Structure of the attracting set of a piecewise linear Hénon mapping

Abstract

Detailed structure of the attracting set of the piecewise linear Hénon mapping (x,y)→(1−a|x|+by,x) with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that (1) the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and (2) the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A. Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.