  The Conley index theory is a powerful topological tool for describing the
basic structure of dynamical systems. One important feature of this theory is
the attractor-repeller decomposition of isolated invariant sets. In this
decomposition, all points in the invariant set belong to the attractor, its
associated dual repeller, or a connecting region. In this connecting region,
points tend towards the attractor in forwards time and the repeller in
backwards time. This decomposition is also, in a certain topological sense,
stable under perturbation. Conley theory is well-developed for flows and
homomorphisms, and has also been extended to some more abstract settings such
as semiflows and relations. In this paper we aim to extend the
attractor-repeller decomposition, including its stability under perturbation,
to continuous time set-valued dynamical systems. The most common of these
systems are differential inclusions such as Filippov systems.
