Influence of Boundary Conditions on Pattern Formation and Spatial Chaos in Lattice Systems

This work elucidates the spatial structure of lattice dynamical systems, which is represented by equilibria of the systems. On a finite lattice, various boundary conditions are imposed. The effect from these boundary conditions on formation of pattern as well as spatial complexity, as the lattice size tends to infinity, is investigated. Two general propositions are proposed as a criteria to demonstrate that this effect is negligible. To illustrate the effectiveness of these criteria, the mosaic patterns in a cellular neural network model on one- and two-dimensional lattices are also studied. On a one-dimensional lattice, the influence of boundary conditions on pattern formation and spatial chaos for mosaic patterns is negligible. This result is justified by verifying the above-mentioned criteria and by using the transition matrices. These appropriately formulated matrices generate all the mosaic patterns on a one-dimensional infinite lattice and on any one-dimensional finite lattice with boundary conditio...

[1]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[2]  John Mallet-Paret,et al.  Pattern Formation and Spatial Chaos in Spatially Discrete Evolution Equations , 1995 .

[3]  A. Mielke.  The complex Ginzburg - Landau equation on large and unbounded domains: sharper bounds and attractors , 1997 .

[4]  Global continuation of branches of nondegenerate solutions , 1986 .

[5]  J. Rogers.  Chaos , 1876 .

[6]  P. Thiran,et al.  Influence of boundary conditions on the behavior of cellular neural networks , 1993 .

[7]  L. Bunimovich,et al.  Density of defects and spatial entropy in extended systems , 1995 .

[8]  Jonq Juang,et al.  Cellular Neural Networks: Mosaic Pattern and Spatial Chaos , 2000, SIAM J. Appl. Math..

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

[10]  Leon O. Chua,et al.  Multiple Layer Cellular Neural Networks: A Tutorial , 1991 .

[11]  Sze-Bi Hsu,et al.  Lectures on Chaotic Dynamical Systems , 2002 .

[12]  Chih-Wen Shih,et al.  Complete Stability for Standard Cellular Neural Networks , 1999 .

[13]  Martin Golubitsky,et al.  Boundary conditions and mode jumping in the buckling of a rectangular plate , 1979 .

[14]  Firth.  Optical memory and spatial chaos. , 1988, Physical review letters.

[15]  New trends in nonlinear dynamics and pattern-forming phenomena : the geometry of nonequilibrium , 1990 .

[16]  Shui-Nee Chow,et al.  Stability and Bifurcation of Traveling Wave Solutions in Coupled Map Lattices , 1994 .

[17]  J. Hale,et al.  Varying Boundary Conditions with Large Diffusivity , 1985 .

[18]  John W. Cahn,et al.  Theory of crystal growth and interface motion in crystalline materials , 1960 .

[19]  John Mallet-Paret,et al.  Pattern Formation and Spatial Chaos in Lattice Dynamical Systems: I , 1995 .

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

[21]  B. Fiedler.  Do global attractors depend on boundary conditions? , 1996, Documenta Mathematica.

[22]  Giulio Casati,et al.  Chaos, Order, and Patterns , 1991 .

[23]  Shui-Nee Chow,et al.  Spatially Discrete Nonlinear Diffusion Equations , 1995 .

[24]  Leon O. Chua,et al.  Pattern formation properties of autonomous Cellular Neural Networks , 1995 .

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

[26]  Shui-Nee Chow,et al.  DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS , 1996 .

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

[28]  Firth Wj,et al.  Optical memory and spatial chaos. , 1988 .

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

[30]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[32]  Patrick K. Simpson.  Neural Networks Applications , 1997 .

[33]  H. E. Cook,et al.  A model for diffusion on cubic lattices and its application to the early stages of ordering , 1969 .

[34]  Leon O. Chua,et al.  CNN: A Vision of Complexity , 1997 .

[35]  Robert Dillon,et al.  Pattern formation in generalized Turing systems , 1994 .

[36]  Chih-Wen Shih,et al.  Pattern Formation and Spatial Chaos for Cellular Neural Networks with Asymmetric Templates , 1998 .

[37]  Leon O. Chua,et al.  Characterization and dynamics of pattern formation in cellular neural networks , 1996 .

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

[39]  Variation of bifurcations along a homotopy from Neumann to Dirichlet problems , 1996 .

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

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