Open loop control of lyapunov exponents at fixed points of nonlinear oscillator
by
Nayyer Iqbal
School of Mathematical Sciences, Government College University, Lahore, Pakistan
Coauthors: Danish Hameed Qureshi

Here we discuss the case of Kapitza dimensional nonlinear damping oscillator driven by sin- or cos- rapidly oscillating periodical force. We use the averaging procedure with respect to the rapidly changing movement and start from the effective potential energy of the pendulum [1]. Our purpose is to investigate how the open-loop (feewforward) control scheme influences on the structure of the Lyapunov spectrum in the fixed points of the dynamical system [2].

The model has one dimensionless parameter depending on the amplitude a and the frequency of the control modulation. Changing its value we construct the phase portraits of the systems in the neighborhood of the fixed points and demonstrate the Lyapunov spectrum under the application of different forms of feedforward control for the both cases of the horizontal and vertical modulation at the origin (0,0), at the inverse position ( ) and at the non-trivial point (the last in the case of the horizontal modulation). We can observe the conversion of the Lyapunov spectrum from the focus to the saddle point.

Our results can be easily extended for the case of non-harmonic modulation.

References

1. P. L. Kapitza, Dynamic stability of a pendulum with an oscillating Point of suspension, Journal of Experimental and Theoretical Physics 21 (1951) 588.

2. L. D. Landau, E. M. Lifshitz, Mechanics, Pergamon Press, Oxford,1960.

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Date received: November 28, 2007