A New Analytical Approach for Solving Van der Pol Oscillator
09 Oct 2019-Science Journal of Applied Mathematics and Statistics (Science Publishing Group)-Vol. 7, Iss: 4, pp 51
TL;DR: In this paper, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator, which gives better result than other existing results and also close to the corresponding numerical result.
Abstract: The Van der Pol oscillator is a nonlinear damping and non-conservative oscillator. Energy is generated at low amplitude and dissipated at high amplitude. This nonlinear oscillator was first introduced by Dutch electrical engineer and physicist B. Van der Pol and it was originally used to investigate vacuum tubes. Nowadays, it is used in both physical and biological sciences. It is also used in sociology and even in economics. It has a limit cycle and in earlier it was determined by the classical perturbation methods when the nonlinear term is small. Then the harmonic balance method was used to determine the limit cycle for stronger nonlinear case. Moreover, many researchers have been analyzed this oscillator by various numerical approaches. In this article, a new analytical approach based on harmonic balance method is presented to determine the limit cycle as well as approximate solutions of this nonlinear oscillator. The frequency as well as the limit cycle obtained by new approach has been compared with those obtained by other existing methods. The present method gives better result than other existing results and also close to the corresponding numerical result (considered to the exact result). Moreover, the present method is simpler than the existing harmonic balance method.
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TL;DR: Based on the modified harmonic balance method, an analytical method has been developed for handling the forced Van der Pol vibration equation, and these results display very nice harmony with the result obtained by the mentioned method as discussed by the authors.
Abstract: Based on the modified harmonic balance method, an analytical method has been developed for handling the forced Van der Pol vibration equation. Usually, a system of nonlinear algebraic equations arises within the unfamiliar coefficients in several harmonics terms and the frequency of the forcing term. A numerical technique has been applied to handle those nonlinear algebraic equations in the classical harmonic balance method. In our study, a system of linear algebraic equations is calculated with the aid of a single nonlinear one. The solutions attained by the suggested scheme have been likened to the results acquired by the well-known Runge-Kutta method, and these results display very nice harmony with the result obtained by the mentioned method. Also, it is noticed that the proposed technique is straightforward and gives the desired results in the whole solution domain.
J. Bangladesh Acad. Sci. 45(2); 231-240: December 2021
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01 Jan 1973
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