Existence of an energy function for 3-dimensional chaotic "sink-source" cascades

TL;DR

This paper proves the existence of an energy function, a type of Lyapunov function, for certain 3D chaotic dynamical systems with specific attractor and repeller structures, advancing understanding of their stability properties.

Contribution

It establishes the existence of an energy function for 3D chaotic systems with specific attractor-repeller configurations, extending previous results in dynamical systems theory.

Findings

Existence of energy function for A-diffeomorphisms on 3D manifolds.

Applicable to systems with chaotic one-dimensional attractors and repellers.

Advances the theoretical framework for analyzing stability in chaotic systems.

Abstract

The paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any $ A $-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of chaotic one-dimensional attractor and repeller.