Breakdown of tori in symplectic maps

Abstract

Most of physical phenomena can be explained in terms of Hamiltonian systems. These continuous dynamical systems can be related with symplectic maps. Under certain hypothesis one can see that these maps present some invariant tori. Then, it is really interesting to understand how these tori behave. One of the most important properties these tori present is that they persist under small perturbation of our initial systems but that for higher perturbations they are going to break down. These perturbations are usually related with equations depending on a parameter, K. For 2D symplectic maps, renormalization techniques allow to understand the mechanisms concerning the destruction of invariant circles. Rotation numbers of these circles play a key role in the analysis of their breakdown. Throughout this work we will show some of the most important tools to deal with these invariant circles.