April  2005, 13(3): 637-658. doi: 10.3934/dcds.2005.13.637

Patterns generation and transition matrices in multi-dimensional lattice models

1. 

The National Center for Theoretical Sciences, Hsinchu 300, Taiwan

2. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

Received  September 2004 Revised  January 2005 Published  May 2005

In this paper we develop a general approach for investigating pattern generation problems in multi-dimensional lattice models. Let $\mathcal S$ be a set of $p$ symbols or colors, $\mathbf Z_N$ a fixed finite rectangular sublattice of $\mathbf Z^d$, $d\geq 1$ and $N$ a $d$-tuple of positive integers. Functions $U:\mathbf Z^d\rightarrow \mathcal S$ and $U_N:\mathbf Z_N\rightarrow \mathcal S$ are called a global pattern and a local pattern on $\mathbf Z_N$, respectively. We introduce an ordering matrix $\mathbf X_N$ for $\Sigma_N$, the set of all local patterns on $\mathbf Z_N$. For a larger finite lattice , , we derive a recursion formula to obtain the ordering matrix of from $\mathbf X_N$. For a given basic admissible local patterns set $\mathcal B\subset \Sigma_N$, the transition matrix $\mathbf T_N(\mathcal B)$ is defined. For each , denoted by the set of all local patterns which can be generated from $\mathcal B$, the cardinal number of is the sum of entries of the transition matrix which can be obtained from $\mathbf T_N(\mathcal B)$ recursively. The spatial entropy $h(\mathcal B)$ can be obtained by computing the maximum eigenvalues of a sequence of transition matrices $\mathbf T_n(\mathcal B)$. The results can be applied to study the set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.
Citation: Jung-Chao Ban, Song-Sun Lin. Patterns generation and transition matrices in multi-dimensional lattice models. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 637-658. doi: 10.3934/dcds.2005.13.637
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