Rank one chaos: Theory and applications

The main purpose of this tutorial is to introduce to a more application-oriented audience a new chaos theory that is applicable to certain systems of differential equations. This new chaos theory, namely the theory of rank one maps, claims a comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain systems of differential equations, the existence of the indicated phenomenon of chaos can be verified through a well-defined computational process. Applications to the well-known Chua's and MLC circuits employing controlled switches are also presented to demonstrate the usefulness of the theory. We try to introduce this new chaos theory by using a balanced combination of examples, numerical simulations and theoretical discussions. We also try to create a standard reference for this theory that will hopefully be accessible to a more application-oriented audience.