Non-autonomous lattice systems with switching effects and delayed recovery

Abstract The long term behavior of a type of non-autonomous lattice dynamical systems is investigated, where these have a diffusive nearest neighborhood interaction and discontinuous reaction terms with recoverable delays. This problem is of both biological and mathematical interests, due to its application in systems of excitable cells as well as general biological systems involving delayed recovery. The problem is formulated as an evolution inclusion with delays and the existence of weak and strong solutions is established. It is then shown that the solutions generate a set-valued non-autonomous dynamical system and that this non-autonomous dynamical system possesses a non-autonomous global pullback attractor.

[1]  R. W. Joyner,et al.  Cellular mechanisms of delayed recovery of excitability in ventricular tissue. , 1991, The American journal of physiology.

[2]  Pullback attractors for a class of nonlinear lattices with delays , 2015 .

[3]  Shui-Nee Chow,et al.  Traveling Waves in Lattice Dynamical Systems , 1998 .

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

[5]  Luis L. Bonilla,et al.  Pulse Propagation in Discrete Systems of Coupled Excitable Cells , 2003, SIAM J. Appl. Math..

[6]  John Mallet-Paret,et al.  The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems , 1999 .

[7]  Exponential attractors for lattice dynamical systems in weightedspaces , 2011 .

[8]  Thomas Erneux,et al.  Propagation failure in arrays of coupled bistable chemical reactors , 1992 .

[9]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[10]  Peter E. Kloeden,et al.  Pullback attractors of reaction-diffusion inclusions with space-dependent delay , 2017 .

[11]  Leon O. Chua,et al.  Edge of Chaos and Local Activity Domain of FitzHugh-Nagumo Equation , 1998 .

[12]  Józef Banaś,et al.  Measures of Noncompactness in Banach Spaces , 1980 .

[13]  Ahmed Y. Abdallah.  Uniform exponential attractors for first order non-autonomous lattice dynamical systems , 2011 .

[14]  Leon O. Chua,et al.  Spiral waves on a 2-D array of nonlinear circuits , 1993 .

[15]  C. Castaing,et al.  Convex analysis and measurable multifunctions , 1977 .

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

[17]  Shengfan Zhou.  Attractors for Second Order Lattice Dynamical Systems , 2002 .

[18]  T. Caraballo,et al.  Attractors for non-autonomous retarded lattice dynamical systems , 2015 .

[19]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[20]  Wenxian Shen,et al.  Lifted Lattices, Hyperbolic Structures, and Topological Disorders in Coupled Map Lattices , 1996, SIAM J. Appl. Math..

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

[22]  Raymond Kapral,et al.  Discrete models for chemically reacting systems , 1991 .

[23]  G. Walter.  Hermite series solutions of differential equations , 1971 .

[24]  Xiaoying Han,et al.  Pullback Exponential Attractors for Non-autonomous Lattice Systems , 2012, Journal of Dynamics and Differential Equations.

[25]  P. Kloeden,et al.  Reaction-diffusion equations with a switched--off reaction zone , 2014 .

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

[27]  Bixiang Wang,et al.  Asymptotic behavior of non-autonomous lattice systems , 2007 .

[28]  M. Shipston,et al.  Alternative splicing of potassium channels: a dynamic switch of cellular excitability. , 2001, Trends in cell biology.

[29]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[30]  T. Caraballo,et al.  On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems , 2013 .

[31]  Ahmed Y. Abdallah.  Uniform global attractors for first order non-autonomous lattice dynamical systems , 2010 .

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

[33]  P. Kloeden,et al.  Non-autonomous attractors for integro-differential evolution equations , 2009 .

[34]  Francisco Morillas,et al.  Peano's Theorem and attractors for Lattice Dynamical Systems , 2009, Int. J. Bifurc. Chaos.

[35]  Wolf-Jürgen Beyn,et al.  Attractors of Reaction Diffusion Systems on Infinite Lattices , 2003 .

[36]  A C SCOTT.  ANALYSIS OF A MYELINATED NERVE MODEL. , 1964, The Bulletin of mathematical biophysics.

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

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

[39]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[40]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

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

[42]  Peter E. Kloeden,et al.  Nonautonomous Dynamical Systems , 2011 .

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

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

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

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

[47]  Shengfan Zhou,et al.  Attractors for first order dissipative lattice dynamical systems , 2003 .