Estimation of the Amplitude of Resonance in the General Standard Map

Abstract

This paper formulates some conjectures about the amplitude of resonance in the General Standard Map. The main idea is to expand the periodic perturbation function in Fourier series. Given any rational rotation number, we choose a finite number of harmonics in the Fourier expansion and we compute the amplitude of resonance of the reduced perturbation function of the map, using a suitable normal form around the resonance, which is valid for asymptotically small values of the perturbation parameter. For this map, we obtain a relation between the amplitude of resonance and the perturbation parameter: the amplitude is proportional to a rational power of the parameter, and so can be represented as a straight line on a log-log graph. The convex hull of these straight lines gives a lower bound for the amplitude of resonance, valid even when the perturbation parameter is of the order of 1. We find that some perturbation functions give rise a phenomenon that we call collapse of resonance; this means that the amplitude of resonance goes to zero for some value of the perturbation parameter. We find an empirical procedure to estimate this value of the parameter related to the collapse of resonance.