Analysis of Singularities of the Standard Map Conjugation Function

Abstract

There are many different approaches to the determination of the breakdown of the invariant curves and the resulting stochastic transition for the area-preserving maps but essentially we can divide them in two classes. Given an invariant curve it is possible to detect its breakdown either analyzing the behaviour of the nearby resonance orbits, using the overlap criteria ¹ or the Greene’s method ², or studying directly the properties of this curve^3,4. The Kam theorem^5,6 tells us that the diophantine rotation number orbits of a slightly perturbed Hamiltonian system are analytically conjugated to the curves with the same winding number of the unperturbed system, until a certain value of the perturbation parameter is reached. The analytical function Φ that transforms the perturbed tori to the unperturbed ones is called conjugation function; it depends on the winding number and it is analytical in ε, strength of the perturbation, until ε _c , critical value. So a method to find the critical value at which the invariant curve with rotation number ω breakdown is to study the loss of analyticity of Φ.