Controlling Chaos in a High Dimensional System with Periodic Parametric Perturbations

K. A. Mirus and J. C. Sprott
Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 19 December 1998; accepted for publication 14 January 1999)

ABSTRACT

The effect of applying a periodic perturbation to an accessible parameter of a high-dimensional (coupled-Lorenz) chaotic system is examined.  Numerical results indicate that perturbation frequencies near the natural frequencies of the unstable periodic orbits of the chaotic system can result in limit cycles or significantly reduced dimension for relatively small perturbations.

Ref: K. A. Mirus and J. C. Sprott, Phys. Lett. A 254, 275-278 (1999)

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Fig. 1. Recurrence plot histogram for the coupled Lorenz system.  The period-one peak occurs at m = 252.5 time steps, or f = 1.014.
[Figure 1]

Fig. 2. Results of perturbing a coupled Lorenz system as evidenced by the Lyapunov dimension (DL) and the leading Lyapunov exponent (LLE).  The solid horizontal lines indicate values for the unperturbed system.  Note that a perturbation frequency f = 1.014 produces a limit cycle for r1 = 4, 10, and 12.
[Figure 2]

Fig. 3. Spatiotemporal plot of xi vs time for the coupled Lorenz equations.  A perturbation amplitude of r1 = 4 and frequency f = 1.014 is turned on at t = 40 and off at t = 160.
[Figure 3]