Attractors and Attracting Neighborhoods for Multiflows

Abstract

We already know a great deal about dynamical systems with uniqueness in forward time. Indeed, flows, semiflows, and maps (both invertible and not) have been studied at length. A view that has proven particularly fruitful is topological: consider invariant sets (attractors, repellers, periodic orbits, etc.) as topological objects, and the connecting sets between them form gradient like flows. In the case of systems with uniqueness in forward time, an attractor in one system is related to nearby attractors in a family of other, "close enough" systems. One way of seeing that connection is through the Conley decomposition (and the Conley index) [2], [13]. This approach requires focusing on isolated invariant sets - that is, invariant sets with isolating neighborhoods. If there is an invariant set I, which has an isolating neighborhood N, we say that I is the invariant set associated to N, and N is an isolating neighborhood associated to I. When the invariant set in question is an attractor or a repeller, then the isolating neighborhood is called an attracting neighborhood or a repelling neighborhood, respectively. A more specialized case may be called an attractor block or a repeller block. This approach was expanded to discrete time systems which lack uniqueness in forward time, using relations, in [7] and [11]. Relations do not rely on uniqueness in forward time, but the graph of any map is a relation; thus they serve to generalize maps. Some of this is reviewed in the next few sections. In addition, I expanded on work done in [7] to show that in compact metric spaces, attractors for closed relations continue (see Section 6.1). On the continuous time side, more work needs to be done. This paper is a step toward a more systematic approach for continuous time systems which lack uniqueness in forward time. This work applies to Filippov systems [4] and in control theory [12]. In the following pages, we establish a tool (multiflows) for discussing the continuous time case and develop a framework for understanding attractors (and therefore stability) in these systems. A crucial part of this work was establishing attractor / attracting neighborhood pairs, which happens in Section 5.5.