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The Supression of Chaos in Periodically Forced Oscillators

  • Steven W. Shaw
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Summary

A method for suppressing chaos in a class of periodically forced oscillators is presented. The method involves the manipulation of higher harmonics in the periodic excitation such that the homoclinic tangles which lie at the heart of the chaos are destroyed. The procedure is demonstrated via a particular example: an inverted pendulum with amplitude constraints subjected to periodic base excitation. The general theory is developed using a Melnikov type method (see [1–3] and is supported by some experimental evidence.

Keywords

Unstable Manifold Homoclinic Orbit Inverted Pendulum Homoclinic Solution Optimal Input 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Melnikov, V.K.: “On the stability of the center for time periodic perturbations,” Trans. Moscow Math. Soc. 12 1 - 57, 1963.Google Scholar
  2. 2.
    Guckenheimer, J.; Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. 2nd printing, Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1986.Google Scholar
  3. 3.
    Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods, Springer, Verlag, New York, Berlin, Heidelberg, Tokyo, 1988.Google Scholar
  4. 4.
    Moon, F.C.; Cusumano, J.; Holmes, P.J.: “Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum,” Physica 24D 383 - 390 1987.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Moon, F.C.: Chaotic Vibrations, Wiley-Interscience, New York, 1987.MATHGoogle Scholar
  6. 6.
    Shaw, S.W.; Rand, R.H.: “The transition to chaos in a simple mechanical system,” Int. J. of Nonlinear Mech. 24 41–56 1988.CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Moore, D.B.; Shaw, S.W.: “The experimental response of an impacting pendulum system,” Int. J. of Nonlinear Mech., to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Steven W. Shaw
    • 1
  1. 1.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

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