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

• Main Authors: Shevchenko, Ivan I, Rollin, Guillaume, Melnikov, Alexander V, Lages, José
• Format: Journal Article
• Language: English
• Published: 26-08-2019
• Subjects: Physics - Chaotic Dynamics; Physics - Earth and Planetary Astrophysics; Physics - Instrumentation and Methods for Astrophysics
• DOI: 10.48550/arxiv.1908.09683

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.