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Failure of chaos control

van De Water W¹, de Weger J

¹Physics Department, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|December 2, 2000

Summary

Controlling chaos in a pendulum experiment failed due to an unmeasured small eigenvalue. Experimental data distribution prevents accurate eigenvalue estimation, hindering chaos control efforts.

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Area of Science:

Nonlinear dynamics
Experimental physics
Chaos theory

Background:

Parametrically excited pendulums can exhibit chaotic behavior.
Imperfections in excitation mechanisms introduce complexities, such as weakly excited degrees of freedom.
Controlling chaos is crucial for understanding and manipulating complex systems.

Purpose of the Study:

Investigate the challenges in controlling chaos in a parametrically excited pendulum with an imperfect excitation mechanism.
Identify the factors hindering successful chaos control in experimental settings.
Propose a condition for obtaining necessary information for chaos control.

Main Methods:

Experimental study of a parametrically excited pendulum with an imperfect excitation mechanism.
Numerical modeling to analyze system dynamics and eigenvalue estimation.
Time series analysis to examine phase-space distribution and waiting times.

Main Results:

Failure to control chaos despite a well-characterized pendulum state and weak perturbation.
Numerical models reveal the critical importance of a small eigenvalue, often overlooked.
Experimental time series data, distributed according to the natural measure, obscure the estimation of small eigenvalues.
Logarithmic oscillations observed in waiting times for control attempts due to phase-space distribution.

Conclusions:

The presence of a small eigenvalue, arising from system imperfections, significantly impacts chaos control.
Accurate estimation of small eigenvalues from experimental time series is challenging due to data distribution.
Chaos must be suppressed before the requisite information for control can be reliably acquired.

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