From Time Series to Symbolic Dynamics: An Algebraic Topological Approach

A new approach to constructing a dynamical systems model from experimental time series data is presented. Using the the ideas of delay reconstruction a multivalued dynamical system is constructed. The multivalued approach is taken to allow for bounded experimental error. This system is then analyzed and algebraic invariants based on the Conley index are computed. These invariants have implications concerning the dynamics which must occur, e.g. symbolic dynamics and horseshoes. Theorems indicating conditions under which these invariants can be lifted back to the original physical system are then proven.