Optimal tracking of parameter drift in a chaotic system: Experiment and theory

We present a method of optimal tracking for chaotic dynamical systems with a slowly drifting parameter. The net drift in the parameter is assumed to be small: this makes detecting and tracking the drift more difficult. The method relies on the existence of underlying deterministic behavior in the dynamical system, yet neither requires a system model nor develops one. We begin by describing an experimental study where a heuristic optimality criterion gave good tracking performance: the tracking method there was based on maximizing smoothness and overall variation in the drift observer, which was found by solving an eigenvalue problem. We then develop a theory, based on simplifying assumptions about the chaotic dynamics, to explain the success of the tracking method for chaotic systems. For signals from deterministic systems that are sufficiently complex in a sense that we make precise, typical drift observers provide poor tracking performance and require the drift to be particularly slow. In contrast, our theory shows that the optimality criterion seeks out a special drift observer that both provides better tracking performance and allows the drift to be appreciably faster. For periodic or quasiperiodic systems (no chaos), good tracking is easily achievable and the present method is irrelevant. For stochastic systems (no determinism), the optimal tracking method does not asymptotically improve tracking performance. Exhaustive numerical simulations of a simple drifting chaotic map, first without and then with stochastic forcing, show agreement with theoretical predictions of tracking performance and validate the theory.