Existence of homoclinic orbits of an area-preserving map with a nonhyperbolic structure

Abstract

Basic phenomena in chaos can be associated with the existence of homoclinic orbits. In this paper we provide a simple mathematical example of an existence of homoclinic orbits embedded on a nonhyperbolic invariant set. To do it, we study a two-dimensional area-preserving piecewise linear map. Exploiting its dynamical behaviors from the point of view of open conservative systems, we analytically show that outside of the existing nonhyperbolic invariant set which is enclosed by heteroclinic saddle connections of an unstable periodic orbit, dynamical behaviors exhibit unbounded, but inside of such invariant set, regular and chaotic motions. Moreover, on such the invariant set, we theoretically present the existence of a homoclinic orbit by proving homoclinic intersections of the stable and unstable manifolds of a saddle point and also provide numerical evidences for geometrical structures of the nonhyperbolic invariant set and dynamical behaviors.