Chaos arising near a topologically transversal homoclinic set

Abstract

A diffeomorphism on a $C^1$-smooth manifold is studied possessing a hyperbolic fixed point. If the stable and unstable manifolds of the hyperbolic fixed point have a nontrivial local topological crossing then a chaotic behaviour of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of invariant manifolds for the perturbed diffeomorphism.