Covariant Lyapunov vectors as global solutions of a partial differential equation on the phase space

TL;DR

This paper introduces covariant Lyapunov fields as solutions to a differential equation on phase space, providing a geometric and gauge-invariant framework for analyzing dynamical systems, exemplified by a geodesic flow model.

Contribution

It establishes a novel differential equation characterization of covariant Lyapunov vectors with gauge invariance and geometric interpretation in ergodic systems.

Findings

Covariant Lyapunov fields are solutions to a gauge-invariant differential equation.

Each foliation containing trajectories corresponds to a Lyapunov exponent.

Explicit solution demonstrated for a geodesic flow model.

Abstract

As a new tool to describe the behaviour of a dynamical system, we introduce the concept of "covariant Lyapunov field", i.e. a field which assigns all the components of covariant Lyapunov vectors at almost all points of the phase space. We focus on the case in which these fields are overall continuous and also differentiable along individual trajectories. We show that in ergodic systems such fields can be characterized as the global solutions of a differential equation on the phase space. Due to the arbitrariness in the choice of a multiplicative scalar factor for the Lyapunov vector at each point of the phase space, this differential equation exhibits a gauge invariance that is formally analogous to that of quantum electrodynamics. Under the hypothesis that the covariant Lyapunov field is overall differentiable, we give a geometric interpretation of our result: each 2-dimensional foliation of the space that contains whole trajectories is univocally associated with a Lyapunov exponent, and the corresponding covariant Lyapunov field is one of the generators of the foliation. In order to show with an example how this new approach can be applied to the study of concrete dynamical systems, we display an explicit solution of the differential equations that we have obtained for the covariant Lyapunov fields in a model involving a geodesic flow.