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Threshold Control for Stabilization of Unstable Periodic Orbits in Chaotic Hybrid Systems

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Analysis and Control of Complex Dynamical Systems

Part of the book series: Mathematics for Industry ((MFI,volume 7))

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Abstract

Stabilization of unstable periodic orbits within a given chaotic hybrid dynamical system is realized by a variable threshold value. In the conventional chaos control methods, a control input is proportional to the difference between the target orbit and the current state and it is added to a specific system parameter or the state as a small perturbation. Thus the whole system consumes certain control energy even if small as the amount of such input values during the transition state. We propose a new control method that changing the threshold value dynamically to stabilize the chaotic orbit. No actual control input is added into the system unlike the OGY method and the delayed feedback control. When the orbit hits the threshold, the state-feedback only determines the next threshold value to convey the controlled orbit to the target unstable periodic orbit eventually. Thus the orbit starting from the current threshold value reaches the next controlled threshold value without any direct control energy. We obtain the variation of the threshold value from the composite Poincaré map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show its control performances with evaluating basins of attraction.

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Acknowledgments

The authors wish to thank International Journal of Bifurcation and Chaos, World Scientific Publishing for granting permission to present here a modified version of the material published in “Controlling chaos of hybrid systems by variable threshold values,” D. Ito, T. Ueta, T. Kousaka, J. Imura, and K. Aihara, International Journal of Bifurcation and Chaos, Vol. 24, No. 10, http://www.worldscientific.com/doi/abs/10.1142/S0218127414501259 2014 World Scientific Publishing Company [30].

Author information

Authors and Affiliations

  1. Advanced Technology and Science, System Innovation Engineering, Tokushima University, 2-1 Minamijyousanjima, Tokushima, 770-8504, Japan

    Daisuke Ito

  2. Center for Administration of Information Technology, Tokushima University, 2-1 Minamijyousanjima, Tokushima, 770-8506, Japan

    Tetsushi Ueta

  3. Faculty of Engineering, Oita University, 700 Dannoharu, Oita, 870-1192, Japan

    Takuji Kousaka

  4. Graduate School of Information Science and Engineering, Tokyo Institute of Technology, 2-12-1 O-Okayama, Meguro, Tokyo, 152-8552, Japan

    Jun-ichi Imura

  5. Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro, Tokyo, 153-8505, Japan

    Kazuyuki Aihara

Authors
  1. Daisuke Ito
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  2. Tetsushi Ueta
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  3. Takuji Kousaka
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  4. Jun-ichi Imura
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  5. Kazuyuki Aihara
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Correspondence to Daisuke Ito .

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Editors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Kazuyuki Aihara

  2. Tokyo Institute of Technology, Tokyo, Japan

    Jun-ichi Imura

  3. Tokushima University, Tokushima, Japan

    Tetsushi Ueta

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Ito, D., Ueta, T., Kousaka, T., Imura, Ji., Aihara, K. (2015). Threshold Control for Stabilization of Unstable Periodic Orbits in Chaotic Hybrid Systems. In: Aihara, K., Imura, Ji., Ueta, T. (eds) Analysis and Control of Complex Dynamical Systems. Mathematics for Industry, vol 7. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55013-6_6

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  • DOI: https://doi.org/10.1007/978-4-431-55013-6_6

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  • Print ISBN: 978-4-431-55012-9

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