On the timescales in the chaotic dynamics of a 4D symplectic map.

In this work, we investigate different timescales of chaotic dynamics in a multi-parametric 4D symplectic map. We compute the Lyapunov time and a macroscopic timescale, the instability time, for a wide range of values of the system's parameters and many different ensembles of initial conditions in resonant domains. The instability time is obtained by plain numerical simulations and by its estimates from the diffusion time, which we derive in three different ways: through a normal and an anomalous diffusion law and by the Shannon entropy, whose formulation is briefly revisited.

Estimation of diffusion time with the Shannon entropy approach.

The present work revisits and improves the Shannon entropy approach when applied to the estimation of an instability timescale for chaotic diffusion in multidimensional Hamiltonian systems. This formulation has already been proved efficient in deriving the diffusion timescale in 4D symplectic maps and planetary systems, when the diffusion proceeds along the chaotic layers of the resonance's web. Herein the technique is used to estimate the diffusion rate in the Arnold model, i.

Diffusion and Lyapunov timescales in the Arnold model.

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.

Stochastic approach to diffusion inside the chaotic layer of a resonance.

We model chaotic diffusion in a symplectic four-dimensional (4D) map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus on the diffusion process in the action I of the FR, obtaining a seminumerical method to compute the diffusion coefficient.