Geometry and stability of dynamical systems

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 we explain that 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, we show that 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.

Figure 1

Trajectories of dynamical systems and their perturbations.

Figure 2

Potentials of different dynamical systems mapped to the same Riemannian manifold.