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 dynamicsof a bi-parametric symplectic map, the so-called rational standard map, at moderate-to-largevalues of the amplitude parameter is addressed. After reviewing the model, a discussionconcerning an analytical determination of the maximum Lyapunov exponent is providedtogether with thorough numerical experiments. The theoretical results are obtained in thelimit of a nearly uniform distribution of the phase values. Correlations among phases lead todepartures from the expected estimates. In this direction, a detailed study of the role of stableperiodic islands of periods 1, 2 and 4 is included. Finally, an experimental relationship betweenthe Lyapunov and instability times is shown, while an analytical one applies when correlationsare irrelevant, which is the case, in general, for large values of the amplitude parameter.