Patterns generation and transition matrices in multi-dimensional lattice models

Abstract

In this paper we develop a general approach for investigating pattern generation problems in multi-dimensional lattice models. Let S be a set of p symbols or colors, ZN a fixed finite rectangular sublattice of Zd, d ≥ 1 and N a d-tuple of positive integers. Functions U : Zd → S and UN : ZN → S are called a global pattern and a local pattern on ZN , respectively. We introduce an ordering matrix XN for ΣN , the set of all local patterns on ZN . For a larger finite lattice ZN˜ , N˜ ≥ N, we derive a recursion formula to obtain the ordering matrix XN˜ of ΣN˜ from XN . For a given basic admissible local patterns set B ⊂ ΣN , the transition matrix TN (B) is defined. For each N˜ ≥ N denoted by ΣN˜ (B) the set of all local patterns which can be generated from B, the cardinal number of ΣN˜ (B) is the sum of entries of the transition matrix TN˜ (B) which can be obtained from TN (B) recursively. The spatial entropy h(B) can be obtained by computing the maximum eigenvalues of a sequence of transition matrices Tn(B). The results can be applied to study the set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.