Topological horseshoe and uniform hyperbolicity of the symplectic coupled Hénon map

Abstract

We analyze the topological horseshoe and the uniform hyperbolicity of a four-dimensional symplectic map, which is introduced by coupling two two-dimensional symplectic Hénon maps via linear terms. Based on the cone field argument following Devaney and Nitecki [1], we first derive a sufficient condition for topological horseshoe and uniform hyperbolicity in the parameter space including the two different anti-integrable limits. We then explore uniformly hyperbolic parameter regions by applying the computer-assisted proof developed by Arai and find that there exist non-trivial uniformly hyperbolic regions in the parameter space that differ from those obtained using the cone condition. The existence of such non-trivial uniformly hyperbolic regions reminds us of the so-called hyperbolic plateaus in the two-dimensional Hénon map.