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The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems

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Abstract

The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems are proposed. The proposed continuation method advances the predictor–corrector continuation framework by coupling the reduced space sequential quadratic programming method with the multi-dimensional harmonic balance method and the gradients required for the continuation problem are derived. In order to determine the stability of quasi-periodic solution, a novel approach based on the analytical formulation of the harmonic balance equations is presented by using the Floquet theory with the perturbation term applied to the known quasi-periodic solution. Sensitivity analysis about the stability factor of quasi-periodic solution is also carried out. Finally, the effectiveness and applicability of the proposed methodology is verified and illustrated by two numerical examples. The proposed approaches have been demonstrated to be able to trace the aperiodic solutions of nonlinear systems and analyze their stabilities.

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Acknowledgements

The research reported here has received funding from the National Science Foundation of China under Grants (No. 11972082), the Beijing Municipal Science and Technology Project (Z181100004118002) and the Beijing Institute of Technology Research Fund Program for Young Scholars which are highly appreciated by the authors.

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Authors and Affiliations

  1. Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing, 100081, China

    Haitao Liao & Quanyue Zhao

  2. Beijing Key Laboratory of Lightweight Multi-functional Composite Materials and Structures, Beijing Institute of Technology, Beijing, 100081, China

    Quanyue Zhao

  3. State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, 100081, China

    Daining Fang

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  1. Haitao Liao
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  2. Quanyue Zhao
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Correspondence to Haitao Liao.

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Liao, H., Zhao, Q. & Fang, D. The continuation and stability analysis methods for quasi-periodic solutions of nonlinear systems. Nonlinear Dyn 100, 1469–1496 (2020). https://doi.org/10.1007/s11071-020-05497-7

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