A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method
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In this paper, the Van der Pol and Duffing oscillator equations were recast as nonlinear eigenvalue problems, and the successive linearization method was used to obtain the limit cycle and bifurcation diagrams of the governing equations.Abstract:
We present a novel application of the successive linearisation method to the classical Van der Pol and Duffing oscillator equations. By recasting the governing equations as nonlinear eigenvalue problems we obtain accurate values of the frequency and amplitude. We demonstrate that the proposed method can be used to obtain the limit cycle and bifurcation diagrams of the governing equations. Comparison with exact and other results in the literature shows that the method is accurate and effective in finding solutions of nonlinear equations with oscillatory solutions, nonlinear eigenvalue problems, and other nonlinear problems with bifurcations.read more
Citations
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References
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Shijun Liao,SA Sherif +1 more
TL;DR: In this paper, a simple bifurcation of a nonlinear problem multiple solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free oscillations with Quadratic nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous flow Boundary-layer Flow Boundarylayer Flow with Exponential Property Boundary Layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGR
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Van der pol's oscillator under delayed feedback
TL;DR: In this paper, the effect of delayed feedback on oscillatory behavior was investigated for the classical van der Pol oscillator and it was shown how the presence of delay can change the amplitude of limit cycle oscillations, or suppress them altogether.
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An analytic approximate approach for free oscillations of self-excited systems
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Peter B. Kahn,Yair Zarmi +1 more
TL;DR: In this article, the authors present a formalism of Perturbation Expansion Problems with Eigenvalues that have Negative Real Part Normal Form Expansion for Conservative Planar Systems Dissipative Planar System Nonautonomous Oscillatory Systems Problems with a Zero Eigenvalue Higher-Dimensional Hamiltonian Systems Higher-dimensional DissIPative Systems Appendix Index.
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An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–van der Pol Equation
TL;DR: In this paper, instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is employed, which does not require a small perturbation parameter or a large asmptotic parameter.
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