Abstract
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.
Highlights
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
Many systems have been studied as models for spatial pattern formation in biology, chemistry, engineering and physics
Notable examples include models arising from biology[7, 8, 21, 22, 23, 33, 34, 35], chemical reaction and phase transitions [4, 5, 11, 12, 13, 14, 24, 41, 43], image processing and pattern recognition [11, 12, 15, 16, 17, 18, 19, 25, 40], as well as materials science[9, 20, 26]
Summary
For a given basic admissible local patterns set B ⊂ ΣN , the transition matrix. For each N ≥ N denote 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).
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