Efficient Techniques to Cope with Chaotic Dynamics in Deterministic Systems

In this work we review and improve two useful techniques to cope with chaotic dynamics in deterministic systems, namely the Mean Exponential Growth factor of Nearby Orbits (MEGNO) and the Shannon entropy. The MEGNO provides a direct measure of the hyperbolic dynamics in an arbitrary small neighborhood of a given point of the phase space in comparatively short motion times and the maximum Lyapunov exponent (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 Shannon entropy yields a confident measure of the diffusion speed in comparison with the usual approach of the action-like variance evolution. It has been successfully applied in different dynamical systems, particularly, in exoplanetary systems. A brief discussion concerning the relationship among the Shannon entropy and the Kolmogorov–Sinai or metric entropy and the topological entropy is also addressed. Both methods allow to get two relevant timescales in chaotic dynamics, the Lyapunov time and the diffusion time. An application to a simple 4D symplectic map illustrates the efficiency of both techniques.