Wavelet Transform Method for Coupled Map Lattices
Cheng Juang, Chin-Lung Li, Y. H. Liang, and Jonq Juang
Abstract—The purpose of this paper is twofold. First, we derive a necessary and sufficient condition for local synchronization in cou-pled map lattices (CMLs) with symmetric coupling. In particular, we also identify the best choice of coupling strength in the sense that such a coupling strength gives the fastest convergence rate of initial values toward the synchronous manifold. Furthermore, such a coupling strength is independent of the choice of the individual chaotic map. In addition, it is demonstrated that the wavelet trans-form method, which is proposed by Wei et al., can greatly increase the applicable ranges of coupling strengths, the parameters of the individual chaotic map, and the number of nodes for local synchro-nization of CMLs.
Index Terms—Connectivity topology, synchronization, wavelet transform.
I. INTRODUCTION
S
IMULATION of natural phenomena is one of the most im-portant research fields, and coupled map lattices (CMLs) are a paradigm for studying fundamental questions in spatially extended dynamical systems. This is because of their wide range of applications such as in turbulence, pattern formation in nat-ural systems, and solitons. They also exhibit a very rich phe-nomenology, including a wide variety of both spatial and tem-poral periodic structures, intermittence, chaos, domain walls, kink dynamics, etc. As a matter of fact, one of the most inter-esting aspects of CMLs is the presence of attracting manifolds. Such attracting manifolds lead to notions such as partial syn-chronization [1], weak and strong synsyn-chronization [2], [3], and (complete) synchronization [4]–[9].In continuous coupled chaotic systems, the stably syn-chronous motion was first studied in [10]. Later, Pecora and Carroll [11] proposed a master stability function (MSF) to symmetrically determine the stability of their synchronous manifold. In the case of full-state coupling, the MSF can be further reduced to a single inequality
[12], [13], where is the largest Lyapunov exponent of the single chaotic oscillator, is the coupling strength, and is the second largest eigenvalue of the coupling matrix . This necessary and sufficient stability condition then allows one to achieve the synchronization of identical
Manuscript received June 13, 2007; revised June 26, 2008. First pub-lished October 31, 2008; current version pubpub-lished April 10, 2009. This work was supported in part by the NSC under Contracts 95-2115-M-009-014-MY3 and 95-2221-E159-016. This paper was recommended by Asso-ciate Editor C. W. Wu.
C. Juang is with the Department of Electronics, Ming Hsin University of Sci-ence and Technology, Hsinchu 300, Taiwan (e-mail: [email protected]).
C.-L. Li is with the Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan.
Y. H. Liang and J. Juang are with the Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: jjuang@math. nctu.edu.tw).
Digital Object Identifier 10.1109/TCSI.2008.2004343
dynamical systems whenever is greater than the critical value . However, for a typical connectivity topology such as the diffusively coupled matrix, its moves closer to the origin as the number of nodes increases. Consequently, synchronization is more difficult to realize as increases. In [8], Wei et al. proposed a wavelet transform method to alter the connectivity topology. In doing so, becomes a quantity depending on wavelet parameter . It is found there that a critical wavelet parameter can be chosen to move away from the origin regardless of the number of nodes. This, in turn, greatly reduces the size of the critical coupling strength . Such phenomena are analytically verified in [14]–[16]. We also remark that, in the case of partial-state coupling for continuous coupled chaotic systems, even though the MSF can no longer be further reduced to the single in-equality, as given earlier, much progresses are still made for both local synchronization theory (see, e.g., [17] and the ref-erences cited therein) and global synchronization theory (see, e.g., [18] and the references cited therein). It should also be noted that a theory for multiresolution signal decomposition by using wavelet representation is addressed in [19].
The development of synchronization theory of the discrete dynamical coupled systems is still at the primitive stage as com-pared to that of their continuous counterparts. The reason for the gap between the theory developed in the lattices of coupled chaotic systems and that of CMLs lies mostly on the fact that it is more natural to have a nonlinear coupling between oscillators in CMLs. This is because a nonlinear coupling with suitable range of coupling strength tends to yield an invariant region for the corresponding CMLs, while linear coupling cannot. It should be noted that there is no such problem for the lattices of coupled chaotic systems. It should also be mentioned that the analytical results of the lattice of the coupled chaotic systems stated earlier are linearly coupled. As a result of such nonlinear coupling, the following different phenomena of CMLs as compared to those of the lattices of coupled chaotic systems are observed (see, e.g., [5]). A stronger coupling strength does not ensure synchroniza-tion, but it will cause the trajectory of the oscillator to grow out of bound. Furthermore, synchronization becomes numerically unobservable even for a modest number of oscillators.
The purpose of this paper is twofold. First, we derive an op-timal synchronization theory for CMLs (1). In particular, we have the following results. A necessary and sufficient condition on coupling strength for local synchronization is obtained. Fur-thermore, the coupling strength giving the fastest convergence rate of initial values toward the synchronous state is explicitly obtained. Such is shown to be independent of the choice of the individual chaotic map. Moreover, the maximum number of os-cillators for which the corresponding network would still yield synchronization can be explicitly obtained. Second, by using such results, we demonstrate that the wavelet transform method can greatly increase the applicable range of coupling strength
and the number of oscillators for local synchronization of cou-pled chaotic map lattices. The family of quadratic maps is used as numerical examples.
II. THEORETICALCONSIDERATIONS
A. Stability Results Without Wavelet Transform Method The case of CMLs can be described in vector form [6], [7]
(1)
where ; is the unit matrix;
is the coupling strength; is a symmetric coupling matrix having zero row sums, with zero being a simple eigenvalue;
and . Here,
de-scribes the chaotic dynamics of an individual oscillator. Let be the eigenvalues of the coupling matrix . It was shown, e.g., [6], that if
(2) for all the nonzero eigenvalues . Note that the second largest eigenvalue alone is not enough to ensure that all other eigen-values satisfied (3). To achieve synchronization of CMLs, we need to find so that the maximum of
is a minimum, i.e., to solve a min–max problem of the form (3) To find such saddle point , let , , be real numbers so that
(4)
Solving for (3), we have or . It
then follows that, for fixed
(5) occurred at . Let be indexes so that
for all . We have that
occurred at . Taking the fact that , , are all negative, we conclude that , which is to be called the synchronization index of the system. The aforementioned results are summarized as follows.
Theorem 1: The min–max problem (3) can be achieved when , . Consequently, system (1) is (lo-cally) synchronized if and only if
(6) If (6) holds, then there exists an optimal neighborhood of so that (1) is (locally) synchronized whenever . Here
(7) The interval (if it exists) is optimal in the sense that if is not in , then system (1) will not acquire (local) synchro-nization. Moreover, , which is independent of the choice
of the individual chaotic map, is the best choice of coupling strength for local synchronization of (1) in the sense that such a coupling strength gives the fastest convergence rate of initial values toward the synchronous manifold.
Remark 1: The interval is first derived in [5]. Our approach here gives the optimal coupling strength , which is independent of the choice of the individual chaotic map.
Since , it is then clear that as
the number of oscillators increases, approaches to one. Consequently, (6) cannot be fulfilled, provided that is large. Hence, to achieve synchronization of (1), one has to place a limit on the number of oscillators considered. Using (6), one should be able to find the maximum number of oscillators satisfying (6). From here on, and are to be called the synchronization interval and the Lyapunov index of the system, respectively. The length of is denoted by . Apparently, a larger gives a better applicable range of coupling strength.
B. Stability Results With Wavelet Transform Method
How the wavelet transform method [12] affects the stability of synchronous manifold of (1) is discussed in this part. Let the number of nodes be equal to . Write as
..
. . .. ...
Here, the dimension of each block matrix is . By an -scale wavelet operator [12], [20], the matrix is trans-formed into of the form
..
. . .. ...
where each entry of is the average of entries of , , . After reconstruction [12], the coupling matrix be-comes . Here, is a wavelet parameter. In summary, the effects of the wavelet transform method can be viewed as the changes of the eigenvalues of the coupling matrix [13] and vary dramatically for different ’s. The eigenvalues of
are denoted by , with
. Clearly, Theorem 1 is still valid for such new coupling matrix. Note that the corresponding , , , and now depend on the wavelet parameter as well. To emphasize such dependence, we shall write , , , and as
, , , and , respectively.
III. NUMERICALRESULTS
A. (No Improvement)
The effects of the wavelet transform method can be viewed as the changes of the eigenvalues of the coupling matrix and vary dramatically for different 's. Consider the quadratic map diffusively coupled with periodic boundary conditions for . Fig. 1(a) shows the calculated eigen-values of the coupling matrix as a function of wavelet pa-rameter . The coupling matrices before and after
reconstruc-(a) (b)
(c) (d)
(e)
Fig. 1. (a) Eigenvalues (K) of the coupling matrix as a function of wavelet parameter K for N = 4. The solid line is (K), while the dotted line is (K). (b) Synchronization index of the coupling matrix as a function of wavelet parameterK. (c) Theoretical predicted synchronization intervals which are in agreement with Fig. 1(d) and (e). (d) Numerically produced intervals of synchronization without the wavelet transform method. (e) Numerically produced intervals of synchro-nization with(k = 0:9) the wavelet transform method. The theoretically predicted synchronization interval is shown in Fig. 1(c). Fig. 1(e) shows synchronization intervals(K = 0) with and without (K = 0:9) the wavelet transform method via computer simulation. The dark, gray, and white regions represent in complete synchronization, partial synchronization, and out of synchronization, respectively. The dark areas around = 3:64 and = 3:85 are caused by periodic windows. They are consistent with the results of Fig. 1(a)–(c), whereK 2 [0; 1].
tion are denoted by , with
and , respectively, where
The solid line is , while the dotted line is . Note that as is increased, a crossing appears at . This crossing makes the analytical identification of a diffi-cult task. Thus, the optimal is numerically determined from Fig. 1(b), where , the synchronization index, is obtained from Fig. 1(a). According to (6), is the number for which is a minimum. Fig. 1(b) shows that
. Thus, it is clear that no enhancement of syn-chronization is expected.
B. (Significantly Improved)
For , the enhancement of synchronization is shown. Fig. 2(a) shows the eigenvalues of the coupling matrix as a function of wavelet parameter , where
if
if otherwise.
(a) (b)
(c) (d)
(e) (f)
(g)
Fig. 2. (a) Eigenvalues (K) of the coupling matrix as a function of wavelet parameter K for N = 8. (b) Synchronization index of the coupling matrix as a function of wavelet parameterK. (c) Lyapunov index versus growth rate. (d) White region gives the synchronization intervals, as stated in (8), for K = 0. (e) White region gives the synchronization intervals, as stated in (8), withk = 1:15. (f) Numerically produced synchronization intervals without the wavelet transform method. (g) Numerically produced synchronization intervals with(K = 1:15) the wavelet transform method for N = 8.
Note that two crossing points appear. Similarly, the synchro-nization index is shown in Fig. 2(b). It is observed that . Thus, using an optimal (in the min region), it is expected to have a significant improvement over
according to Theorem 1.
Given the optimal , the effects of the wavelet transform method on synchronization with different growth rates and cou-pling strengths are further investigated. The quantities obtained in Theorem 1 produced those shown Fig. 2(b) and (c). Fig. 2(c) shows the Lyapunov index of system (1) with
and . These two graphs are identical with a vertical shift. It is seen, via Fig. 2(c), that if , so as , then the local synchronization is lost without the wavelet transform method. However, with the wavelet transform method , for all 's, and so, the local synchronization of (1) is preserved. Figs. 2(d) and (e) shows the optimal length of the coupling strength as a function of growth rate without the transform and with the optimal trans-form , respectively.
The numerical simulation for obtaining the interval of syn-chronization recorded in Fig. 2(f) and (g) again confirms our the-oretical prediction earlier. Without the transform , there is a narrow region for complete synchronization. In dark areas around and 3.85, each cell shows a periodic window type of behavior. In gray areas, different initial conditions give rise to different scenarios of partial synchronization (for ex-ample, even/odd cells are synchronized). With the wavelet trans-form method, there is a very significant increase in dark areas as compared to those in Fig. 2(f). The applicable ranges of cou-pling strengths and growth rates are significantly improved. The numerically produced Fig. 2(f) and (g) is in agreement with our theoretically predicted Fig. 2(d) and (e).
C. (Effects on Large )
The wavelet transform method is most dramatic for a large number of oscillators. Fig. 3(a) shows the maximum number of oscillators for which the local synchronization of the system with or without the wavelet transform method can still be sus-tained. The numbers are obtained by solving (6). It is seen, via Fig. 3(a), that the good improvement on the maximum number of oscillators allowed is there even without choosing the op-timal . The graphs in Fig. 2(a) are decreasing with respect to the growth rate of the map, except at those ’s yielding the window behavior.
As increases, the dominant eigenvalue approaches zero. Hence, local synchronization becomes unobservable. Further-more, the change of the dominant eigenvalue due to the wavelet transform method is very significant. Fig. 3(b) shows that if , then system (1) ac-quires synchronization with and . However, it is easily verified from (6) that if , then the maximum number of oscillators allowed for synchronization without the wavelet transform method is . From Fig. 3(b), it is also seen that if , then system (1) achieves synchronization with . It should be noted that in pro-ducing Fig. 3(b). only the end points of synchronization inter-vals are recorded. For those ’s, where , exhibiting the window behavior, the end points of synchronization inter-vals lie outside the interval (0.1, 0.5).
With the wavelet transform method, global synchronization can be achieved for . In the case of , the trans-form enhances the synchronization effect. In this case, there is a very significant region, as shown in the dark areas in Fig. 3(c). Without the transform, synchronization for such a large number of oscillators would not be possible. This demonstrates the dra-matic effects of the transform with relatively large .
(a)
(b)
(c)
Fig. 3. (a) Maximum number of oscillators allowed for which (1) acquires syn-chronization. (b) Lyapunov index versus growth rate. Here, = 3:5704 and = 3:57085. (c) Numerically produced synchronization interval with K = 1 andN = 40.
IV. CONCLUSION
The optimal coupling strength of CMLs with symmetric cou-pling can be analytically obtained. In particular, we also iden-tify the best choice of coupling strength in the sense that such a coupling strength gives the fastest convergence rate of ini-tial values toward the synchronous manifold. Furthermore, such a coupling strength is independent of the choice of the indi-vidual chaotic map. It should be noted that due to the nonlinear coupling for CMLs, both the second largest and the smallest values play a role in determining the synchronization interval.
This, in turn, places a limit on the number of oscillators for ac-quiring synchronization, no matter how we choose the coupling strengths. Based on those results, the theory of the wavelet trans-form method on system (1) can be predicted numerically. The family of quadratic maps is then used to demonstrate that the wavelet transform method can greatly increase the applicable ranges of coupling strengths, the parameters of the individual chaotic map, and the number of nodes for local synchronization of CMLs.
REFERENCES
[1] M. Hasler, Y. Maistrenko, and O. Popovych, “Simple example of partial synchronization of chaotic systems,” Phys. Rev. E, Stat. Phys.
Plasmas Fluids Relat. Interdiscip. Top., vol. 58, no. 5, pp. 6843–6846,
Nov. 1998.
[2] K. Pyragas, “Weak and strong synchronization of chaos,” Phys. Rev. E,
Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 54, no. 5, pp.
4508–4511, Nov. 1996.
[3] M. d. S. Vieira and A. J. Lichtenberg, “Nonuniversality of weak syn-chronization in chaotic systems,” Phys. Rev. E, Stat. Phys. Plasmas
Fluids Relat. Interdiscip. Top., vol. 56, no. 4, pp. 3741–3744, Oct. 1997.
[4] J. Juang and Y.-H. Liang, “Synchronous chaos in coupled map lattices with general connectivity topology,” SIAM J. Appl. Dyn. Syst., vol. 7, no. 3, pp. 755–765, 2008.
[5] X. Li and G. Chen, “Synchronization and desynchronization of com-plex dynamical networks: An engineering view point,” IEEE Trans.
Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 11, pp. 1381–1390,
Nov. 2003.
[6] G. Rangarajan and M. Ding, “Stability of synchronized chaos in cou-pled dynamical systems,” Phys. Lett. A, vol. 296, no. 4/5, pp. 204–209, Apr. 2002.
[7] C. W. Wu, “Global synchronization in coupled map lattices,” in Proc.
ISCAS, 1998, vol. 3, pp. 302–305.
[8] C. W. Wu, Synchronization in Coupled Chaotic Circuits and
Sys-tems. , Singapore: World Scientific, 2002.
[9] C. W. Wu, “Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling,” IEEE Trans. Circuits Syst. I, Fundam.
Theory Appl., vol. 50, no. 2, pp. 294–297, Feb. 2003.
[10] H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Prog. Theor. Phys., vol. 69, no. 1, pp. 32–47, 1983.
[11] L. M. Pecora and T. L. Carroll, “Master stability functions for syn-chronized coupled systems,” Phys. Rev. Lett., vol. 80, no. 10, pp. 2109–2112, Mar. 1998.
[12] G. W. Wei, M. Zhan, and C. H. Lai, “Tailoring wavelets for chaos con-trol,” Phys. Rev. Lett., vol. 89, no. 28, pp. 284 103–4, Dec. 2002. [13] C. W. Wu, “Perturbation of coupling matrices and its effect on
syn-chronizability in arrays of coupled chaotic systems,” Phys. Lett. A, vol. 319, no. 5, pp. 495–503, Dec. 2003.
[14] J. Juang and C. H. Li, “Eigenvalue problems and their application to the wavelet method of chaotic control,” J. Math. Phys., vol. 47, no. 7, pp. 072 704–16, Jul. 2006.
[15] J. Juang, C. H. Li, and J. W. Chang, “Perturbed block circulant matrices and their application to the wavelet method of chaotic control,” J. Math.
Phys., vol. 47, no. 12, pp. 122 702–11, Dec. 2006.
[16] S. F. Shieh, Y. Q. Wang, G. W. Wei, and C. H. Lai, “Mathematical analysis of the wavelet method of chaos control,” J. Math. Phys., vol. 47, no. 8, pp. 082 701–10, Aug. 2006.
[17] M. Y. Chen, “Some simple synchronization criteria for complex dy-namical networks,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 11, pp. 1185–1189, Nov. 2006.
[18] J. Juang, C.-L. Li, and Y.-H. Liang, “Global synchronization in lattices of coupled chaotic systems,” Chaos, vol. 17, no. 3, pp. 033 111–11, Sep. 2007.
[19] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 11, no. 7, pp. 674–693, Jul. 1989.
[20] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992.
Cheng Juang received the Ph.D. degree in electrical engineering from the University of Washington, Seattle, in 1990.
He is currently a Full Professor with the Depart-ment of Electronics, Ming Hsin University of Science and Technology, Hsinchu, Taiwan.
Chin-Lung Li received the Ph.D. degree in applied mathematics from National Chiao Tung University, Hsinchu, Taiwan, in 2007.
He is currently an Assistant Professor with the De-partment of Applied Mathematics, National Hsinchu University of Education, Hsinchu.
Y. H. Liang received the M.S. degree in applied mathematics from National Chiao Tung University, Hsinchu, Taiwan, in 2007, where he is currently working toward the Ph.D. degree in applied mathe-matics in the Department of Applied Mathemathe-matics.
Jonq Juang received the Ph.D. degree in mathe-matics from Texas Tech University, Lubbock, in 1988.
He is currently a Full Professor with the Depart-ment of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan.
