Bringing Order to Disorder: A method for stabilising a chaotic system around an arbitrary unstable periodic orbit

Abstract

This paper introduces a two-step procedure for stabilising a chaotic system around an arbitrary periodic orbit in a Poincaré map. The periodic orbit may be a period-n orbit traversing the Poincaré plane an integer n times before completing the periodic cycle. The order of traversing can also be chosen arbitrarily. Chaotic systems lack predictability due to positive Lyapunov exponents. For this reason, predictive control methods tend to perform poorly because the predictable horizon may be short. Current methods for stabilising chaotic systems work by controlling existing unstable periodic orbits. This paper proposes a method for stabilising the system on an arbitrary point in the Poincaré section. The method in this paper consists of two steps. In the first step, we pose an optimisation problem that computes an input signal that introduces a desired unstable or stable periodic orbit. In the second step, we use existing methods, such as the method using delay coordinate embedding developed by Ott, Grebogi and Yorke in 1990, to stabilise the system around the newly introduced periodic orbit. As an example, we demonstrate the method on the resistively and capacitively shunted driven Josephson junction.