Analytical solutions for limit cycles of the forced Van der Pol Duffing oscillator
17 Oct 2013-Vol. 1558, Iss: 1, pp 2187-2192
TL;DR: In this article, the authors studied limit cycle solutions of the forced Van der Poling oscillator under the condition that the external frequency is equal to the resultant frequency of the nonlinear oscillator.
Abstract: We study limit cycle solutions of the forced Van der Pol Duffing oscillator under the condition that the external frequency is equal to the resultant frequency of the nonlinear oscillator. The effects of the damping parameter (μ), the nonlinear term (β) and the magnitude of the external forcing (g) are discussed. We use the homotopy analysis method (HAM) to obtain closed form solutions. We note that without forcing we obtain solutions reported previously in the literature. We minimize the square residual error of this problem to obtain solutions by HAM. Finally, a comparison of numerical and analytical solutions has been carried out.
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TL;DR: In this article, the authors proposed a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators and applied it to the forced Van der Pol oscillator and the forced van der Pol Duffing oscillator.
Abstract: In this paper we propose a technique to obtain limit cycles and quasi-periodic solutions of forced nonlinear oscillators. We apply this technique to the forced Van der Pol oscillator and the forced Van der Pol Duffing oscillator and obtain for the first time their limit cycles (periodic) and quasi-periodic solutions analytically. We introduce a modification of the homotopy analysis method to obtain these solutions. We minimize the square residual error to obtain accurate approximations to these solutions. The obtained analytical solutions are convergent and agree well with numerical solutions even at large times. Time trajectories of the solution, its first derivative and phase plots are presented to confirm the validity of the proposed approach. We also provide rough criteria for the determination of parameter regimes which lead to limit cycle or quasi-periodic behaviour.
20 citations
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TL;DR: In this article, the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are analyzed by using Floquet theory.
Abstract: Based on the homotopy analysis method (HAM), the high accuracy frequency response curve and the stable/unstable periodic solutions of the Van der Pol-Duffing forced oscillator with the variation of the forced frequency are obtained and studied. The stability of the periodic solutions obtained is analyzed by use of Floquet theory. Furthermore, the results are validated in the light of spectral analysis and bifurcation theory.
15 citations
TL;DR: Comparisons of the obtained solutions and numerical results show that the homotopy analysis method (HAM) is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error.
Abstract: In this paper, the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator are studied by combining the homotopy analysis method (HAM) with the multi-scale method analytically. Comparisons of the obtained solutions and numerical results show that this method is effective and convenient even when t is large enough, and the convergence of the approximate solutions is discussed by the so-called discrete square residual error. This method is a capable tool for solving this kind of nonlinear problems.
9 citations
Cites background from "Analytical solutions for limit cycl..."
...In addition to the right applications of the physical model, the jump phenomena, limit cycles, period-doubling solutions, and quasi-periodic solutions of the forced Van der Pol oscillator and the forced Van der Pol-Duffing oscillator have been paid attention and researched wildly these years [9, 10]....
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