Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction

TL;DR

This paper investigates the local structure of pseudohyperbolic chaotic attractors using angle distributions and introduces instant Lyapunov exponents to analyze volume expansion and contraction on infinitesimal time scales.

Contribution

It introduces a method to verify pseudohyperbolicity via angle distributions and defines instant Lyapunov exponents to study local volume dynamics in chaotic attractors.

Findings

Pseudohyperbolic attractors lack tangencies between tangent subspaces.

Instant Lyapunov exponents reveal local volume expansion and contraction.

Average properties of pseudohyperbolicity are often violated on infinitesimal time scales.

Abstract

In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite time Lyapunov exponents we demonstrate that specific to the pseudohyperbolicity average expanding and contracting properties are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.