Much of the focus of dynamical systems is on the existence and structure of invariant sets. What we have learned over the past century is that this is an extremely rich subject, one need only think of chaotic dynamics and/or bifurcation theory. I will argue that from the perspective of applications this richness may, in fact, be detrimental. Consider for example multiscale models where the nonlinearities are at best heuristic descriptions and parameters are at best poorly known. In this case detailed information about invariant sets can be irrelevant or misleading.
With this in mind I will describe our efforts to develop a computationally efficient, but mathematically rigorous framework for extracting combinatorial/algebraic topological descriptions of the global dynamical structures of multi-parameter nonlinear systems.
I will demonstrate these ideas in the context of some simple models from population biology.