Asymptotic solution of the van der Pol oscillator with small fractional damping

Xie, Feng; Xueyuan, Lin

doi:10.1088/0031-8949/2009/t136/014033

Cited by 30 publications

(12 citation statements)

References 13 publications

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“…w Àa cos wdw ¼ Cð1 À aÞ sin ap 2 and Z 1 0 w Àa sin wdw ¼ Cð1 À aÞ cos ap 2 ð47Þ After [28,27] we approximate I c and I s in Eqs. (45) and (46) by their values in Eq.…”

Section: Fractional Mathieu Equationmentioning

confidence: 99%

Fractional Mathieu equation

Rand

Sah

Suchorsky

2010

Communications in Nonlinear Science and Numerical Simulation

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“…w Àa cos wdw ¼ Cð1 À aÞ sin ap 2 and Z 1 0 w Àa sin wdw ¼ Cð1 À aÞ cos ap 2 ð47Þ After [28,27] we approximate I c and I s in Eqs. (45) and (46) by their values in Eq.…”

Section: Fractional Mathieu Equationmentioning

confidence: 99%

Fractional Mathieu equation

Rand

Sah

Suchorsky

2010

Communications in Nonlinear Science and Numerical Simulation

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“…the second-order equation (52), or (53), or (54) is equal to the fractional oscillation Equation (31), or (42), or (43), respectively. Obviously, once we discover the equivalent equations of the fractional oscillation Equations (52), or (53), and (54), all problems stated previously can be readily solved.…”

Section: Research Thoughtsmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

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“…Eq. (3) has been studied in [12,27]. Barbosa [12] showed the steady state time responses and amplitude Fourier spectra of Eq.…”

Section: Fractional Order Oscillator Equationmentioning

confidence: 99%