interesting pattern, the spatially periodic and temporally chaotic pattern (SPTCP), i.e., a state chaotic in time while periodic in space in a strict sense, was observed by Xiao et al. for the first time in [3]. The coupled systems, schematically shown in Fig. 1, are built out of three components of spatio-temporal systems: the drive component X and two response components Y and Z. Each of these has a one-way coupled structure and Z is an identical auxiliary system of Y; the system size of X is L, while that of Y, N, can be infinite. In this context, it is worth noting that the topology of either the ring [4–7], or a linear array [8], has been the main coupling model in the studies of chaos synchronization [9– 11]. In Ref. [3], the authors extended the investigation of secure communication by focusing on the problem of single signal transmission using low-dimensional chaotic systems as transmitters and receivers. They successfully realized the synchronization of two spatio-temporal chaotic systems (Y and Z)—the synchronizations of all of the corresponding sites—using a single driving key (i.e. only one site information from the drive component X). Hence, the multichannel
*
Corressponding author. Tel.: +65 6874 8751; fax: +65 6872 1763. E-mail address: [email protected] (C.-H. Lai).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.105
768
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
Fig. 1. Schematic diagram of the coupled systems constructed by the coupled ring and linear arrays systems, each of which is one-way coupled and driven.
spread-spectrum communication method based on two spatio-temporal systems is effective (one key and many channels); meanwhile, due to the hyperchaoticity with very high dimensions of both key and transmission signals, it is safe against attacks. Although in [3] the authors were primarily interested in the synchronization of Y and Z, through this constructed system, they also discovered the synchronization of corresponding sites in X and Y with a very wide parameter region, and the resulting SPTCP in Y with the size of X, L, as the spatial period [12,13]. It is apparent that the appearance of the SPTCP in Y is only the sufficient, but not the necessary, condition for the synchronization of Y and Z. In Ref. [3], however, a very important question of the stability of the synchronization of Y and X, that is, the condition under which the SPTCP state in Y is stable (or unstable), is unclear. Very recently, the phenomenon of the SPTCP with the same model was rediscovered in Ref. [14]. From the chaos synchronization viewpoint [9], it is a type of nonlocal synchronization, that is, the chaos of contiguous units of the linear array do not synchronize, but that of noncontiguous ones synchronize with a spatial periodicity. Again the stability question has not been well studied. In addition to the natural connection with application in chaos-based communication and image storage [3], its potential applications in neural processes and information processing systems were also emphasized in Ref. [14]. In the present work, the SPTCP in coupled nonidentical systems with this network, in which the coupled lattices have parameter mismatches, is extensively studied, and the condition of the stability of the SPTCP (especially, the relationship with the generalized synchronization), is focused on. The motivation of this investigation is very direct. First of all, the model proposed in Refs. [3,14] has very strong potential for applications, as indicated above. Next, the exact parameter matches of the local dynamic units are usually unavailable under realistic conditions, and no doubt the model with nonidentical parameters may be more general and common for practical situations. Most importantly, in the recent secure communication scheme based on the synchronization of spatio-temporal chaotic systems [15], the parameters are intentionally chosen as additional keys to increase security; from this application point of view, the model with parameter mismatches is worth studying. Our model is made up of the following: the drive component X, which can be expressed as X_ i X0
¼ ¼
f ðX i ; P i Þ þ eðX i1 X i Þ; X L ; i ¼ 1; 2; . . . ; L;
ð1Þ
and the response component Y, which is described by Y_ j Y0
¼ ¼
f ðY j ; Qj Þ þ eðY j1 Y j Þ; X k ; j ¼ 1; 2; . . . ; N;
ð2Þ
where X_ ¼ f ðX Þ denotes the local nonlinear dynamics, e is the coupling strength, and Pi and Qj are the parameter sets for systems X and Y, respectively. Both of these systems are connected by one-way couplings (using one element to drive the next one) with the only difference that X is autonomous with periodic condition, while Y is driven by one unit of X, Xk. k can be arbitrarily chosen from 1 to L. In Fig. 1, we only choose one particular example: k = L. Because of the inhomogeneity of the parameters in X, Pi, the situation would change when a different driving signal, Xk, is used. An important question is: how can we observe a stable SPTCP in Y? Equivalently, can we observe the synchronization of the corresponding nodes in X and Y, Y j ðtÞ ¼ X ½ðjþkÞ=L ðtÞ;
j ¼ 1; 2; . . . ; N;
ð3Þ
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
769
where [. . .] denotes the modulo function? Only when all of nodes synchronize can the complex pattern in X be transferred to Y, and beyond, thus the periodicity of the SPTCP sustains. This is determined by two separate conditions: one is the parameter matches of the corresponding nodes in X and Y, i.e., Qj ¼ P ½ðjþkÞ=L ;
j ¼ 1; 2; . . . ; N
ð4Þ
although each parameters Pi in X is still not equal to each other; another is the stability of chaos synchronization for all of the corresponding nodes. To establish the synchronization between the ring and linear arrays, the first step is to synchronize the first node, e.g. 1 0 in X and 1 in Y in Fig. 1 (k = L). The dynamics of these two sites (X1 and Y1) are determined by X_ 1 Y_ 1
¼ ¼
f ðX 1 ; P 1 Þ þ eðX L X 1 Þ; f ðY 1 ; P 1 Þ þ eðX L Y 1 Þ:
ð5Þ
Both of them have the same local dynamic expression and are driven by XL. The stability of synchronization of X1 and Y1 is controlled by the linearized equation, df ðX 1 ; P 1 Þ d_ 1 ¼ ð6Þ e d1 ; d1 ¼ Y 1 X 1 : dðX 1 Þ Based on Eq. 6 and the variable X1, the maximum conditional Lyapunov exponent (MCLE) k1 can be computed. Thus the necessary condition for the synchronization between X1 and Y1 is the negativeness of k1. This analysis is similar to the traditional auxiliary method of generalized synchronization [16–18] defined in drive–response systems (using XL to drive X1), except that now the drive system (XL in Eq. 5) is nonautonomous due to feedback of the ring. To maintain the synchronization of the corresponding elements, the next step is the synchronization of X2 and Y2, then X3 and Y3, and so on, whose stabilities are determined by the corresponding MCLEÕs, from k2 to kL, respectively. Thus the stability of the particular pattern, the SPTCP, is controlled by the maximum conditional Lyapunov exponent spectrum, {k1,k2, . . . ,kL}, schematically shown in Fig. 2(b). It can be observed when all of ki < 0;
i ¼ 1; 2; . . . ; L
ð7Þ
under certain driving strength e. In fact, very recently, the concept of generalized synchronization has been extended from the usual drive–response systems (only two systems with one-way driving) [16–18] to an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators [19]. The basic idea is to choose another auxiliary system having the same environment as the corresponding one in the coupled systems, schematically shown in Fig. 2(a), and to study its stability. By increasing the coupling strengths, the system behavior undergoes a cascade of transitions from partial to global generalized synchronization, i.e., different elements are gradually entrained through a clustering process. The similarity of the definition of the MCLE spectra in the mutually coupled systems [Fig. 2(a)] and in the one-way coupled ring systems [Fig. 2(b)] is very clear. Using the definition in [19], ki in our analysis above just corresponds to the MCLE of the generalized synchronous manifold. So, the pattern in the response component Y is controlled by the distribution of critical coupling strengths for generalized synchronization in the array X, while the formation of the SPTCP exactly corresponds to the global generalized synchronous state. Note that the analysis of generalized synchronization in one-way circular loop structure directly relates to the above pattern analysis and has a simple physical picture. Clearly if we use another node (k5L) in X to
(a)
(b)
Fig. 2. The schematic approach of the auxiliary method to study the generalized synchronization in mutually coupled chaotic systems (a) and one-way coupled systems (b).
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M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
drive Y, the condition for the stability of the SPTCP is the same (the condition in Eq. 7 does not change) with the only difference of the sequences of synchronization with X[(k+1)/L], X[(k+2)/L],. . .. To check the above analysis, we adopt the x-coupled Lorenz oscillators as our example (the coupling is added on the equations of the components x). The equations of motion of a single Lorenz oscillator are written as dx dt dy dt dz dt
¼
rðy xÞ
¼
ci x y xz
¼
xy bz;
ð8Þ
with the classical parameter values: r = 10.0 and b = 8/3, and ci being variable for different oscillators. Let us start with considering a small system size L = 3. It is well known that c relates to the Rayleigh number in studies of turbulence, and the chaoticity strength of a single Lorenz oscillator is approximately determined by the value of c. Without losing generality, c1 = 50, c2 = 55, and c3 = 60 are chosen. Three tests (I, II, III) using different driving signals (x03 , x01 , and x02 , respectively), shown in Fig. 3, are performed. In Fig. 4, the MCLE spectrum {k1, k2, k3} as the functions of the driving strength e are exhibited, and the sequence of generalized synchronizations is indicated by the transition from positive to negative of ki,i = 1, . . . , L, at the critical coupling strengths ec,1 12.0, ec,2 15.5, and ec,3 21.1 emphasized by three arrows in Fig. 4. To measure the degree of synchronization of the corresponding nodes between X and Y, the average difference, Z 1 T Dj ðtÞdt; ð9Þ Dj ¼ lim T !1 T 0 is defined with the instantaneous distance as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dj ðtÞ ¼ kY j ðtÞ X i ðtÞk ¼ ðxj x0i Þ2 þ ðy j y 0i Þ2 þ ðzj z0i Þ2 ;
i ¼ ½ðj þ kÞ=L:
ð10Þ
The dependence of Dj (j = 1,2,3) on the coupling strength e in the test I, II, and III, are plotted in Fig. 5 (a), (b) and (c), respectively. We choose a very long time interval, T = 10,000, in the numerical calculations. In all of the three tests, it is only after the former oscillator transits to the synchronous state that the other oscillators, whose spatial positions are behind, are able to synchronize. For example, in Fig. 5(a), it is oscillator 1 that synchronizes first, followed by oscillator 2, and finally oscillator 3. In Fig. 5(b), it is oscillator 1 first, then oscillators 2 and 3 simultaneously. Finally in Fig. 5(c), all three oscillators synchronize simultaneously. Moreover, an excellent agreement with Fig. 4, where the complete synchronization thresholds coincide with the theoretical estimation of the critical coupling strengths of generalized synchronization, is found. Although the response systems show distinct sequence of synchronization when different driving signals (different k) are used in Fig. 5, they transit to global synchronization at the same coupling strength, ec,3, the largest one among ec,i. To confirm the observation of a stable SPTCP, the patterns of the variables xj(t) under ea = 14.0, eb = 19.0, and ec = 22.0 for test I, II, and III are plotted in Fig. 6(a–c), (d–f), and (g–i), respectively. Here we have used N = 30 = 10L. Fig. 6(c), (f), and (i), ec>(ec,i)max = ec,3 21.1, apparently show the SPTCP with spatial period L = 3. Before that, no SPTCP can be found.
Fig. 3. Schematic plot of three tests (I, II, and III) using different sites (k = 3, 1, and 2, respectively) in the ring as driving signals of the linear array systems. L = 3.
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
771
Fig. 4. The dependences of k1 (square), k2 (circle), and k3 (up-triangle) on e. At different coupling threshold (ec,1, ec,2, or ec,3), the individual oscillator (No. 1, 2, or 3 one in the ring) transits to generalized synchronous state progressively.
Fig. 5. The average differences, D1 (square), D2 (circle), and D3 (up-triangle) vs e for test I (a), II (b), and III (c).
In the above, the analysis involved a drive component of small size. It is interesting to explore the possibility for more general SPTCP states with a larger system size. In Fig. 7 (with L = 50, and ciÕs are the random numbers with uniform distribution from 28.0 to 100.0), the generalized synchronization thresholds for all specific oscillators, ec,i, i = 1, 2, . . . ,L, are obtained. Only above the ec,i, can we find the generalized synchronization for the individual oscillator, i. The disorder of the distribution of ec,i with large fluctuations certainly shows strong competition between the coupled oscillators due to the inhomogeneity of the parameter sets. In Fig. 8, N = 150 = 3L, k = L, the spatio-temporal patterns are plotted with the variables of xj(t) and the variable differences, d j ðtÞ ¼ xj ðtÞ x0j ðtÞ, for ea = 22.0 [(a) and (b)] and eb = 33.0 [(c) and (d)]. The values of the chosen coupling strength are exemplified in Fig. 7 with dash and dot lines, respectively. ea = 22.0 is an approximate average value, and eb = 33.0>ec,49 32.26 (ec,49 is the largest among the ec,i). Comparing with Fig. 8(a), Fig. 8(c) shows clear characteristics of the SPTCP, namely, periodic in space with a period L = 50 and chaotic in time. At the same time the static pattern with dj(t) = 0 for all driven oscillators [shown in Fig. 8(d)] indicates a nice establishment of synchronization between systems Y and X. Furthermore, a partial synchronization phenomenon with the first 8 oscillators (denoted by the arrow) is found in Fig. 8(b) for ea = 22.0 before the global one in Fig. 8(d). The result coincides with the analysis of the MCLE spectrum distribution in Fig. 7 and the particular spatial locations for ec,i, i.e., ea>ec,i, i = 1,2, . . . , 8, while ea
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M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
Fig. 6. The patterns of variable xj(t) under different coupling strengths ea = 14.0, eb = 19.0, and ec = 22.0 (ec>(ec,i)max = ec,3 21.1), for test I [(a)–(c)], II [(d)–(f)], and III [(g)–(i)]. L = 3, and N = 30.
Fig. 7. The distribution of the critical coupling strengths, ec,i, i = 1,2, . . ., L, for generalized synchronization of specific oscillator. L = 50.
once the node ahead synchronizes with the corresponding one in the ring, a new stable driving source at this new spatial position will form. As a result, a rough linear function of synchronization time versus the spatial sites (at least in the scale of spatial period of L) can be expected. This point is confirmed by the shape of linear relationship with little fluctuation in Fig. 9, under the test condition of the time average of the instantaneous distance, Dj ðtÞ < 108 , in a small time interval of 10. Comparing to e = 35.0, a larger driving strength e = 40.0 leads to a faster synchronization behavior. Thus, the synchronization can propagate reasonably far provided the conditions of parameter matching (Eq. 4) and stability (Eq. 7) are satisfied. In [8], a similar phenomenon of synchronization waves in arrays of driven chaotic systems has been studied. In conclusion, we have investigated the SPTCP in detail. This pattern can occur in the coupled ring and linear array systems, regardless of the array size. The mechanism underlying this seemingly strange pattern has been made clear: under the condition of stability of the global generalized synchronization in the ring systems, the response systems
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
773
Fig. 8. The patterns of variables xj(t) and the variable differences dj(t) under different coupling strengths ea [(a) and (b)], and eb [(c) and (d)]. ea = 22.0>ec,i, i = 1,2, . . . , 8, while ea(ec,i)max = ec,49 32.26. L = 50, and N = 150. Partial synchronization (before j = 9) and global synchronization are observed in (b) and (d), respectively.
Fig. 9. The synchronization time sj vs the site number j for e = 35.0 (square points) and e = 40.0 (circle points) with approximately linear relationship.
can copy the pattern of the driving system (which maybe a spatio-temporal chaotic state), transfer it in its one-way structure, and establish the SPTCP with multiple copies of the ring in space. In Refs. [20–22], authors studied the chaotic itinerancy phenomenon between attractor ruins. Here our analysis mainly concerns the formation condition of a specific pattern, SPTCP.
References [1] Cross M, Hohenberg P. Pattern formation outside of equilibrium. Rev Mod Phys 1993;65:851. [2] Kuramoto Y. Chemical oscillations, waves, and turbulence. New York: Springer-Verlag; 1984. [3] Xiao JH, Hu G, Qu ZL. Synchronization of spatiotemporal chaos and its application to multichannel spread-spectrum communication. Phys Rev Lett 1996;77:4162;
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M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774 Hu G, Xiao JH, Yang JZ, Xie FG, Qu ZL. Synchronization of spatiotemporal chaos and its applications. Phys Rev E 1997;56:2738. Heagy JF, Carroll TL, Pecora LM. Synchronous chaos in coupled oscillator systems. Phys Rev E 1994;50:1874; Pecora LM, Carroll TL. Master stability functions for synchronized coupled systems. Phys Rev Lett 1998;80:2109. Matias MA, Munuzuri VP, Lorenzo MN, Marino IP, Villar VP. Observation of a fast rotating wave in rings of coupled chaotic oscillators. Phys Rev Lett 1997;78:219; Matias MA, Guemez J. Transient periodic rotating waves and fast propagation of synchronization in linear arrays of chaotic systems. ibid 1998;81:4124. Hu G, Yang JZ, Ma WQ, Xiao JH. Hopf bifurcation from chaos and generalized winding numbers of critical modes. Phys Rev Lett 1998;81:5314. Liu ZH, Lai YC, Hoppensteadt FC. Phase clustering and transition to phase synchronization in a large number of coupled nonlinear oscillators. Phys Rev E 2001;63:055201. Lorenzo MN, Marino IP, Munuzuri VP, Matias MA, Villar VP. Synchronization waves in arrays of driven chaotic systems. Phys Rev E 1996;54:R3094. Pikovsky AS, Rosenblum MG, Kurths J. Synchronization-A unified approach to nonlinear science. Cambridge, England: Cambridge University Press; 2001; Pecora LM, Carroll TL, Johnson GA, Mar DJ, Heagy JF. Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 1997;7:520. Tao C, Du G. Determinate relation between two generally synchronized spatiotemporal chaotic systems. Phys Lett A 2003;311:158. Afraimovich V, Chazottes JR, Cordonet A. Nonsmooth functions in generalized synchronization of chaos. Phys Lett A 2001;283:109. Zhou C, Lai C-H. Synchronization with positive conditional Lyapunov exponents. Phys Rev E 1998;58:5188. Shuai JW, Wong KW, Cheng LM. Synchronization of spatiotemporal chaos with positive conditional Lyapunov exponents. Phys Rev E 1997;56:2272. Deng XL, Huang HB. Spatial periodic synchronization of chaos in coupled ring and linear arrays of chaotic systems. Phys Rev E 2002;65:055202. Wang SH, Kuang JY, Li JH, Luo YL, Lu HP, Hu G. Chaos-based secure communications in a large community. Phys Rev E 2002;66:065202. Rulkov NF, Sushchik MM, Tsimring LS, Abarbanel HDI. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 1995;51:980; Abarbanel HDI, Rulkov NF, Sushchik MM. Generalized synchronization of chaos: the auxiliary system approach. Phys Rev E 1996;53:4528. Kocarev L, Parlitz U. Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys Rev Lett 1996;76:1816; Pyragas K. Weak and strong synchronization of chaos. Phys Rev E 1996;54:R4508. Zhan M, Wang XG, Gong XF, Wei GW, Lai C-H. Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Phys Rev E 2003;68:036208. Zheng ZG, Wang XG, Cross MC. Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators. Phys Rev E 2002;65:056211. Kaneko K. Dominance of Milnor attractors and noise-induced selection in a multiattractor system. Phys Rev Lett 1997;78:2736. Otsuka K. Self-induced phase turbulence and chaotic itinerancy in coupled laser systems. Phys Rev Lett 1990;65:329. Cencini M, Falcioni M, Vergni D, Vulpiani A. Macroscopic chaos in globally coupled maps. Physica D 1999;130:58.
Spatially periodic and temporally chaotic pattern in coupled nonidentical chaotic systems Meng Zhan a, Xingang Wang a, Xiaofeng Gong a, C.-H. Lai a b
b,*
Temasek Laboratories, National University of Singapore, 117508 Singapore Department of Physics, National University of Singapore, 117542 Singapore Accepted 14 September 2004
Abstract A particular spatio-temporal pattern, the spatially periodic and temporally chaotic pattern (SPTCP), can be observed in coupled one-way ring and linear array systems. This is a state chaotic in time while periodic in space in a strict sense. In this work, a driven system, of coupled nonidentical chaotic elements, which supports this structure is studied. We find that the appearance of the pattern is closely connected with the cascade of generalized synchronization in the ring. In particular, the establishment of the spatial periodicity of the SPTCP is determined by the condition that all of the coupled sites in the ring stay in the generalized synchronous state. Ó 2004 Elsevier Ltd. All rights reserved.
For several decades pattern formation [1,2] has become a hot research topic in the studies of several disciplines of science, including physics, chemistry, biology, hydrodynamics, materials science, and pattern recognition. Usually the formation of a specific pattern, or the establishment of coherent behavior in a spatio-temporal system, determines the systemÕs global features. The studies of pattern formation are mainly concerned with the conditions (the system structure, the proper system parameters, the external environment, etc.) under which a particular pattern can be realized, the relationship between the patterns, and the possibility for realistic applications. It is also of significance to exploit the emergence of interesting patterns to study the underlying dynamics. An interesting pattern, the spatially periodic and temporally chaotic pattern (SPTCP), i.e., a state chaotic in time while periodic in space in a strict sense, was observed by Xiao et al. for the first time in [3]. The coupled systems, schematically shown in Fig. 1, are built out of three components of spatio-temporal systems: the drive component X and two response components Y and Z. Each of these has a one-way coupled structure and Z is an identical auxiliary system of Y; the system size of X is L, while that of Y, N, can be infinite. In this context, it is worth noting that the topology of either the ring [4–7], or a linear array [8], has been the main coupling model in the studies of chaos synchronization [9– 11]. In Ref. [3], the authors extended the investigation of secure communication by focusing on the problem of single signal transmission using low-dimensional chaotic systems as transmitters and receivers. They successfully realized the synchronization of two spatio-temporal chaotic systems (Y and Z)—the synchronizations of all of the corresponding sites—using a single driving key (i.e. only one site information from the drive component X). Hence, the multichannel
*
Corressponding author. Tel.: +65 6874 8751; fax: +65 6872 1763. E-mail address: [email protected] (C.-H. Lai).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.09.105
768
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
Fig. 1. Schematic diagram of the coupled systems constructed by the coupled ring and linear arrays systems, each of which is one-way coupled and driven.
spread-spectrum communication method based on two spatio-temporal systems is effective (one key and many channels); meanwhile, due to the hyperchaoticity with very high dimensions of both key and transmission signals, it is safe against attacks. Although in [3] the authors were primarily interested in the synchronization of Y and Z, through this constructed system, they also discovered the synchronization of corresponding sites in X and Y with a very wide parameter region, and the resulting SPTCP in Y with the size of X, L, as the spatial period [12,13]. It is apparent that the appearance of the SPTCP in Y is only the sufficient, but not the necessary, condition for the synchronization of Y and Z. In Ref. [3], however, a very important question of the stability of the synchronization of Y and X, that is, the condition under which the SPTCP state in Y is stable (or unstable), is unclear. Very recently, the phenomenon of the SPTCP with the same model was rediscovered in Ref. [14]. From the chaos synchronization viewpoint [9], it is a type of nonlocal synchronization, that is, the chaos of contiguous units of the linear array do not synchronize, but that of noncontiguous ones synchronize with a spatial periodicity. Again the stability question has not been well studied. In addition to the natural connection with application in chaos-based communication and image storage [3], its potential applications in neural processes and information processing systems were also emphasized in Ref. [14]. In the present work, the SPTCP in coupled nonidentical systems with this network, in which the coupled lattices have parameter mismatches, is extensively studied, and the condition of the stability of the SPTCP (especially, the relationship with the generalized synchronization), is focused on. The motivation of this investigation is very direct. First of all, the model proposed in Refs. [3,14] has very strong potential for applications, as indicated above. Next, the exact parameter matches of the local dynamic units are usually unavailable under realistic conditions, and no doubt the model with nonidentical parameters may be more general and common for practical situations. Most importantly, in the recent secure communication scheme based on the synchronization of spatio-temporal chaotic systems [15], the parameters are intentionally chosen as additional keys to increase security; from this application point of view, the model with parameter mismatches is worth studying. Our model is made up of the following: the drive component X, which can be expressed as X_ i X0
¼ ¼
f ðX i ; P i Þ þ eðX i1 X i Þ; X L ; i ¼ 1; 2; . . . ; L;
ð1Þ
and the response component Y, which is described by Y_ j Y0
¼ ¼
f ðY j ; Qj Þ þ eðY j1 Y j Þ; X k ; j ¼ 1; 2; . . . ; N;
ð2Þ
where X_ ¼ f ðX Þ denotes the local nonlinear dynamics, e is the coupling strength, and Pi and Qj are the parameter sets for systems X and Y, respectively. Both of these systems are connected by one-way couplings (using one element to drive the next one) with the only difference that X is autonomous with periodic condition, while Y is driven by one unit of X, Xk. k can be arbitrarily chosen from 1 to L. In Fig. 1, we only choose one particular example: k = L. Because of the inhomogeneity of the parameters in X, Pi, the situation would change when a different driving signal, Xk, is used. An important question is: how can we observe a stable SPTCP in Y? Equivalently, can we observe the synchronization of the corresponding nodes in X and Y, Y j ðtÞ ¼ X ½ðjþkÞ=L ðtÞ;
j ¼ 1; 2; . . . ; N;
ð3Þ
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
769
where [. . .] denotes the modulo function? Only when all of nodes synchronize can the complex pattern in X be transferred to Y, and beyond, thus the periodicity of the SPTCP sustains. This is determined by two separate conditions: one is the parameter matches of the corresponding nodes in X and Y, i.e., Qj ¼ P ½ðjþkÞ=L ;
j ¼ 1; 2; . . . ; N
ð4Þ
although each parameters Pi in X is still not equal to each other; another is the stability of chaos synchronization for all of the corresponding nodes. To establish the synchronization between the ring and linear arrays, the first step is to synchronize the first node, e.g. 1 0 in X and 1 in Y in Fig. 1 (k = L). The dynamics of these two sites (X1 and Y1) are determined by X_ 1 Y_ 1
¼ ¼
f ðX 1 ; P 1 Þ þ eðX L X 1 Þ; f ðY 1 ; P 1 Þ þ eðX L Y 1 Þ:
ð5Þ
Both of them have the same local dynamic expression and are driven by XL. The stability of synchronization of X1 and Y1 is controlled by the linearized equation, df ðX 1 ; P 1 Þ d_ 1 ¼ ð6Þ e d1 ; d1 ¼ Y 1 X 1 : dðX 1 Þ Based on Eq. 6 and the variable X1, the maximum conditional Lyapunov exponent (MCLE) k1 can be computed. Thus the necessary condition for the synchronization between X1 and Y1 is the negativeness of k1. This analysis is similar to the traditional auxiliary method of generalized synchronization [16–18] defined in drive–response systems (using XL to drive X1), except that now the drive system (XL in Eq. 5) is nonautonomous due to feedback of the ring. To maintain the synchronization of the corresponding elements, the next step is the synchronization of X2 and Y2, then X3 and Y3, and so on, whose stabilities are determined by the corresponding MCLEÕs, from k2 to kL, respectively. Thus the stability of the particular pattern, the SPTCP, is controlled by the maximum conditional Lyapunov exponent spectrum, {k1,k2, . . . ,kL}, schematically shown in Fig. 2(b). It can be observed when all of ki < 0;
i ¼ 1; 2; . . . ; L
ð7Þ
under certain driving strength e. In fact, very recently, the concept of generalized synchronization has been extended from the usual drive–response systems (only two systems with one-way driving) [16–18] to an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators [19]. The basic idea is to choose another auxiliary system having the same environment as the corresponding one in the coupled systems, schematically shown in Fig. 2(a), and to study its stability. By increasing the coupling strengths, the system behavior undergoes a cascade of transitions from partial to global generalized synchronization, i.e., different elements are gradually entrained through a clustering process. The similarity of the definition of the MCLE spectra in the mutually coupled systems [Fig. 2(a)] and in the one-way coupled ring systems [Fig. 2(b)] is very clear. Using the definition in [19], ki in our analysis above just corresponds to the MCLE of the generalized synchronous manifold. So, the pattern in the response component Y is controlled by the distribution of critical coupling strengths for generalized synchronization in the array X, while the formation of the SPTCP exactly corresponds to the global generalized synchronous state. Note that the analysis of generalized synchronization in one-way circular loop structure directly relates to the above pattern analysis and has a simple physical picture. Clearly if we use another node (k5L) in X to
(a)
(b)
Fig. 2. The schematic approach of the auxiliary method to study the generalized synchronization in mutually coupled chaotic systems (a) and one-way coupled systems (b).
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drive Y, the condition for the stability of the SPTCP is the same (the condition in Eq. 7 does not change) with the only difference of the sequences of synchronization with X[(k+1)/L], X[(k+2)/L],. . .. To check the above analysis, we adopt the x-coupled Lorenz oscillators as our example (the coupling is added on the equations of the components x). The equations of motion of a single Lorenz oscillator are written as dx dt dy dt dz dt
¼
rðy xÞ
¼
ci x y xz
¼
xy bz;
ð8Þ
with the classical parameter values: r = 10.0 and b = 8/3, and ci being variable for different oscillators. Let us start with considering a small system size L = 3. It is well known that c relates to the Rayleigh number in studies of turbulence, and the chaoticity strength of a single Lorenz oscillator is approximately determined by the value of c. Without losing generality, c1 = 50, c2 = 55, and c3 = 60 are chosen. Three tests (I, II, III) using different driving signals (x03 , x01 , and x02 , respectively), shown in Fig. 3, are performed. In Fig. 4, the MCLE spectrum {k1, k2, k3} as the functions of the driving strength e are exhibited, and the sequence of generalized synchronizations is indicated by the transition from positive to negative of ki,i = 1, . . . , L, at the critical coupling strengths ec,1 12.0, ec,2 15.5, and ec,3 21.1 emphasized by three arrows in Fig. 4. To measure the degree of synchronization of the corresponding nodes between X and Y, the average difference, Z 1 T Dj ðtÞdt; ð9Þ Dj ¼ lim T !1 T 0 is defined with the instantaneous distance as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dj ðtÞ ¼ kY j ðtÞ X i ðtÞk ¼ ðxj x0i Þ2 þ ðy j y 0i Þ2 þ ðzj z0i Þ2 ;
i ¼ ½ðj þ kÞ=L:
ð10Þ
The dependence of Dj (j = 1,2,3) on the coupling strength e in the test I, II, and III, are plotted in Fig. 5 (a), (b) and (c), respectively. We choose a very long time interval, T = 10,000, in the numerical calculations. In all of the three tests, it is only after the former oscillator transits to the synchronous state that the other oscillators, whose spatial positions are behind, are able to synchronize. For example, in Fig. 5(a), it is oscillator 1 that synchronizes first, followed by oscillator 2, and finally oscillator 3. In Fig. 5(b), it is oscillator 1 first, then oscillators 2 and 3 simultaneously. Finally in Fig. 5(c), all three oscillators synchronize simultaneously. Moreover, an excellent agreement with Fig. 4, where the complete synchronization thresholds coincide with the theoretical estimation of the critical coupling strengths of generalized synchronization, is found. Although the response systems show distinct sequence of synchronization when different driving signals (different k) are used in Fig. 5, they transit to global synchronization at the same coupling strength, ec,3, the largest one among ec,i. To confirm the observation of a stable SPTCP, the patterns of the variables xj(t) under ea = 14.0, eb = 19.0, and ec = 22.0 for test I, II, and III are plotted in Fig. 6(a–c), (d–f), and (g–i), respectively. Here we have used N = 30 = 10L. Fig. 6(c), (f), and (i), ec>(ec,i)max = ec,3 21.1, apparently show the SPTCP with spatial period L = 3. Before that, no SPTCP can be found.
Fig. 3. Schematic plot of three tests (I, II, and III) using different sites (k = 3, 1, and 2, respectively) in the ring as driving signals of the linear array systems. L = 3.
M. Zhan et al. / Chaos, Solitons and Fractals 24 (2005) 767–774
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Fig. 4. The dependences of k1 (square), k2 (circle), and k3 (up-triangle) on e. At different coupling threshold (ec,1, ec,2, or ec,3), the individual oscillator (No. 1, 2, or 3 one in the ring) transits to generalized synchronous state progressively.
Fig. 5. The average differences, D1 (square), D2 (circle), and D3 (up-triangle) vs e for test I (a), II (b), and III (c).
In the above, the analysis involved a drive component of small size. It is interesting to explore the possibility for more general SPTCP states with a larger system size. In Fig. 7 (with L = 50, and ciÕs are the random numbers with uniform distribution from 28.0 to 100.0), the generalized synchronization thresholds for all specific oscillators, ec,i, i = 1, 2, . . . ,L, are obtained. Only above the ec,i, can we find the generalized synchronization for the individual oscillator, i. The disorder of the distribution of ec,i with large fluctuations certainly shows strong competition between the coupled oscillators due to the inhomogeneity of the parameter sets. In Fig. 8, N = 150 = 3L, k = L, the spatio-temporal patterns are plotted with the variables of xj(t) and the variable differences, d j ðtÞ ¼ xj ðtÞ x0j ðtÞ, for ea = 22.0 [(a) and (b)] and eb = 33.0 [(c) and (d)]. The values of the chosen coupling strength are exemplified in Fig. 7 with dash and dot lines, respectively. ea = 22.0 is an approximate average value, and eb = 33.0>ec,49 32.26 (ec,49 is the largest among the ec,i). Comparing with Fig. 8(a), Fig. 8(c) shows clear characteristics of the SPTCP, namely, periodic in space with a period L = 50 and chaotic in time. At the same time the static pattern with dj(t) = 0 for all driven oscillators [shown in Fig. 8(d)] indicates a nice establishment of synchronization between systems Y and X. Furthermore, a partial synchronization phenomenon with the first 8 oscillators (denoted by the arrow) is found in Fig. 8(b) for ea = 22.0 before the global one in Fig. 8(d). The result coincides with the analysis of the MCLE spectrum distribution in Fig. 7 and the particular spatial locations for ec,i, i.e., ea>ec,i, i = 1,2, . . . , 8, while ea
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Fig. 6. The patterns of variable xj(t) under different coupling strengths ea = 14.0, eb = 19.0, and ec = 22.0 (ec>(ec,i)max = ec,3 21.1), for test I [(a)–(c)], II [(d)–(f)], and III [(g)–(i)]. L = 3, and N = 30.
Fig. 7. The distribution of the critical coupling strengths, ec,i, i = 1,2, . . ., L, for generalized synchronization of specific oscillator. L = 50.
once the node ahead synchronizes with the corresponding one in the ring, a new stable driving source at this new spatial position will form. As a result, a rough linear function of synchronization time versus the spatial sites (at least in the scale of spatial period of L) can be expected. This point is confirmed by the shape of linear relationship with little fluctuation in Fig. 9, under the test condition of the time average of the instantaneous distance, Dj ðtÞ < 108 , in a small time interval of 10. Comparing to e = 35.0, a larger driving strength e = 40.0 leads to a faster synchronization behavior. Thus, the synchronization can propagate reasonably far provided the conditions of parameter matching (Eq. 4) and stability (Eq. 7) are satisfied. In [8], a similar phenomenon of synchronization waves in arrays of driven chaotic systems has been studied. In conclusion, we have investigated the SPTCP in detail. This pattern can occur in the coupled ring and linear array systems, regardless of the array size. The mechanism underlying this seemingly strange pattern has been made clear: under the condition of stability of the global generalized synchronization in the ring systems, the response systems
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Fig. 8. The patterns of variables xj(t) and the variable differences dj(t) under different coupling strengths ea [(a) and (b)], and eb [(c) and (d)]. ea = 22.0>ec,i, i = 1,2, . . . , 8, while ea
Fig. 9. The synchronization time sj vs the site number j for e = 35.0 (square points) and e = 40.0 (circle points) with approximately linear relationship.
can copy the pattern of the driving system (which maybe a spatio-temporal chaotic state), transfer it in its one-way structure, and establish the SPTCP with multiple copies of the ring in space. In Refs. [20–22], authors studied the chaotic itinerancy phenomenon between attractor ruins. Here our analysis mainly concerns the formation condition of a specific pattern, SPTCP.
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