THE MEASURE OF CHAOS BY THE NUMERICAL-ANALYSIS OF THE FUNDAMENTAL FREQUENCIES - APPLICATION TO THE STANDARD MAPPING

The method of analysis of the chaotic behaviour of a dynamical system by the numerical analysis of the fundamental frequencies developed for the study of the stability of the solar system (J. Laskar, Icarus 88, 1990) is presented here with application to the standard mapping. This method is well suited for weakly chaotic motion with any number of degrees of freedom and is based on the analysis of the variations with time of the fundamental frequencies of an hamiltonian system. It allows to give an analytical representation of the solution when it is regular, to detect if an orbit is chaotic over a smaller time span than with the Lyapunov exponents and gives also an estimate of the size of the chaotic zones in the frequency domain. The frequency analysis also provides a numerical criterion for the destruction of invariant curves. Its application to the standard mapping shows that the golden curve does not survive for a = 0.9718 which is very close and compatible with Greene's value a(c) = 0.971635.