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Dynamic Robust Bifurcation Analysis

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Analysis and Control of Complex Dynamical Systems

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

Abstract

We propose a new concept of dynamic robust bifurcation analysis for uncertain dynamical systems. An uncertain system is described by a feedback form composed of a nonlinear dynamical system and a dynamic uncertainty that is defined by a set of differential equations. For such an uncertain system, a bifurcation point can be uncertain as well. Therefore, we formulate a dynamic robust bifurcation analysis problem of identifying the set of all potential bifurcation points. To this end, first, we study equilibrium analysis to evaluate the existence and location of equilibria. Next, we derive a condition for robust hyperbolicity of the evaluated set of potential equilibrium points. On the basis of the condition, we propose a method for identifying the set of potential bifurcation points. Finally, illustrative examples for robustness analysis of normal forms for various bifurcations are presented.

This research was done when M.I. was with FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency/Graduate School of Information Science and Engineering, Tokyo Institute of Technology.

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Acknowledgments

The authors gratefully acknowledge Takayuki Arai, Masayasu Suzuki, and Takayuki Ishizaki for their comments and fruitful discussion on this research.

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8521, Japan

    Masaki Inoue

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

    Jun-ichi Imura

  3. Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, Japan

    Kenji Kashima

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

    Kazuyuki Aihara

Authors
  1. Masaki Inoue
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  2. Jun-ichi Imura
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  3. Kenji Kashima
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  4. Kazuyuki Aihara
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Corresponding author

Correspondence to Masaki Inoue .

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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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Inoue, M., Imura, Ji., Kashima, K., Aihara, K. (2015). Dynamic Robust Bifurcation Analysis. 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_1

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  • DOI: https://doi.org/10.1007/978-4-431-55013-6_1

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  • Print ISBN: 978-4-431-55012-9

  • Online ISBN: 978-4-431-55013-6

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