Shilnikov chaos: how to characterize homoclinic and heteroclinic behavior

We introduce the concepts of Shil'nikov chaos and competing instabilities in a nonlinear dynamics including at least a saddle focus and a saddle point, a parameter change induces a smooth transition from a homoclinic to a heteroclinic trajectory. In terms of the return map to a given Poincaré section, the two trajectories have the following characterization. In the homoclinic case, the global behavior is recovered from the local linear dynamics within a unit box around the saddle focus. The heteroclinic case requires the composition of two linearized maps around the two unstable points. By an exponential transformation the geometrical map yields the return map of the orbital times. This new map represents the most appropriate indicator for experimental situations whenever a symbolic dynamics built on geometric position does not offer a sensitive test. Furthermore the time maps display a large sensitivity to noise. This offers a criterion to discriminate between a simulation (either analog or digital) with a few variables and experiment dealing with the physical variables embedded in the real world and thus acted upon by noise.