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An approach combining periodicity ratio and secondary Poincaré map for characteristics diagnosis of nonlinear oscillatory systems

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Abstract

A secondary Poincaré map approach is developed in this research for diagnosing the nonlinear characteristics such as quasiperiodic and chaotic responses of a dynamic system. With the secondary Poincaré map approach developed, an approach combining the periodicity ratio method and the secondary Poincaré map approach is established such that all the dynamical characteristics of a nonlinear dynamic system can be systemically and completely identified. An example of an ecological oscillatory system is presented in the research to demonstrate the application of the combined approach. Periodic–quasiperiodic–chaotic region diagrams are generated with the employment of the approach, for a global characterization of this system with consideration of large ranges of system parameters. The approach developed in this research demonstrates effectiveness and efficiency in completely diagnosing the complex dynamical characteristics of nonlinear oscillatory systems, such as periodic, quasiperiodic, chaotic responses of the systems together with those in between periodic and chaotic responses.

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Acknowledgments

The authors would like to acknowledge with great gratitude for the supports of Chinese Natural Science Foundation (Project 39560023 to H.Z.), National Special Water Programs (No. 2009ZX07210-009, No. 2015ZX07203-011, No. 2015ZX07204-007), Department of Environmental Protection of Shandong Province (SDHBPJ-ZB-08), the China Scholarship Council (No. 201206730024), and the University of Regina.

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Authors and Affiliations

  1. Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing, 102206, People’s Republic of China

    Tousheng Huang & Huayong Zhang

  2. Industrial Systems Engineering, University of Regina, Regina, SK, S4S 0A2, Canada

    Tousheng Huang & Liming Dai

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  1. Tousheng Huang
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  2. Liming Dai
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Correspondence to Liming Dai.

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Huang, T., Dai, L. & Zhang, H. An approach combining periodicity ratio and secondary Poincaré map for characteristics diagnosis of nonlinear oscillatory systems. Nonlinear Dyn 84, 959–975 (2016). https://doi.org/10.1007/s11071-015-2542-y

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