A Condition for the Existence of Homoclinic Intersection in C2 Standard-Like Mappings

Abstract

Area-preserving twist mappings1)-3) on the cylinder possess many important properties. We are interested in the structure of stable and unstable manifolds of Birkhoff saddles.4) Chaotic behavior appears due to the existence of Smale horseshoes5) produced by the transversal intersection of stable and unstable manifolds. The proof of the existence of homoclinic intersections is one of the most fundamental issues in this field. If the analytical form of the mapping is given, we can apply several methods to prove the existence of the homoclinic intersection of stable and unstable manifolds of the fixed point.6)-10) Without using the explicit expression of the function in the mapping, can we prove the existence of a homoclinic intersection? In order to consider this problem, we study C2 mappings. In C2 mappings, we can use information on the first and second derivatives of the mapping function. We give a simple sufficient condition for the existence of the homoclinic intersection of stable and unstable manifolds of the fixed point. We use the method developed by Brown,11) who proved the existence of a homoclinic intersection in the conservative Henon mapping.12) He used the mapping of the curvature of the unstable manifold. In this paper, we use the mappings of the first and second derivatives of the unstable manifold, since these mappings are simpler than that of the curvature. We also use the symmetrical structure of stable and unstable manifolds.13) In §1.1, we introduce the system and state a theorem. Discussion is given in §1.2. We give the proof of the theorem in §2.