Deterministic point processes generated by threshold crossings: Dynamics reconstruction and chaos control

We consider chaotic point processes generated by threshold crossings in deterministic dynamical systems. Specifically, we assume that an event occurs every time a scalar function of a continuous;chaotic trajectory crosses a preset threshold from a given direction. It is shown that, although the times between successive crossings, called interspike intervals here, are not sufficient to fully reconstruct the global dynamics of the original continuous system, they nevertheless provide enough information for an estimate of dynamical invariants such as correlation dimension. In addition, it is argued that, with a suitably chosen threshold, converting a continuous trajectory to an interspike interval time series preserves periodic orbit information and facilitates the implementation and achievement of chaos control. We demonstrate the ideas by conducting numerical experiments on two systems: the Lorenz model and a coupled Duffing oscillator.