Application of the Slater criteria to localize invariant tori in Hamiltonian mappings

Abstract

We investigate the localization of invariant spanning curves for a family of two-dimensional area-preserving mappings described by the dynamical variables I and θ by using Slater's criterion. The Slater theorem says there are three different return times for an irrational translation over a circle in a given interval. The returning time, which measures the number of iterations a map needs to return to a given periodic or quasi periodic region, has three responses along an invariant spanning curve. They are related to a continued fraction expansion used in the translation and obey the Fibonacci sequence. The rotation numbers for such curves are related to a noble number, leading to a devil's staircase structure. The behavior of the rotation number as a function of invariant spanning curves located by Slater's criterion resulted in an expression of a power law in which the absolute value of the exponent is equal to the control parameter γ that controls the speed of the divergence of θ in the limit the action I is sufficiently small.