Geometry and stability of dynamical systems

TL;DR

This paper explores the geometric aspects of stability in dynamical systems, proposing new definitions for local stability using geometric structures and analyzing their implications for various types of systems.

Contribution

It introduces a geometric framework for defining local stability in dynamical systems, emphasizing the role of linear connections and intrinsic structures in Lagrangian systems.

Findings

Global Lyapunov stability depends on seminorm choices

A new geometric definition of local stability is proposed

Intrinsic stability notions are established for Lagrangian systems

Abstract

We reconsider both the global and local stability of solutions of continuously evolving dynamical systems from a geometric perspective. We clarify that an unambiguous definition of stability generally requires the choice of additional geometric structure that is not intrinsic to the dynamical system itself. While global Lyapunov stability is based on the choice of seminorms on the vector bundle of perturbations, we propose a definition of local stability based on the choice of a linear connection. We show how this definition reproduces known stability criteria for second order dynamical systems. In contrast to the general case, the special geometry of Lagrangian systems provides completely intrinsic notions of global and local stability. We demonstrate that these do not suffer from the limitations occurring in the analysis of the Maupertuis-Jacobi geodesics associated to natural Lagrangian systems.