Chaos and Diffusion in Dynamical Systems Through Stable-Unstable Manifolds

The phase-space structure of conservative non-integrable dynamical systems is characterized by a mixture of stable invariant sets and unstable structures which possibly support diffusion. In these situation, many practical and theoretical questions are related to the problem of finding orbits which connect the neighbourhoods of two points A and B of the phase-space. Hyperbolic dynamics has provided in the last decades many tools to tackle the problem related to the existence and the properties of the so called stable and unstable manifolds, which provide natural paths for the diffusion of orbits in the phase-space. In this article we review some basic results of hyperbolic dynamics, through the analysis of the stable and unstable manifolds in basic mathematical models, such as the symplectic standard map, up to more complicate models related to the Arnold diffusion.