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

Massimo Marino, Doriano Brogioli

TL;DR

This work introduces covariant Lyapunov fields (CLFs), continuous vector fields that assign covariant Lyapunov vectors (CLVs) consistent with a Lyapunov exponent $\\lambda$ across a domain of phase space. It proves that, for ergodic systems, CLFs are global solutions of a first-order differential equation on phase space, $\\mathcal{L}_{\\boldsymbol{F}}\\boldsymbol{v} + b\\boldsymbol{v} = 0$, with $\\lambda = \langle b \rangle$, and demonstrates a gauge invariance under rescaling of $\\boldsymbol{v}$ analogous to electromagnetism. When CLFs are normalized by a metric, they satisfy a nonlinear equation $\\mathcal{L}_{\\boldsymbol{F}}\\boldsymbol{w} + c\\boldsymbol{w} = 0$ with $c = \frac{1}{2} w^{\\mu} (\\mathcal{L}_{\\boldsymbol{F}} g_{\\mu\\nu}) w^{\\nu}$, yielding $\\lambda = \langle c \rangle$; the paper also provides a geometric interpretation via 2D foliations and an explicit Hadamard-Gutzwiller geodesic-flow example. This framework offers a norm-free, metric-invariant perspective on CLVs, connects Lyapunov exponents to phase-space geometry, and suggests potential new definitions of LE alongside applications to concrete dynamical systems. The results bridge dynamical systems theory with differential geometry, offering a novel lens to study stability structures beyond traditional LE calculations.

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.

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

TL;DR

This work introduces covariant Lyapunov fields (CLFs), continuous vector fields that assign covariant Lyapunov vectors (CLVs) consistent with a Lyapunov exponent across a domain of phase space. It proves that, for ergodic systems, CLFs are global solutions of a first-order differential equation on phase space, , with , and demonstrates a gauge invariance under rescaling of analogous to electromagnetism. When CLFs are normalized by a metric, they satisfy a nonlinear equation with , yielding ; the paper also provides a geometric interpretation via 2D foliations and an explicit Hadamard-Gutzwiller geodesic-flow example. This framework offers a norm-free, metric-invariant perspective on CLVs, connects Lyapunov exponents to phase-space geometry, and suggests potential new definitions of LE alongside applications to concrete dynamical systems. The results bridge dynamical systems theory with differential geometry, offering a novel lens to study stability structures beyond traditional LE calculations.

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.
Paper Structure (13 sections, 11 theorems, 94 equations, 2 figures)

This paper contains 13 sections, 11 theorems, 94 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of covariant Lyapunov fields
  4. A global equation characterizing the covariant Lyapunov fields
  5. Discussion of the results
  6. The equation for normalized covariant Lyapunov fields
  7. Geometrical interpretation
  8. An example of application of the differential equation for the CLFs
  9. Conclusion
  10. Conflict of interest
  11. Data availability statement
  12. Ethics statement
  13. Acknowledgments

Key Result

Lemma 4.1

Let $\boldsymbol{F}(\boldsymbol{x})$ be a differentiable vector field on a manifold $X$, and let $\gamma$ be a trajectory defined by a function $\boldsymbol{x}(t)$ satisfying Eq. (eq:xp) for $-\infty<t<+\infty$. Let us suppose that $\boldsymbol{v}(t)$ and $b(t)$ are respectively a vector and a scala Then, for any arbitrary nonvanishing smooth scalar function $a(t)$, the vector function satisfies

Figures (2)

  • Figure 1: Covariant Lyapunov vectors of the geodesic flow on a genus-2 hyperbolic surface of constant negative curvature. The graph shows the section at $\Re{z}=0$, with coordinates $\Im{z}$, $\vartheta$. The arrows are the projections on the section of the covariant Lyapunov vectors with maximum exponent (normalized with respect to the Euclidean norm).
  • Figure 2: Covariant Lyapunov vectors of the Hénon-Heiles system. The graph shows the Poincaré section at $x=0$ with coordinates $y$, $p_y$, for energy $H=0.037$ and $\Lambda=2$. The black line is the contour of the constant energy surface. The magenta and red lines are invariant tori. The arrows are the projections on the section of the covariant Lyapunov vectors with maximum exponent (normalized with respect to the Euclidean norm).

Theorems & Definitions (24)

  • Definition 2.1: covariant Lyapunov vector
  • Definition 3.1: covariant Lyapunov field
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Proposition 4.4
  • proof
  • ...and 14 more