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Upper semicontinuity of attractors and synchronization. (English) Zbl 0907.34039

Summary: The authors prove that diffusively coupled abstract semilinear parabolic systems synchronize. The abstract results obtained are applied to a class of ordinary differential equations and to reaction diffusion problems. The technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem, explicitly exhibited, in the diagonal.

MSC:

34D45 Attractors of solutions to ordinary differential equations
35K57 Reaction-diffusion equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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[1] V. S. Afraimovich, H. M. Rodrigues, Uniform ultimate boundedness and synchronization for nonautonomous equations, CDSNS94-202, Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 1994; V. S. Afraimovich, H. M. Rodrigues, Uniform ultimate boundedness and synchronization for nonautonomous equations, CDSNS94-202, Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 1994
[2] Afraimovich, V. S.; Verichev, N. N.; Rabinovich, M. I., Stochastic synchronization of oscillation in dissipative systems, Izv. Vysshi. Uchebn. Zaved. Radiofiz., 29, 1050-1060 (1986)
[3] Alexander, J. C.; Kan, I.; Yorke, J. A.; You, Z., Riddled basins, Internat. J. Bifurcation Chaos, 2, 795 (1992) · Zbl 0870.58046
[4] Anishenko, V. S.; Vadivasova, T. E.; Postnov, D. E.; Safonova, M. A., Synchronization of chaos, Internat. J. Bifurcation Chaos, 2, 633-644 (1992) · Zbl 0876.34039
[5] Carroll, T. L.; Pecora, L. M., Synchronizing nonautonomous chaotic circuits, IEEE Trans. Circuits and Systems, 38, 453-456 (1991)
[6] Carvalho, A. N.; Oliveira, L. A., Delay-partial differential equations with some large diffusion, Nonlinear Anal., 22, 1057-1095 (1994) · Zbl 0809.35148
[7] de Carvalho, A. N.; Ruas-Filho, J. G., Global attractors for parabolic problems in fractional power spaces, SIAM J. Math. Anal., 26, 415-427 (1995) · Zbl 0824.35009
[8] Carvalho, A. N., Contracting sets and dissipation, Proc. Roy. Soc. Edinburgh Sect. A, 125, 1305-1329 (1995) · Zbl 0846.35066
[9] Carvalho, A. N.; Cholewa, J. W.; Dlotko, T., Examples of global attractors in parabolic problems, Hokkaido Math. J., 26, 1-27 (1997)
[10] Chua, L. O.; Itoh, M.; Kocarev, L.; Eckert, K., Chaos synchronization in Chuas’s circuit, J. Circuits Systems Comput., 3, 93-108 (1993)
[11] Cuomo, K. M.; Oppenheim, A. V., Chaotic signals and systems for communications, ICASSP (1993)
[12] Fabiny, L.; Colet, P.; Roy, R., Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993)
[13] Feller, W., An Introduction to Probability Theory and Its Applications (1968), Wiley: Wiley New York · Zbl 0155.23101
[14] Friedman, A., Partial Differential Equations (1969), Holt, Rinehart & Winston: Holt, Rinehart & Winston New York · Zbl 0224.35002
[15] Fujisaka, H.; Yamada, T., Stability theory of synchronized motion in coupled-oscillator systems, Progr. Theoret. Phys., 69, 32-47 (1983) · Zbl 1171.70306
[16] Gantmacher, F. R., The Theory of Matrices (1960), Chelsea: Chelsea New York · Zbl 0088.25103
[17] Hale, J. K., Asymptotic Behavior of Dissipative Systems (1988), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0642.58013
[18] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1981), Springer-Verlag: Springer-Verlag Berlin · Zbl 0456.35001
[19] Mirollo, R. E.; Strogatz, S. H., Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50, 1645-1662 (1990) · Zbl 0712.92006
[20] Ostrovskii, L. A.; Soustova, I. A., Phase-locking effects in a system of nonlinear oscillators, Chaos, 1, 224-231 (1991) · Zbl 0906.34025
[21] Rodrigues, H. M., Abstract methods for synchronization and applications, Appl. Anal., 62, 263-296 (1996) · Zbl 0890.34052
[22] Rul’kov, N. F.; Volkovskii, A. R., Synchronized chaos in electronic circuits, Proceedings of SPIE Conference on Exploiting Chaos and Nonlinearities, San Diego (July 1993)
[23] Rul’kov, N. F.; Volkovskii, A. R., Threshold synchronization of chaotic relaxation oscillations, Phys. Lett. A, 179, 332-336 (1993)
[24] Strogatz, S. H.; Stewart, I., Coupled oscillators and biological synchronization, Sci. Amer. (1993)
[25] Verichev, N. N., Mutual synchronization of chaotic oscillations in the Lorenz systems, Methods Qual. Theory Differential Equations Gorky, 47-57 (1986)
[26] Walter, W., Differential and Integral Inequalities (1970), Springer-Verlag: Springer-Verlag Berlin · Zbl 0252.35005
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