Journal ArticleDOI
Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator
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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.read more
Citations
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Journal ArticleDOI
On the limit cycles, period-doubling, and quasi-periodic solutions of the forced Van der Pol-Duffing oscillator
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.
Journal ArticleDOI
Study on a Multi-Frequency Homotopy Analysis Method for Period-Doubling Solutions of Nonlinear Systems
H. X. Fu,Y. H. Qian +1 more
TL;DR: A modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system and finds that the result obtained is closer to reality than previously thought.
Journal ArticleDOI
Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis
Galal M. Moatimid,T. S. Amer +1 more
TL;DR: In this article , a cubic-quintic Duffing equation of magnetic spherical pendulum was analyzed by grouping nonlinear expanded frequency, homotopy perturbation method (HPM), and Laplace transforms.
Journal ArticleDOI
On the dynamics of a Van der Pol–Duffing snap system
Vinícius Wiggers,Paulo C. Rech +1 more
Journal ArticleDOI
Analytical solutions of nonlinear system of fractional-order Van der Pol equations
TL;DR: In this article, the double-well, in-phase and out-of-phase periodic solutions of the system of fractional-order Van der Pol equations and the exact solution of the nonlinear fractionalorder VDPDO with independent initial profiles are investigated.
References
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Journal ArticleDOI
Impulses and Physiological States in Theoretical Models of Nerve Membrane
TL;DR: Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle, which qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve.
Book
Homotopy Analysis Method in Nonlinear Differential Equations
TL;DR: In this paper, a convergence series for Divergent Taylor Series is proposed to solve nonlinear initial value problems and nonlinear Eigenvalue problems with free or moving boundary in heat transfer.
Book
The Duffing Equation: Nonlinear Oscillators and their Behaviour
Ivana Kovacic,Michael J. Brennan +1 more
TL;DR: In this article, the authors present a survey of the literature on nonlinear dynamics of pendulum and nonlinear oscillators, including a brief biography of Georg Duffing, and some of the most relevant works.
Journal ArticleDOI
On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder
TL;DR: In this paper, a totally analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder is obtained using homotopy analysis method (HAM), and the series solution is developed and the recurrence relations are given explicitly.
Journal ArticleDOI
Chaotically transitional phenomena in the forced negative-resistance oscillator
Yoshisuke Ueda,Norio Akamatsu +1 more
TL;DR: In this paper, Chaotically transitional processes in the forced negative-resistance oscillator were investigated using analog and digital computers and the difference between the almost periodic oscillations and the Chaotic Transition Process (CTP) was clarified.