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Apr 25, 2017
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Jun 1, 2008
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Abstract

We study synchronization in the framework of invariant manifold theory for systems with a time lag.  Normal hyperbolicity and its persistence in infinite dimensional dynamical systems in Banach spaces is applied to give general results on synchronization and its stability. 

 

 

Keywords

Normal hyperbolicity Synchronization Robustness Lyapunov numbers.

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References

  1. AFRAIMOVICH, V.S, VERCHEV, N.N. and RIABONVICH, M.I. 1986. Stochastic Synchronization of oscillations in dissipative systems. Radio Phy. Quantum Electron. 29: 31-45.
  2. BATES, P.W., LU, K., ZENG, C. 1998. Existence and persistence of invariant manifold for semiflows in Banach spaces, Mem. Amer. Math. Soc. 135(645): 305-317.
  3. BATES, P.W., LU, K., ZENG, C. 1999. Persistence of overflowing Manifolds for semiflows, Comm. Pure Appl. Math. LII: 983-1046.
  4. BATES, P.W., LU, K., ZENG, C. 2000. Invariant Foliations near Normally Hyperbolic Invariant Manifolds For Semiflows, Trans. Amer. Math. Soc. 352(10): 4641-4676.
  5. BOSE, F.G. 1986. Stability Conditions for the General Linear Difference-Differential Equation with Constant Coefficients and One Constant Delay. J. Math. Anal. Appl. 140:136-176.
  6. CHOW, S.N. and LIU, W. 1997. Synchronization, stability and normal hyperbolicity. Resenhas IME-USP}. 3: 139-158.
  7. CHOW, S.N. 2000. Lattice Dynamical Systems, Dynamical Systems Springer. 1822: 1-99.
  8. DIEKMANN, O., VAN GILS, S. A.,VERDUYN LUNEL, S.M. and WALTHER, H.-O. 1995. Delay Equations. Springer-Verlag, Berlin, Heidelberg, New York.
  9. FENICHEL, N. 1971. Persistence and smoothness of invariant manifolds for flows. Ind, Univ. Math. J., 21(3): 193-225.
  10. FUJISAKA, H. and YAMADA, T. 1983. Stability Theory of Synchronized Motion in Coupled-Oscillator System. Prog. Theor. Phys. 69: 32-47.
  11. GRASMAN, J., and JANSAN, M.J.W. 1979. Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J. math. Biology 7: 171-197.
  12. HALANAY, A. 1967. Invariant manifolds for systems with Time lag, Differential Equations and Dynamical systems. Academic Press. 199-213.
  13. HALE, J. 1997. Diffusive coupling, dissipation and synchronization J. Dyn. Differ. Equ. 9:1-52.
  14. HALE, J.K, and VERDUYN LUNEL. 1993. Introduction to functional Differential Equation; Springer-Verlag: New York.
  15. HENRY, D. 1983. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. No. 840, Springer-Verlag: Berlin.
  16. HIRSCH, M.W., PUGH, C.C., SHUB, M., Invariant manifolds. Lecture Notes in Mathematics Vol 583. Springer-Verlag: Berlin, Heidelberg, New York, 1977.
  17. JOSICA, K. 2000. Synchronization of chaotic systems and invariant manifolds, Nonlinearity 13: 1321-1336.
  18. KURZWEIL, J. 1967. Invariant Manifolds for flows, Differential Equations and Dynamical systems. Academic Press. 431-468.
  19. PYRAGAS, K. 1998. Synchronization of coupled time-delay systems: Analytical estimations Physical Review E. 53(3): 3067-3071.
  20. ROSSONI, E., CHEN, Y., DING, M., FENG, J. 2005. Stability of synchronous oscillations in systems of HH neurons with delayed diffusive and pulsed Coupling. Phys. Review E. 71: 819-823.
  21. WASIKE, A.A.M. 2002. Periodic solutions of Systems of delay differential equations Indian Journal of Mathematics, 44(1): 95-117.
  22. WASIKE, A.A.M. 2003. Synchronization and oscillator death in diffusively coupled lattice oscillators International Journal of Mathematical Science, 2(1): 67-82.
  23. WASIKE, A.A.M., Rotich, K.T. 2007. Synchronization, Persistence in Diffusively coupled Lattice oscillators Sultan Qaboos University Journal of Science, 12(1): 41-52 .
  24. WIGGINS, S. 1994. Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer-Verlag: Berlin, Heidelberg, New York.