Lyapunov Analysis of Strange Pseudohyperbolic Attractors: Angles Between Tangent Subspaces, Local Volume Expansion and Contraction

Abstract

Pseudohyperbolic attractors are genuine strange chaotic attractors. They do not contain stable periodic orbits and are robust in the sense that such orbits do not appear under variations. The tangent space of these attractors is split into a direct sum of volume expanding and contracting subspaces and these subspaces never have tangencies with each other. Any contraction in the first subspace, if it occurs, is weaker than contractions in the second one. In this paper we analyze the local structure of several chaotic attractors recently suggested in the 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 average expanding and contracting properties specific to pseudohyperbolicity 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.