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Chaotic dynamics analysis of double inverted pendulum with large swing angle based on Hamiltonian function

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Abstract

In this paper, the chaotic dynamics of double inverted pendulum with big swing angle is studied based on the Hamiltonian canonical equation. At the same time, since Hamiltonian system has the property of symplectic geometry, the basic tools of symplectic geometry are used to deal with the Jacobian matrix in the system. So far, there are few outcomes from this method in the study of chaotic dynamics of double inverted pendulum, so the present work fills the blank in this field. In this paper, analytical mechanics is used to establish the Lagrange function for double inverted pendulum at first, then the corresponding Hamiltonian function is obtained through Legendre transformation, and finally the Hamilton canonical equation is obtained. Then, the Hamiltonian canonical equation of the inverted pendulum with big swing angle is analyzed by methods such as Lyapunov exponent, bifurcation diagram and Poincare section. In the course of analysis, because of the symplectic property of Hamiltonian system, the Jacobian matrix obtained also has the symplectic property. Symplectic algorithm maintains the structure of symplectic transformation in numerical calculation, so it has high stability and is most suitable for classical mechanical systems. In analyzing the equilibrium points of the system, it is found that the system has an infinite number of equilibrium points, so as long as the swing angle of one of the swing arms is 0 degree, the inverted pendulum can maintain a state of equilibrium. The study can provide new ideas and theoretical basis for analysis of chaotic dynamics in fields of aerospace, electronic technology and biological manufacturing.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China under Grant 62171182 and the Natural Science Foundation of Hunan Province No. 2019JJ50183.

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Authors and Affiliations

  1. Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Jingxiu He, Li Cui, Pan Huang & Yuan Huang

  2. College of Computer Science and Electronic Engineer, Hunan University, Changsha, 410082, China

    Jingru Sun

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  1. Jingxiu He
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  2. Li Cui
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  3. Jingru Sun
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  4. Pan Huang
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  5. Yuan Huang
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Correspondence to Li Cui.

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He, J., Cui, L., Sun, J. et al. Chaotic dynamics analysis of double inverted pendulum with large swing angle based on Hamiltonian function. Nonlinear Dyn 108, 4373–4384 (2022). https://doi.org/10.1007/s11071-022-07455-x

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  • DOI: https://doi.org/10.1007/s11071-022-07455-x

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