Chaotic orbits for differentiable maps near anti-integrable limits

Abstract

In this work, we study the dynamical system which can be transformed into a difference equation of the form epsilon g(x(t-n2),...,x(t),..,x(t+n1) ;epsilon) + go (x(t)) = 0 where epsilon is an element of R is a parameter, g : Rn1+n2+1 -> R and go : R -> R. are both C-1 functions and n(1), n(2) are both nonnegative integers. Suppose that the function go has k simple zeros where k >= 2. We give criteria of existence of some kinds of chaotic orbits, and construct infinite ones explicitly. As applications of these results, we establish snap-back repellers and heteroclinical repellers for the modified Mira maps and transversal homoclinic orbits or transversal heteroclinic orbits for the Henon maps, the Arneodo-Coullet-Tresser maps and the quadratic volume preserving maps. (C) 2015 Elsevier Inc. All rights reserved.