Séance Séminaire

Séminaire Gaston Darboux

Conley Index for Filippov Invariant Sets

Filippov vector field theory was developed to study dynamical systems with discontinuities, which naturally appear in models with impacts, friction, switching, or control mechanisms. To handle the ambiguity of trajectories at discontinuity surfaces, the Filippov convention is adopted, and the system is analyzed using multivalued flows, allowing a well-defined generalized dynamics across singularities. The Conley index is a topological invariant used to characterize isolated invariant sets, providing information about the existence and persistence of equilibria, periodic orbits, and more complex invariant structures in smooth dynamical systems. Extending the Conley index to Filippov vector fields allows the study of non-smooth dynamics, offering a robust tool to detect and classify invariant sets, understand bifurcations, and explore global qualitative behavior in systems where classical linearization fails. This generalization bridges smooth and discontinuous dynamics, providing a unified perspective on stability and global structure in complex systems.