Chaotic difference equations in two variables and their multidimensional perturbations

Abstract

We consider difference equations Φλ(yn, yn+1, …, yn+m) = 0, , of order m with parameter λ close to that exceptional value λ0 for which the function Φ depends on two variables: with 0 ⩽ N, N + L ⩽ m. It is also assumed that for the equation ξ(x, y) = 0, there is a branch y = φ(x) with positive topological entropy htop(φ). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with λ in some neighbourhood of λ0, there is a closed (in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to htop(φ)/|L|, and moreover, orbits of this invariant set depend continuously on λ not only in the product topology but also in the uniform topology. We then apply this result to establish chaotic behaviour for Arneodo–Coullet–Tresser maps near degenerate ones, for quadratic volume preserving automorphisms of and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua's circuit.