Article Sidebar

Submitted
Apr 25, 2017
Published
Jun 1, 2007
Download
PDF
Statistic
Read Counter : 309 Download : 224

Main Article Content

Abstract

We consider the synchronization and persistence of a system of identical lattice oscillators that are diffusively coupled to their nearest neighbours. Each subsystem has a compact global attractor. This is done in the framework of invariant manifold theory. Normal hyperbolicity and its persistence are applied to obtain general conditions for the stability and robustness of the synchronization manifold. AMS(MOS) Subject classifications: 37C80, 37D10.

 

Keywords

Lattice oscillators bravais lattice synchroniza

Article Details

References

  1. AFRAIMOVICH, V.S., BYKOV, V. and SHILNKOV, L. 1983. On Structurally Unstable Attracting Limit sets of Lorenz attractor Type, {it Tran. of Moscow Math. Soc. 44:153-216.
  2. CAHN, J.W. 1960. Theory of growth and interface motion in crystalline material, Acta Metallurgica 8: 554-562.
  3. CHOW, S.N., LIU, W. 1997. Synchronization, stability and normal hyperbolicity. Resenhas IME-USP. 3: 139-158.
  4. CUOMO, K.M. and OPPENHEIM, A.V. 1992. Synchronized Chaotic Circuits and systems for communications, MIT Research Laboratory of electronics technical report 575 (unpublished).
  5. CUOMO, K.M., OPPENHEIM, A.V., 1993. Circuit implementation of synchronized chaos with applications to communications, it Phys. Rev. Lett. 71(1): 65-68.
  6. CUOMO, K.M., OPPENHEIM, A.V. 1993. Chaotic signals and systems for communication, ICASSP 3: 137-140.
  7. CHUA, L.O. and ROSKA, T. 1993. The CNN Paradigm. IEEE Trans. Circuits Syst., 40: 147-156.
  8. FENICHEL, N. 1971. Persistence and smoothness of invariant manifolds for flows. Ind., Univ. Math. J., Vol. 21 3: 193-225.
  9. FUJISAKA, H., YAMADA, T. 1983. Stability theory of synchronized motion in coupled-oscillators Systems. Prog. Theor. Phys. 69: 32-47.
  10. HALE, J. 1997. Diffusive coupling, dissipation and synchronization J. Dyn. Differ. Equ. 9: 1-52.
  11. HIRSCH, M.W., PUGH, C.C., SHUB, M. 1977. Invariant manifolds, Lecture Notes in Mathematics Vol 583. Springer-Verlag: Berlin, Heidelberg, New York.
  12. JOSIĆA, K. 2000. Synchronization of chaotic systems and invariant manifolds, Nonlinearity 13:1321-1336.
  13. LANCASTER, P. 1985. Tismenetsky, M.; The theory of Matrices, Academic press.
  14. MANÉ, R. 1977. Persistent manifolds are normally hyperbolic, Trans. Amer. Math. Soc. 246: 261-283.
  15. WASIKE, A.A.M. 2003. Synchronization and oscillator death in diffusively coupled lattice oscillators International Journal of Mathematical science, 2(1): 67-82.
  16. WIGGINS, S. 1994. Normally Hyperbolic Invariant Manifolds in Dynamical Systems; Springer-Verlag: Berlin, Heidelberg, New York.