Numerical and Theoretical Studies on the Rational Standard Map at Moderate-to-Large Values of the Amplitude Parameter

Abstract

In this work an exhaustive numerical and analytical investigation of the dynamics of a bi-parametric symplectic map, the so-called rational standard map, at moderate-to-large values of the amplitude parameter is addressed. After reviewing the model, a discussion concerning an analytical determination of the maximum Lyapunov exponent is provided together with thorough numerical experiments. The theoretical results are obtained in the limit of a nearly uniform distribution of the phase values. Correlations among phases lead to departures from the expected estimates. In this direction, a detailed study of the role of stable periodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship between the Lyapunov and instability times is shown, while an analytical one applies when correlations are irrelevant, which is the case, in general, for large values of the amplitude parameter.