Selective stabilisation of very long unstable periodic orbits in chaotic systems

Abstract

We propose a novel extension of Pyragas' method to stabilise arbitrarily long unstable periodic orbits (UPOs) in chaotic systems. No restrictions are imposed on the dimensionality of the chaotic attractor and the system is sampled using time series of a single scalar variable. Ways to estimate all of the necessary control parameters are presented. The method is tested stabilising UPOs in the Mackey-Glass system that has an infinite dimensional state space due to an inherent time delay. The effective attractor dimension can be tuned to any convenient value by changing the delay, which allows one to test for effects of high dimensional chaos.