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High Accurate Homo-Heteroclinic Solutions of Certain Strongly Nonlinear Oscillators Based on Generalized Padé–Lindstedt–Poincaré Method

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Abstract

Background

In this work, the generalized Padé–Lindstedt–Poincaré method is applied to obtain high accurate homoclinic and heteroclinic solutions of the entire \(\Phi^{6}\)-Van der Pol oscillator with asymmetric potential and the generalized Duffing-Harmonic-Van der Pol oscillator. Meanwhile, the critical values of bifurcation parameter are also predicted

Methods

To improve the efficiency and accuracy, several new types of generalized Padé approximant are constructed and introduced into the procedure of the present method. In this method, the perturbation equations obtained from the Lindstedt–Poincaré procedures are solved by generalized Padé approximation method instead of the traditional integration method. Consequently, the proposed method does not involve the cumbersome calculations of derivation and integration compared to traditional perturbation-based methods. It means that the high order perturbation solutions can be easily obtained via the present method.

Results

To demonstrate the feasibility of the method, the oscillators with small and big parameters are both solved. All solutions and critical values obtained in this paper are compared to the results obtained from the Runge–Kutta method. It shows high agreement between the present results and the numerical ones.

Conclusion

In general, this method has characters of simple calculation, high accuracy and wide applicability, which can be regarded as a supplement and improvement of existing perturbation-based method.

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Acknowledgements

The authors acknowledge support by the National Natural Science Foundation of China (Grant No. 11747147), the Natural Science Foundation of Hunan Province of China (Grant No. 2019JJ50515) and the Research Foundation of Education Bureau of Hunan Province of China (Grant No. 21B0419).

Funding

This work was funded by National Natural Science Foundation of China (Grant no. 11747147); Natural Science Foundation of Hunan Province (Grant no. 2019JJ50515); Research Foundation of Education Bureau of Hunan Province of China (Grant no. 21B0419).

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Authors and Affiliations

  1. School of Mathematics and Physics, University of South China, Hengyang, 421001, China

    Zhenbo Li

  2. College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, China

    Jiashi Tang

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  1. Zhenbo Li
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  2. Jiashi Tang
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Correspondence to Zhenbo Li.

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Li, Z., Tang, J. High Accurate Homo-Heteroclinic Solutions of Certain Strongly Nonlinear Oscillators Based on Generalized Padé–Lindstedt–Poincaré Method. J. Vib. Eng. Technol. 10, 1291–1308 (2022). https://doi.org/10.1007/s42417-022-00446-7

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