Estimating hyperbolicity of chaotic bidimensional maps

TL;DR

This paper applies a numerical method to estimate hyperbolicity in bidimensional chaotic maps by analyzing covariant Lyapunov vectors and their angles, highlighting how coordinate changes affect transversality measures.

Contribution

It demonstrates the application of Ginelli et al.'s method to bidimensional maps and investigates how coordinate transformations influence transversality statistics.

Findings

Angles between covariant Lyapunov vectors can quantify hyperbolicity.

Coordinate transformations deform splitting angles and alter their probability densities.

Complete tangencies remain invariant despite coordinate changes.

Abstract

We apply to bidimensional chaotic maps the numerical method proposed by Ginelli et al. to approximate the associated Oseledets splitting, i.e. the set of linear subspaces spanned by the so called covariant Lyapunov vectors (CLV) and corresponding to the Lyapunov spectrum. These subspaces are the analog of linearized invariant manifolds for non-periodic points, so the angles between them can be used to quantify the degree of hyperbolicity of generic orbits; however, being such splitting non invariant under smooth transformations of phase space, it is interesting to investigate the properties of transversality when coordinates change, e.g. to study it in distinct dynamical systems. To illustrate this issue on the Chirikov-Taylor standard map we compare the probability densities of transversality for two different coordinate systems; these are connected by a linear transformation that deforms splitting angles through phase space, changing also the probability density of almost-zero angles although complete tangencies are in fact invariant. This is completely due to the PDF transformation law and strongly suggests that any statistical inference from such distributions must be generally taken with care.