Rigorous Validation of Isolating Blocks for Flows and Their Conley Indices

Isolated invariant sets and their associated Conley indices are valuable tools for studying dynamical systems and their global invariant structures. Through their design, they aim to capture invariant behavior which is robust under small perturbations, and this in turn makes them amenable to a computational treatment. Over the years, a number of algorithms have been proposed to find index pairs for an isolated invariant set, and then to use an index pair to compute the associated Conley index. Nevertheless, most of these methods are restricted to discrete, albeit possibly multi-valued, dynamical systems. Only relatively few general methods exist for dynamical systems generated by differential equations. In the current paper, we present a new method for finding and rigorously verifying a special type of index pairs, namely isolating blocks and their exit sets. Our method makes use of a recently developed adaptive algorithm for rigorously determining the location of nodal sets of smooth functions, which combines an adaptive subdivision technique with interval arithmetic. By characterizing an exit set as a nodal domain, we are able to determine a valid index pair and proceed to compute its Conley index. Our method is illustrated using several examples for three-dimensional flows.