Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems

TL;DR

This paper applies differential geometry and mechanics to analyze chaotic dynamical systems, deriving analytical expressions for slow manifolds and introducing a new singular manifold to better understand attractor structures.

Contribution

It introduces a novel geometric approach using curvature and torsion to analytically determine slow manifolds and proposes a new singular manifold concept for chaotic systems.

Findings

Analytical expressions for slow manifolds derived from geometric properties.

Characterization of attractor structures using the singular manifold.

Applications demonstrated on Van der Pol, Chua, Lorenz, and Volterra-Gause models.

Abstract

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables velocity, acceleration and over-acceleration or jerk. The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincar\'e. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra-Gause are proposed.