In this paper, we study solutions of difference equations λ (y n , y n+1 ,. .. , y n+m) = 0, n ∈ Z, of order m with parameter λ, and consider the case when λ has a singular limit depending on a single variable as λ → λ 0 , i.e. λ 0 (y 0 ,. .. , y m) = ϕ(y N), where N is an integer with 0 N m and ϕ is a function. We prove that if ϕ has k simple zeros then for λ close enough to λ 0 , the difference equation has a k-horseshoe among its solutions, that is, the dynamics is conjugate to the full shift with k symbols. Moreover, we show that these horseshoes change continuously in the uniform topology as λ varies. As applications of these results, we establish the horseshoe structure in families of generalized Hénon-like maps and of Arneodo-Coullet-Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.

We consider difference equations λ (y n , y n+1 ,. .. , y n+m) = 0, n ∈ Z, of order m with parameter λ close to that exceptional value λ 0 for which the function depends on two variables: λ 0 (x 0 ,. .. , x m) = ξ(x N , x N +L) 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 h top (ϕ). Under these assumptions we prove that in the set of bi-infinite solutions of the difference equation with λ in some neighbourhood of λ 0 , there is a closed (in the product topology) invariant set to which the restriction of the shift map has topological entropy arbitrarily close to h top (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 R 3 and for several lattice models including the generalized cellular neural networks (CNNs), the time discrete version of the CNNs and coupled Chua's circuit.

We study hyperbolic dynamics and bifurcations for generalized Hénon maps in the form x = y, y = γy(1 − y) − bx + αxy (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.