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

Abstract

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 conditions. On a two-dimensional lattice, two illustrative examples demonstrate that the boundary effect can be dominant. The results and analysis in this investigation have significant implications for circuit design in cellular neural networks.