Dynamical Systems Analysis Using Differential Geometry

This paper aims to analyze trajectories behavior and attractor structure of chaotic dynamical systems with the Differential Geometry and Mechanics formalism. Applied to slow-fast autonomous dynamical systems (S-FADS), this approach provides: on the one hand a kinematics interpretation of the trajectories motion, and on the other hand, a direct determination of the slow manifold equation. The attractivity of this manifold established with a new criterion makes it possible to ensure attractors stability. Then, a qualitative description of the geometrical structure of the attractor is presented. It consists in considering it as the deployment in the space phase of a special submanifold that is called singular manifold. The attractor can be obtained by integration of initial conditions taken on this singular manifold. Applications of this method are made for the following models: cubic-Chua, and Volterra-Gause. Introduction In the Mechanics formalism the solution of a dynamical system is considered as the coordinates of a moving point M at the instant t. Then, three kinematics variables are attached to this point which represents the “trajectory curve”: X(t) : parametric representation of chaotic orbit, V(t) : instantaneous velocity vector, γ(t) : instantaneous acceleration vector. The Differential Geometry allows to use the Frénet frame [2] which is moving with the “trajectory curve” and directed towards its motion, consists in, a unit tangent vector to the “trajectory curve”, a unit normal vector, directed towards the interior of the concavity of the curve and a unit binormal vector to the trajectory curve so that the trihedron ( ) , , τ β ν is direct (Cf. Fig.1.). Dynamical Systems Analysis Using Differential Geometry 2 Fig. 1. Frénet frame and osculating plane. In this moving frame the instantaneous acceleration vector may be decomposed in a tangential and normal component both depending on instantaneous velocity and acceleration vectors directions. γ.V V (1) γ V V τ