doi: 10.3850/S201042862015001258

A Time Domain Collocation Method for Obtaining the Periodic Solutions
to the Van Der Pol Oscillator

Xuechuan Wang¹, Xiaokui Yue^2,a and Honghua Dai^{2,b 1}School of Astronautics, Northwestern Polytechnical University, Xi'an, 710021, China xcwang91@gmail.com ²National Key Laboratory of Aerospace Flight Dynamics, Xi'an, China ^axkyue@nwpu.edu.cn ^bhhdai@nwpu.edu.cn

ABSTRACT

In this paper, the time domain collocation method (TDC) is applied to solve the van der Pol equation. The results obtained by the TDC are compared with the results by the traditional harmonic balance method (HB) and high dimensional harmonic balance method (HDHB). We strictly prove that the HDHB is equivalent to the TDC and both of them can be regarded as special cases of the extended TDC method, where least squares method is used to handle an over-determined system. For the extended TDC method, appropriately increasing the number of collocation points can significantly relive the nonphysical solution phenomenon and generate a better solution. Compared with the HDHB, the TDC method is derived more strictly, and more convenient to implement. The extended TDC method with more collocation points can not only reduce the computational cost of the HB method, but also improve the resulting accuracy of the HDHB method. Numerical results verify the simplicity, accuracy and efficiency of the present TDC method.

Keywords: Time domain collocation method, High dimensional harmonic balance method, Period solution, Van der Pol equation, Limit cycle.

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