Topological horseshoes for perturbations of singular difference equations

Abstract

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.

Habit formation and chaotic dynamics in an n-dimensional cash-in-advance economy

“…When the persistent habits last for 2 periods, we analyze the possibility of chaos in the horseshoe structure which provides a sufficient condition for the occurrence of entropic chaos. When persistent habits last for n periods, no graphic methodology can be used and we utilize the theorems developed by Juang et al [17] and Li and Malkin [19] to provide a sufficient condition for the existence of entropic chaos. This result in the last case can be applied to any positive integer n. …”

“…It shows that the critical value θ * * depends on the length of the persistent habits (n). To study the possibility of chaos for an n-D dynamical system, we modify the theorems of [17] and [19] to prove the occurrence of entropic chaos. 10 …”

“…Then on [B, C] n+1 , the function Φ η is C 1 for each η and is continuous in η 10 A simple version of the theorems of [17] and [19] is given in the Appendix. and so are the partial derivatives ∂ i Φ η for 1 ≤ i ≤ n + 1.…”

“…Our goal in the present paper is to give rigorous proofs (instead of numerical experiments) for conditions generating entropic chaos. Hence, we adopt a newly-developed method by Juang et al [17] and Li and Malkin [19] to approximate an n-D dynamical system by using a 1-D dynamical system and study the possibility of entropic chaos; also refer to Li et al [20].…”

Chaotic dynamics in an overlapping generations model with myopic and adaptive expectations

“…Because it is much more difficult to study a 2D dynamical system than a 1D dynamical system, we apply a recent method developed by Juang et al (2005) and Li and Malkin (2006) to approximate a 2D dynamical system by using a 1D dynamical system and identify the existence of entropic chaos (positive topological entropy) when both the inter-temporal elasticity of substitution and the total factor productivity are sufficiently high and when agents rely heavily on current information to form their expectations.…”