American Institute of Mathematical Sciences

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 ${\mathbf Z}_{\scriptsize\tilde{N}}$, ${\small \tilde{N}\geq N}$, we derive a recursion formula to obtain the ordering matrix ${\mathbf X}_{\scriptsize\tilde{N}}$ of $\Sigma_{\scriptsize\tilde{N}}$ 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 $\scriptsize{\tilde{N}\geq N}$, denoted by $\Sigma_{\scriptsize\tilde{N}} (\mathcal B)$ the set of all local patterns which can be generated from $\mathcal B$, the cardinal number of $\Sigma_{\scriptsize\tilde{N}} (\mathcal B)$ is the sum of entries of the transition matrix ${\mathbf T}_{\scriptsize\tilde{N}} (\mathcal B)$ 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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