Massive evaluation and analysis of Poincar\'e recurrences on grids of initial data: a tool to map chaotic diffusion

TL;DR

This paper introduces a new numerical method based on Poincaré recurrence statistics for analyzing chaotic diffusion in Hamiltonian systems, offering a simple way to map local diffusion timescales and compare well with existing techniques.

Contribution

The paper presents a novel Poincaré recurrence method (PRM) for characterizing global chaotic behavior, providing a simpler alternative to existing tools like Lyapunov exponents.

Findings

PRM exposes global dynamics similarly to Lyapunov exponents.

PRM can construct approximate charts of local diffusion timescales.

The method is simple, straightforward, and applicable to both bounded and unbounded phase spaces.

Abstract

We present a novel numerical method aimed to characterize global behaviour, in particular chaotic diffusion, in dynamical systems. It is based on an analysis of the Poincar\'e recurrence statistics on massive grids of initial data or values of parameters. We concentrate on Hamiltonian systems, featuring the method separately for the cases of bounded and non-bounded phase spaces. The embodiments of the method in each of the cases are specific. We compare the performances of the proposed Poincar\'e recurrence method (PRM) and the custom Lyapunov exponent (LE) methods and show that they expose the global dynamics almost identically. However, a major advantage of the new method over the known global numerical tools, such as LE, FLI, MEGNO, and FA, is that it allows one to construct, in some approximation, charts of local diffusion timescales. Moreover, it is algorithmically simple and straightforward to apply.