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A subdomain synthesis method for global analysis of nonlinear dynamical systems based on cell mapping

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Abstract

Unaffordable memory requirement becomes a bottleneck for the cell mapping method when global analysis is carried out in a chosen domain of the state space with a specific computer in order to depict complex invariant sets with high resolution, and/or in high dimensions. This may also be often encountered even though the subdivision technique is adopted. Thus a subdomain synthesis method for cell mapping is proposed in this paper in order to release this difficulty. The proposed method first divides the chosen domain into smaller subdomains with affordable memory requirements, then carries the cell mapping analysis out on each subdomain independently meanwhile records the message on inter-subdomain mappings, finally recovers the global structure in the complete chosen domain based on the dynamical information revealed in the subdomains. The method can effectively avoid the loss of the invariant sets in the chosen domain that are unavoidably sliced into pieces by the state space partition without prior knowledge on them. Furthermore, both the subdivision technique and the parallel computing based on graphical processing units can also be incorporated into the proposed method that enhances the computation efficiency. Three examples of application on global analysis of nonlinear dynamical systems are presented to demonstrate the basic idea and the potential of the proposed method.

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Acknowledgements

The author acknowledges financial supports from the National Natural Science Foundation of China through the Grants (11332008, 11702213 and 11772243) and the Natural Science Foundation of Shaanxi Province through the Grant (2018JQ1061).

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Authors and Affiliations

  1. Department of Mechanics, Xi’an University of Science and Technology, Xi’an, 710054, People’s Republic of China

    Zigang Li & Ming Li

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, 28, Xianning West Road, Xi’an, 710049, Shaanxi, People’s Republic of China

    Jun Jiang & Ling Hong

  3. School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Jing Li

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  1. Zigang Li
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  2. Jun Jiang
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  3. Jing Li
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  4. Ling Hong
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Correspondence to Jun Jiang.

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Li, Z., Jiang, J., Li, J. et al. A subdomain synthesis method for global analysis of nonlinear dynamical systems based on cell mapping. Nonlinear Dyn 95, 715–726 (2019). https://doi.org/10.1007/s11071-018-4592-4

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  • DOI: https://doi.org/10.1007/s11071-018-4592-4

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