  A combinatorial framework for dynamical systems provides an avenue for
connecting classical dynamics with data-oriented, algorithmic methods.
Combinatorial vector fields introduced by Forman and their recent
generalization to multivector fields have provided a starting point for
building such a connection. In this work, we strengthen this relationship by
placing the Conley index in the persistent homology setting. Conley indices are
homological features associated with so-called isolated invariant sets, so a
change in the Conley index is a response to perturbation in an underlying
multivector field. We show how one can use zigzag persistence to summarize
changes to the Conley index, and we develop techniques to capture such changes
in the presence of noise. We conclude by developing an algorithm to track
features in a changing multivector field.
