Infinitesimal Lyapunov functions for singular flows

TL;DR

This paper extends infinitesimal Lyapunov functions to singular flows, providing new characterizations of hyperbolic sets and uniform hyperbolicity using derivative-based conditions.

Contribution

It introduces a novel approach to analyze singular flows through infinitesimal Lyapunov functions, enabling characterization of hyperbolic structures without singularities.

Findings

Characterizes partial and sectional hyperbolic sets using infinitesimal Lyapunov functions.

Provides conditions for uniform hyperbolicity in the absence of singularities.

Reduces hyperbolicity verification to checking positive definiteness of a symmetric operator.

Abstract

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.