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Proceedings ArticleDOI: 10.1063/1.4825972

Analytical solutions for limit cycles of the forced Van der Pol Duffing oscillator

17 Oct 2013-Vol. 1558, Iss: 1, pp 2187-2192
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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Topics: Van der Pol oscillator (65%), Duffing equation (58%), Homotopy analysis method (53%) ...read more
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Journal ArticleDOI: 10.1088/0031-8949/89/7/075202
08 May 2014-Physica Scripta
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.

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Topics: Van der Pol oscillator (66%), Duffing equation (59%), Limit cycle (56%) ...read more

16 Citations


Open accessJournal ArticleDOI: 10.1016/J.ASEJ.2014.11.010
Abstract: The present paper deals with the influence of thermophoretic particle deposition on the MHD mixed convective heat and mass transfer flow in a vertical channel in the presence of radiative heat flux with thermal-diffusion and diffusion-thermo effects. The resulting nonlinear coupled equations are solved under appropriate boundary conditions using the homotopy analysis method. The influence of involved parameters on heat and mass transfer characteristics of the fluid flow is presented graphically. It is noted that fluid velocity is an increasing function of radiation parameter, Dufour number, Buoyancy ratio parameter and mixed convection parameter whereas the magnetic parameter, thermophoresis constant, Soret number and Schimidt number lead to suppress the velocity. The fluid temperature increases with increasing radiation parameter and Dufour number. The convergence of homotopy analysis method (HAM) solutions is discussed and a good agreement is found between the analytical and the numerical solution.

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Topics: Convective heat transfer (61%), Homotopy analysis method (56%), Fluid dynamics (56%) ...read more

14 Citations


Journal ArticleDOI: 10.1088/0031-8949/91/1/015201
Jifeng Cui1, Jiaming Liang1, Zhiliang Lin1Institutions (1)
01 Jan 2016-Physica Scripta
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.

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Topics: Van der Pol oscillator (64%), Floquet theory (59%), Homotopy analysis method (55%) ...read more

9 Citations


Journal ArticleDOI: 10.1007/S11075-017-0420-Z
Jifeng Cui1, Wenyu Zhang1, Zeng Liu2, Jianglong Sun2Institutions (2)
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.

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4 Citations


Open accessJournal ArticleDOI: 10.1016/J.CPC.2018.05.009
Abstract: Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package — InSyDE. The package is designed, apart from materializing the algorithms presented, to provide a set of tools to allow the user to analyse the intermediary steps of the process. Computer programs in physics Program Title: InSyDE — Invariants and Symmetries of (racional second order ordinary) Differential Equations. Program Files doi: http://dx.doi.org/10.17632/4ytft6zgk7.1 Licensing provisions: GPLv3 Programming language: Maple 17. Nature of problem Search for first integrals of rational 2ODEs. Solution method: The method of solution is based on an algorithm described in this paper. Restrictions: If the rational 2ODE that is being analysed presents a very high degree in ( x , y , z ), then the method may not work well. Unusual features: Our implementation can find first integrals in many cases where the rational 2ODE under study cannot be reduced by other powerful solvers. Besides that, the package presents some useful research commands.

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3 Citations