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Unstable Invariant Sets in Nonlinear Dynamical Systems

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Global Analysis of Nonlinear Dynamics

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Abstract

Unstable invariant sets, such as unstable periodic orbits and chaotic saddles, can have great effect on the global behaviors of dynamical systems. In this chapter, we investigate how to use the generalized cell mapping (GCM) method to locate these sets. To this end, we improve the GCM method in the following three aspects. First, a new effective algorithm is presented to locate complete self-cycling sets in the linear time. Secondly, the refinement technique is used to accurately locate coexisting unstable invariant sets and also to reveal their detailed structures. In order to make the refinement process robust, two different sampling techniques are presented. Finally, the transient analysis is also investigated. An optimal order for analyzing transient cells which leads to the minimal computational work is presented. We presented some examples to show the effectiveness of the improved GCM method, in particular, for the cases of coexisting unstable invariant sets.

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Acknowledgements

We would like to thank Professors Jun Jiang, Ling Hong, Yanmei Kang and Yong Xie for valuable discussions.

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Authors and Affiliations

  1. Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore, 117543, Singapore

    Hai-Lin Zou

  2. Aerospace School, Xi’an Jiaotong University, Xi’an, 710049, China

    Jian-Xue Xu

Authors
  1. Hai-Lin Zou
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  2. Jian-Xue Xu
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Corresponding author

Correspondence to Hai-Lin Zou .

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Editors and Affiliations

  1. School of Engineering, University of California, Merced, N. Lake Road 5200, Merced, 95344, California, USA

    Jian-Qiao Sun

  2. Edwardsville, School of Engineering, Southern Illinois University, Room: EB20, Edwardsville, 62025, USA

    Albert C. J. Luo

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Zou, HL., Xu, JX. (2012). Unstable Invariant Sets in Nonlinear Dynamical Systems. In: Sun, JQ., Luo, A. (eds) Global Analysis of Nonlinear Dynamics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3128-2_6

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  • DOI: https://doi.org/10.1007/978-1-4614-3128-2_6

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-3127-5

  • Online ISBN: 978-1-4614-3128-2

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