Much of the fascination and challenge of studying nonlinear dynamical systems arises from the complicated spatial, temporal and even spatio-temporal behavior that they exhibit. On the level of mathematics this complicated behavior can occur at all scales, both in state space and in parameter space. This points to the need for a coherent set of mathematical techniques that is capable at a given scale of extracting coarse but robust information about the structure of these systems. Furthermore, since most of our understanding of explicit nonlinear systems comes from numerical simulations and data driven models are becoming ever more relevant, it is important that these techniques be based on finite information and computationally efficient.
Algebraic topology is the classical mathematical tool for the global analysis of nonlinear spaces and functions, within which homology is perhaps the most computable subset. In particular, it provides a well understood framework through which the information hidden in large datasets can be reduced to compact algebraic expressions that provide insight into underlying geometric structures and properties.
The material described through these web pages represents our ongoing effort to develop and apply efficient and effective topologically based methods to the analysis of nonlinear systems.