Revisiting the relation between the Lyapunov time and the instability time

Abstract

In this effort we focus on the so-called TL−Tinst relationship i.e., any relation among the Lyapunov time and a characteristic instability time of a given dynamical system. By means of extensive numerical simulations with a high-dimensional dynamical system, a 4D symplectic map, we investigate a possible correlation between both time-scales. Herein the instability time is the one associated to diffusion along the homoclinic tangle of the resonances of the system. We found that different laws could fit the computed values, depending mostly on the dynamics of the system when varying the involved parameters; in some small domain of the parameter space a power law appears while in a larger one an exponential relation fits quite well the computed values of TL and Tinst. We compare the obtained functional forms of the relationships with those known for lower-dimensional systems and identify typical functional dependences, confirmed analytically.