Spatio-temporal chaotic synchronization for modes coupled two Ginzburg-Landau equations
Abstract
On the basis of numerical computation, the conditions of the modes coupling are proposed, and the high-frequency modes are coupled, but the low frequency modes are uncoupled. It is proved that there exist an absorbing set and a global finite dimensional attractor which is compact and connected in the function space for the high-frequency modes coupled two Ginzburg-Landau equations (MGLE). The trajectory of driver equation may be spatio-temporal chaotic. One associates with MGLE, a truncated form of the equations. The prepared equations persist in long time dynamical behavior of MGLE. MGLE possess the squeezing properties under some conditions. It is proved that the complete spatio-temporal chaotic synchronization for MGLE can occur. Synchronization phenomenon of infinite dimensional dynamical system (IFDDS) is illustrated on the mathematical theory qualitatively. The method is different from Liapunov function methods and approximate linear methods.
Key words
complete synchronization Ginzberg-Landau equations attractor spatio-temporal chaosChinese Library Classification
O175.292000 Mathematics Subject Classification
35B99Preview
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References
- [1]Pecora L M, Corrol T L. Synchronization in chaotic in chaotic systems[J]. Phys Rev Lett, 1990, 64(8):821–824.MATHMathSciNetCrossRefGoogle Scholar
- [2]Abarbane H D, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: the auxiliary system approach[J]. Phys Rev E, 1996, 53(5):4528–4533.CrossRefGoogle Scholar
- [3]Maistrenko Y, Kapitaniak T. Different type of chaos synchronization in two coupled piecewise linear maps[J]. Phys Rev E, 1999, 54(4):3285–3289.MathSciNetCrossRefGoogle Scholar
- [4]Codreanu S. Synchronization of spatiotemporal nonlinear dynamical systems by an active control[J]. Chaos Solitons Fractals, 2003, 15(3):507–510.MATHMathSciNetCrossRefGoogle Scholar
- [5]Duane G S, Tribbia J J. Synchronized chaos in geophysic dynamics[J]. Phys Rev Lett, 2001, 86(19):4298–4301.CrossRefGoogle Scholar
- [6]Wei G W. Synchronization of single-side locally averaged adaptive coupling and its application to shock capturing[J]. Phys Rev Lett, 2001, 86(16):3542–3545.CrossRefGoogle Scholar
- [7]Wu S G, He K F, Huang Z G. Controlling spatio-temporal chaos via small external forces[J]. Phys Lett A, 1999, 260(5):345–351.MATHMathSciNetCrossRefGoogle Scholar
- [8]Junge L, Parlitz U. Phase synchronization of coupled Ginzburg-Landau equations[J]. Phys Rev E, 2000, 62(1):438–441.CrossRefGoogle Scholar
- [9]Temam R. Infinite Dimensional System in Mechanics and Physics Applied Mathematics Series[M]. Springer-Verlag, New York, 1988.Google Scholar
- [10]Li Y, Mclaughlin D W Q, Shatan J, Wiggins S. Persistent homoclinic orbits for perturbed nonlinear Schrodinger equations[J]. Comm Pure App Math, 1996, 49(1):1175–1255.MATHCrossRefGoogle Scholar