這篇論文主要研究當藕合強度ｃ變化時藕合網格的動態行為。主要結果如下
１. 當ｃ很小的時候，我們可以儘可能的找到snap-back repellers。
２. 我們可以刻劃出當藕合網格有局部同步現象時ｃ的範圍，而此範圍可以用中心點、左半徑以及右半徑描述出來，而當ｃ等於此中心點的值時，收斂到同步區域的速度將會最快。除此之外，我們也可驗證，此中心點只跟藕合網格上質點的個數，以及藕合矩陣Ｇ有關，而跟未藕合前的動態行為ｆ無關。

We study the dynamics of coupled map lattices (CMLs) as the coupling strength c varies. Our main results are as follows. First, we show that for small c, as many snap-back repellers as possible can be obtained. Second, we study the “intermediate” range of c. In particular, the optimal range of c for getting local synchronization of CMLs can be explicitly constructed by indentifying its center, left and right radii. The center of coupling strength for local synchronization gives the fastest convergence rate of the initial values toward synchronous manifold. Moreover, we prove that the center depends on the number of nodes and the connectivity topology G, and is independent of the choice of uncoupled dynamics f

[1] K. Kaneko, Spatial periodic-doubling in open flow, Phys. Lett. A 111(1985), 321-325

[2] K. Kaneko, Simulating physics with coupled map lattices---pattern
dynamics, information flow, and thermodynamics of spatiotemporal
chaos, in Formation, Dynamics and Statics of Patterns, edited by K.Kawasaki, A. Onuki and M. Suzuki (World Scientific, Singapore, 1990), Vol. 1 1-54. On the other hands there are works that pursue a theorectical approach.

[3] V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diffusively coupled maps, Int, J. Bif. Chaos 4 (1994), 631-637.

[4] S. P. Kuznetsov, in Theory and Applications of Coupled Map Lattices, edited by K. Kaneko (Wiley, New York 1993); S. P. Kuznetsov, Pis a Zh. Tekh Fiz. 9, 94 (1983) [Sov. Tech. Phys, Lett. 9, 41 (1983)].

[5] H. Kook, F. H. Ling, and G. Schmidt, Universal behavior of coupled nonlinear systems, Phys. Rev. A 43 (1991), 2700-2708.

[6] F. Kaspar and H.G. Shuster, Scaling at the onset of spatial disorder in coupled
piecewise linear maps, Phys. Lett. A 113 (1986), 451-453.

[7] T. Bohr and O. B. Christensen, Size dependence, coherence, and scaling in turbulent coupled-map lattices Phys. Rev. Lett. 63, (1989), 2161-2164.

[8] J. Losson and M. C. Mackey, Phys. Rev. E, Coupled map lattices as models of deterministic and stochastic differential delay equations 52 (1995), 115-128; Evolution of probability densities in stochastic coupled map lattices 52 (1995), 1403-1417.

[9] M. S. Bourzutschky and M. C. Cross, Coupled map models for chaos in extended systems, CHAOS 2 (1992), 173-181.

[10] J. Juang and Y-H Liang, Synchronous chaos in coupled map lattices with general connectivity topology, SIAM Appli. Dynam. Syst., to appear.

[11] L.M. Pecora, Synchronization conditions and desynchronization patterns in coupled limit-cycle and chaotic systems, Phys. Rev. E, Vol. 58 Num. 1(1998), 347-360.

[12] L.M. Pecora, and T.L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., Vol. 80 Num. 10(1998), 2109-2112.

[13] A. Pogromsky, and H. Nijmeijer, Cooperative oscillatory behavior of mutually coupled dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl. Vol. 48 Num. 2(2001), 152-162.

[14] G. Rangarajan, and M. Ding, Stability of synchronized chaos in coupled dynamical systems, Phys. Lett. A, Vol. 296 Num. 4(2002), 204-209.

[15] M. Vidyasagar, Nonlinear Systems Analysis, Prentice Hall Inter., Inc, first edition, 1978.

[16] P. M. Gade and R. E. Amritkar, Spatially periodic orbits in coupled-map lattices, Phys. Rev. E 47 (1993), 143-154.

[17] R. E. Amritkar, P. M. Gade, A. D. Gangal, and V. M. Nandkumaran, Satbility of periodic orbits of coupled-map lattices, Phys. Rev. A 44 (1991), 3407-3410.

[18] C. Giberti and C. Vernia, Normally attracting manifolds and periodic behavior in
one-dimensional and two-dimensional coupled map lattices, Chaos 4, 4 (1994), 651-663.

[19] N. Belykh and E. Mosekilde, One-dimensional map lattices: synchronization, bifurcations, and chaotic structures} Phys. Rev. E 54 (1996), 3196-3203.

[20] M. Hasler, Yu. Maistrenko and O. Popovych, Simple example of partial synchronization of chaotic systems, Phys. Rev. E 58 (1998), 6843 - 6846.

[21] K. Pyragas, Weak and strong synchronization of chaos, Phys. Rev. E 54 (1996), 4508-4511.

[22] M. de Sousa Vieira and A. J. Lichtenberg, Nonuniversality of weak synchronization in chaotic systems, Phys. Rev. E 56 (1997), 3741-3744.

[23] G. Rangarajan and M. Ding, Stability of synchronized chaos in coupled dynamical systems, Phys. Lett. A 296 (2002) 204-209.

[24] C. W. Wu, Global Synchronization in coupled map lattices, IEEE ISCAS (1998), 302-305

[25] W. W. Lin, C. C. Peng and C. S. Wang, Synchronization in coupled map lattices with periodic boundary condition, Internat. J. Bifur. Chaos 9, 8 (1999), 1635-1652.

[26] W. W. Lin and Y. Q. Wang, Chaotic synchronization in coupled map lattices with
periodic boundary conditions, SIAM J. Appl. Dyn. Syst. 1, 2 (2002), 175-189

[27] F. R. Marotto, Snap-back repellers imply chaos in R^n, J. Math. Anal. Appl. 63 (1978) 199-223.

[28] G. Chen, S. -B. Hsu and J. Zhou, Snap-back repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy
injection at the middle of the span, J.Math. Phys., 39 (1998), 6459-6489.

[29] F. R. Marotto, On redefining a snap-back repeller, Chaos Solitons Fractals 25 (2005) 25-28.

[30] Y. -W. Chang, J. Juang and C. -L. Li, Snap-back repellers and chaotic traveling waves in one-dimensional celluar neural networks, Inter. J. Bifur. Chaos. (2007).

[31] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, CA., 1989

[32] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math 547 (2002), 51-68.

[33] Y. Chen, G. Rangarajan and M. Ding, General stability analysis of synchronized dynamics in coupled systems, Phys. Rev. E 67, 026209 (2003).