Chaotic characteristics of a double pendulum system based on the Melnikov method

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Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (14) : 92-98.

LIU Dingyang1，JIAN Kailin1,2，ZHANG Liang1,2

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Abstract

The chaotic characteristics of the double pendulum system are studied. First, the double pendulum is approximated from the Hamilton system to the quasi-Hamilton system, and then the Melnikov method with two degrees of freedom is used to predict the energy threshold of the chaotic motion of the quasi-Hamilton system, and the necessary condition of chaos of Hamiltonian system is obtained indirectly. The necessary conditions are used to analyze the motion state of the system under some specific conditions, and combine the maximum Lyapunov exponent diagram, bifurcation diagram, Poincaré section and time series diagram of the system under different parameters to verify the correctness of the theory. At the same time, two kinds of exceptions due to the limitations of the model are found, and the reasons for the exceptions are analyzed theoretically. The results show that the energy threshold is closely related to the length and weight of the pendulum, and the length and weight of the pendulum also affect the energy, which means that there is a complex connection between energy and chaos in the system, instead of the low-energy quasi-period and high-level chaos generally considered.
Key words: double pendulum; Melnikov method; quasi-period; chaos; maximum Lyapunov exponent

Key words

double pendulum / Melnikov method / quasi-period / chaos / maximum Lyapunov exponent

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LIU Dingyang1，JIAN Kailin1,2，ZHANG Liang1,2. Chaotic characteristics of a double pendulum system based on the Melnikov method[J]. Journal of Vibration and Shock, 2022, 41(14): 92-98

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