Copyright © 2012 Qingdu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on Smale's famous example, provide a rigorous way to study chaos in dynamical systems. The paper is devoted to constructing them in two-dimensional diffeomorphisms with the existence of transversal homoclinic saddles. We first propose an algorithm to automatically construct “horizontal” and “vertical” sides of the curvilinear rectangle near to segments of the stable and of the unstable manifolds, respectively, and then apply it to four classical chaotic maps (the Duffing map, the Hénon map, the Ikeda map, and the Lozi map) to verify its effectiveness.