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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)
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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.

Fig. 2. Results of perturbing a coupled Lorenz system as evidenced by
the Lyapunov dimension (D_{L}) 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 *r*_{1} = 4, 10, and 12.

Fig. 3. Spatiotemporal plot of *x*_{i} vs time for the
coupled Lorenz equations. A perturbation amplitude of *r*_{1}
= 4 and frequency *f* = 1.014 is turned on at *t* = 40 and off
at *t* = 160.