Conley Index Theory for Multivalued Dynamical Systems and Piecewise-Continuous Differential Equations

Abstract

Modern dynamical systems–particularly heuristic models–often take the form of piecewise- continuous differential equations. In order to better understand the behavior of these systems we generalize aspects of Conley index theory to this setting. Because nons- mooth models do not generally have unique solutions, this process involves organizing the solution set of the piecewise-continuous equation into a set-valued object called a multiflow. We prove several properties of this object, providing us with a foundation for extending Conley’s techniques. This framework allows us to define isolating neigh- borhoods and demonstrate that they are stable under perturbation. We also provide an attractor-repeller pair decomposition of compact invariant sets for multiflows which helps us to understand the limiting behavior of solutions in such sets. This decomposi- tion is shown to continue under perturbation. Because we assume very little structure in proving these results we are able to connect them to many different existing formu- lations of the Conley index for multivalued dynamical systems. Therefore we are able to identify isolating neighborhoods in a large class of differential inclusions, decompose the associated isolated invariant sets into an attractor-repeller pair, and provide the index of of the original isolated invariant set, the attractor, and the repeller; all of this information is stable under small perturbations. This process is carried out on a piecewise-continuous model from oceanography as an example.