Synchronization in lattices of coupled oscillators. (English) Zbl 1194.34056

Summary: We consider coupled nonlinear oscillators with external periodic forces and the Dirichlet boundary conditions. We prove that synchronization occurs provided that the coupling is dissipative and the coupling coefficients are sufficiently large. The synchronization here is of an obvious type - the size of an attractor is comparable to the difference of the subsystems.


34C11 Growth and boundedness of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
70K20 Stability for nonlinear problems in mechanics
Full Text: DOI


[1] Fujisaka, H.; Yamada, T., Stability theory of synchronized motion in coupled oscillator systems, Progr. Theoret. Phys., 69, 32-47 (1983) · Zbl 1171.70306
[2] Afraimovich, V. S.; Verichev, N. N.; Rabinovich, M. I., Sov. Radiophys., 29, 795 (1986)
[3] Carrol, T. L.; Pecora, L. M., Synchronization in nonautonomous chaotic circuits, IEEE Trans. Circuits Systems, 38, 453-456 (1991)
[4] Carrol, T. L.; Pecora, L. M., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[5] Chua, L. O.; Itoh, M.; Kosarev, L.; Eckert, K., Chaos synchronization in Chua’s circuits, J. Circuits Systems Comput., 3, 93-108 (1993)
[6] Fabiny, L.; Colet, P.; Roy, R., Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993)
[7] Rul’kov, N. F.; Volkovsky, A. R., Threshold synchronization of chaotic relaxation oscillations, Phys. Lett. A, 179, 332-336 (1993)
[8] Brown, R.; Rul’kov, N.; Tufillaro, N., Synchronization in chaotic systems: The effects of additive noise and drift in the dynamics of driving, Phys. Rev. E, 50, 4488-4508 (1994)
[9] Rodrigues, H. M., Uniform ultimate boundedness and synchronization, (preprint CDSNS 94-160 (1994), Georgia Tech)
[10] Afraimovich, V. S.; Rodrigues, H., Uniform ultimate boundedness and synchronization in nonautonomous equations, (preprint CDSNS 94-202 (1994), Georgia Tech)
[11] Heagy, J. F.; Carrol, T. L.; Pecora, L. M., Synchronous chaos in coupled oscillator systems, Phys. Rev. E, 50, 1874-1885 (1994)
[12] Wu, C. W.; Chua, L. O., A unified framework for synchronization and control of dynamical systems, Int. J. Bifur. and Chaos, 4, 979-988 (1994) · Zbl 0875.93445
[13] Babin, A. V.; Vishik, M. I., Attractors of Evolution Equations (1991), North-Holland: North-Holland Amsterdam · Zbl 0765.35023
[14] Hale, J. K., Asymptotic behavior of dissipative systems, Math. Surveys and Monographs. Amer. Math. Soc., 25 (1988) · Zbl 0642.58013
[15] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, Appl. Math. Sci., 68 (1988) · Zbl 0662.35001
[16] J.K. Hale, Diffusive coupling, dissipation and synchronization, J. Dyn. Diff. Eqs., to be published.; J.K. Hale, Diffusive coupling, dissipation and synchronization, J. Dyn. Diff. Eqs., to be published. · Zbl 1091.34532
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.