Efficient Techniques to Cope with Chaotic Dynamics in Deterministic Systems

Abstract

In this work we review and improve two useful techniques to cope with chaotic dynamics in deter-ministic systems, namely the Mean Exponential Growth factor of Nearby Orbits (MEGNO) and the Shan-non entropy. The MEGNO provides a direct measure of the hyperbolic dynamics in an arbitrary small neigh-borhood of a given point of the phase space in comparatively short motion times and the maximum Lyapunovexponent (or its spectrum) can be easily derived from this fast dynamical indicator which has become a wide-spread tool in the investigation of the global dynamics in planetary systems. The time derivative of the Shan-non entropy yields a confident measure of the diffusion speed in comparison with the usual approach of theaction-like variance evolution. It has been successfully applied in different dynamical systems, particularly, inexoplanetary systems. A brief discussion concerning the relationship among the Shannon entropy and theKolmogorov–Sinai or metric entropy and the topological entropy is also addressed. Both methods allow toget two relevant timescales in chaotic dynamics, the Lyapunov time and the diffusion time. An application toa simple 4D symplectic map illustrates the efficiency of both techniques.