Dynamical Behavior of the Almost-Periodic Discrete Fitzhugh-nagumo Systems

In this paper, we study the dynamical behavior of nonautonomous, almost-periodic discrete FitzHugh–Nagumo system defined on infinite lattices. We prove that the nonautonomous infinite-dimensional system has a uniform attractor which attracts all solutions uniformly with respect to the translations of external terms. We also establish the upper semicontinuity of uniform attractors when the infinite-dimensional system is approached by a family of finite-dimensional systems. This paper is based on a uniform tail method, which shows that, for large time, the tails of solutions are uniformly small with respect to bounded initial data as well as the translations of external terms. The uniform tail estimates play a crucial role for proving the uniform asymptotic compactness of the system and the upper semicontinuity of attractors.

[1]  Athanasios N. Yannacopoulos,et al.  Global existence and compact attractors for the discrete nonlinear Schrödinger equation , 2005 .

[2]  Shengfan Zhou,et al.  Attractors for lattice systems corresponding to evolution equations , 2002 .

[3]  Erik S. Van Vleck,et al.  Spatially Discrete FitzHugh--Nagumo Equations , 2005, SIAM J. Appl. Math..

[4]  Bixiang Wang,et al.  Attractors for reaction-diffusion equations in unbounded domains , 1999 .

[5]  Xinfu Chen,et al.  Traveling Waves of Bistable Dynamics on a Lattice , 2003, SIAM J. Math. Anal..

[6]  John M. Ball,et al.  Erratum to: Continuity Properties and Global Attractors of Generalized Semiflows and the Navier-Stokes Equations , 1997 .

[7]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[8]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[9]  Jonathan G. Bell,et al.  Some threshold results for models of myelinated nerves , 1981 .

[10]  Bixiang Wang,et al.  Dynamics of systems on infinite lattices , 2006 .

[11]  Shui-Nee Chow,et al.  Pattern formation and spatial chaos in lattice dynamical systems. II , 1995 .

[12]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[13]  Dugald B. Duncan,et al.  Solitons on lattices , 1993 .

[14]  Scott,et al.  Classical and quantum analysis of chaos in the discrete self-trapping equation. , 1990, Physical review. B, Condensed matter.

[15]  Wenxian Shen,et al.  Dynamics in a Discrete Nagumo Equation: Spatial Topological Chaos , 1995, SIAM J. Appl. Math..

[16]  V. V. Zhikov,et al.  Almost Periodic Functions and Differential Equations , 1983 .

[17]  Erik S. Van Vleck,et al.  Analysis and computation of travelling wave solutions of bistable differential-difference equations , 1999 .

[18]  Chris Cosner,et al.  Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons , 1984 .

[19]  Peter W. Bates,et al.  A Discrete Convolution Model¶for Phase Transitions , 1999 .

[20]  V. V. Chepyzhov,et al.  Attractors of non-autonomous dynamical systems and their dimension , 1994 .

[21]  B. Zinner,et al.  Existence of traveling wavefront solutions for the discrete Nagumo equation , 1992 .

[22]  Thomas Erneux,et al.  Propagating waves in discrete bistable reaction-diffusion systems , 1993 .

[23]  Erik S. Van Vleck,et al.  Attractors for lattice Fitzhugh-Nagumo systems , 2005 .

[24]  Vladimir I. Nekorkin,et al.  CHAOS OF TRAVELING WAVES IN A DISCRETE CHAIN OF DIFFUSIVELY COUPLED MAPS , 1994 .

[25]  Alain Haraux,et al.  Attractors of asymptotically compact processes and applications to nonlinear partial differential equations , 1988 .

[26]  Peter S. Lomdahl,et al.  The discrete self-trapping equation , 1985 .

[27]  Peter W. Bates,et al.  Attractors for Lattice Dynamical Systems , 2001, Int. J. Bifurc. Chaos.

[28]  Ahmed Y. Abdallah.  Upper semicontinuity of the attractor for a second order latticedynamical system , 2005 .

[29]  J. Keener,et al.  The effects of discrete gap junction coupling on propagation in myocardium. , 1991, Journal of theoretical biology.

[30]  Standing wave instabilities in a chain of nonlinear coupled oscillators , 2001, nlin/0104025.