Controlling chaos in high dimensions: Theory and experiment

Abstract

The main contribution of this work is the development of a high-dimensional chaos control method that is effective, robust against noise, and easy to implement in experiment. Assuming no knowledge of the model equations, the method achieves control by stabilizing a desired unstable periodic orbit with any number of unstable directions, using small time-dependent perturbations of a single system parameter. Specifically, our major results are as follows. First, we derive explicit control laws for time series produced by discrete maps. Second, we show how to apply this control law to continuous-time problems by introducing straightforward ways to extract from a continuous-time series a discrete time series that measures the dynamics of some Poincaré map of the original system. Third, we illustrate our approach with two examples of high-dimensional ordinary differential equations, one autonomous and the other periodically driven. Fourth, we present the result on our successful control of chaos in a high-dimensional experimental system, demonstrating the viability of the method in practical applications.