针对双摆系统的混沌特性展开研究，首先将双摆从Hamilton系统近似为拟Hamilton系统，然后应用双自由度的Melnikov法预测拟Hamilton系统出现混沌运动的能量阈值，间接得到Hamilton系统混沌的必要条件。通过必要条件分析某些特定情况下系统的运动状态，结合不同参数下系统的最大Lyapunov指数图、分岔图、Poincaré截面图和时间历程图，验证了理论的正确性；同时也发现了因模型局限而产生的两种例外情况，并从理论角度对产生例外的原因进行分析。结果表明：能量阈值与摆长、摆重密切相关，而摆长、摆重又影响能量大小，意味着能量与系统混沌之间存在复杂的联系，而不是一般认为的低能级拟周期、高能级混沌。
关键词：双摆；Melnikov法；拟周期；混沌；最大Lyapunov指数
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
关键词
双摆 /
Melnikov法 /
拟周期 /
混沌 /
最大Lyapunov指数
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Key words
double pendulum /
Melnikov method /
quasi-period /
chaos /
maximum Lyapunov exponent
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