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一类环上的三自由度耦合van der Pol方程的同伦分析方法
The Homotopy Analysis Method for a Class of Three-Degree-of-Freedom Coupled van der Pol Oscillators

DOI: 10.12677/DSC.2020.91003, PP. 22-39

Keywords: 同伦分析方法，耦合van der Pol振子，周期解
Homotopy Analysis Method, Coupled van der Pol Oscillator, Periodic Solution

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Abstract:

本文主要运用同伦分析方法研究一类三自由度耦合van der Pol振子环系统周期解的近似表达式。我们先给出此系统的一般方程，再将此一般方程分成四类来讨论。第一，所有振子都同步运动；第二，两个振子同步运动，而第三个振子以一无关的方式运动(除它与第二个振子有相同周期的振动外)；第三，环上相邻的振子之间彼此都相位差1/3周期的运动；第四，两个振子相位差1/2周期，而第三个振子2倍于它们的频率振动。利用四种不同类型的van der Pol振子来说明同伦分析方法的有效性与广泛应用性，且将此方法与数值积分法进行了比较，结果发现得到的解析解与数值解具有很高的吻合性。
In this paper, the approximate expression of periodic solution for a class of 3-DOF coupled van der Pol oscillator ring system is studied by homotopy analysis method. We first give the general equation of the system, and then divide the general equation into four categories: First, all oscillators move synchronously; second, two oscillators move synchronously, while the third vibrator moves in an independent manner (except that it oscillates in the same period as the second); third, the phase difference between adjacent oscillators on the ring is one third-period motion. Fourth, the two oscillators are 1/2 cycle apart, and the third vibrates at twice their frequency. Four different types of van der Pol oscillators are used to illustrate the validity and wide application of the homotopy analysis method.

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