Diffusion and Lyapunov timescales in the Arnold model

Abstract

In the present work, we focus on two dynamical timescales in the Arnold Hamiltonian model: the Lyapunov time and the diffusion time when the system is confined to the stochastic layer of its dominant resonance (guiding resonance). Following Chirikov's formulation, the model is revisited, and a discussion about the main assumptions behind the analytical estimates for the diffusion rate is given. On the other hand, and in line with Chirikov's ideas, theoretical estimations of the Lyapunov time are derived. Later on, three series of numerical experiments are presented for various sets of values of the model parameters, where both timescales are computed. Comparisons between the analytical estimates and the numerical determinations are provided whenever the parameters are not too large, and those cases are in fact in agreement. In particular, the case in which both parameters are equal is numerically investigated. Relationships between the diffusion time and the Lyapunov time are established, like an exponential law or a power law in the case of large values of the parameters.