Isolating Neighborhoods and Filippov Systems: Extending Conley Theory to Differential Inclusions

Abstract

The Conley index theory is a powerful topological tool for obtaining information about invariant sets in continuous dynamical systems. A key feature of Conley theory is that the index is robust under perturbation; given a continuous family of flows $\{\varphi_\lambda\}$, the index remains constant over a range of parameter values, avoiding many of the complications associated with bifurcations. This theory is well-developed for flows and homomorphisms, and has even been extended to certain classes of semiflows. However, in recent years mathematicians and scientists have become interested in differential inclusions, often called Filippov systems, and Conley theory has not yet been developed in this setting. This paper aims to begin extending the index theory to this larger class of dynamical systems. In particular, we introduce a notion of perturbation that is suitable for this setting and show that isolating neighborhoods, a fundamental object in Conley index theory, are stable under this sense of perturbation. We also discuss how perturbation in this sense allows us to provide rigorous results about "nearby smooth systems" by analyzing the discontinuous Filippov system.