Targeting and Control of Chaos

Abstract

The “control of chaos” refers to a procedure in which a saddle fixed point in a chaotic attractor is stabilized by means of small time dependent perturbations. Control may be switched between different saddle periodic orbits, but it is necessary to wait for the trajectory to enter a small neighborhood of the saddle point before the control algorithm can be applied.

This paper describes an extension of the control idea, called “targeting.” By targeting, we mean a process in which a typical initial condition can be steered to a prespecified point on a chaotic attractor using a sequence of small, time dependent changes to a convenient parameter. We show, using a 4-dimensional mapping describing a kicked double rotor, that points on a chaotic attractor with two positive Lyapunov exponents can be steered between typical saddle periodic points extremely rapidly—in as little 12 iterations on the average. Without targeting, typical trajectories require 10,000 or more iterations to reach a small neighborhood of saddle periodic points of interest.