Stability and chaos in a class of finite-dimensional discrete spatiotemporal systems

This paper is concerned with a class of finite-dimensional discrete spatiotemporal systems of the form {(x₁ (m + 1, n) = f₁ (x₁ (m, n - 1), x₁ (m, n), x₂ (m, n), ..., x_k (m, n), x₁ (m, n + 1)); x₂ (m + 1, n) = f₂ (x₂ (m, n - 1), x₁ (m, n), x₂ (m, n), ..., x_k (m, n), x₂ (m, n + 1)); ⋯ ⋯ ⋯ ⋯; x_k (m + 1, n) = f_k (x_k (m, n - 1), x₁ (m, n), x₂ (m, n), ..., x_k (m, n), x_k (m, n + 1)),) where k > 0 is an integer, f_i : R^{k + 2} → R is a real function for all i = 1, 2, ..., k, m ∈ N₀ = {0, 1, 2, ...} and n ∈ Z = {..., - 1, 0, 1, ...} (or, n ∈ N₀ in some special cases). Definitions of chaos of this system in the sense of Devaney and of Li-Yorke are given. Some sufficient conditions for this system to be stable and some illustrative examples for this system to be chaotic in the sense of Devaney and of Li-Yorke, respectively, are derived. © 2008 Elsevier Ltd. All rights reserved.