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Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method

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  • Xiao, Min
  • Zheng, Wei Xing
  • Cao, Jinde

Abstract

Although the Van der Pol oscillator, which was originally proposed as a model of vacuum tube circuits, has been widely used in electronics, biology and acoustics, its characteristics in fractional order formulations are not clearly explained even now. This paper is interested in gaining insights of approximate expressions of the periodic solutions in a fractional order Van der Pol oscillator. The presence of fractional derivatives requires the use of suitable criteria, which usually makes the analytical work much hard. Most existing methods for studying the nonlinear dynamics fail when applied to such a class of fractional order systems. In this paper, based on the residue harmonic balance method, a detailed analysis on approximations to the periodic oscillations of the fractional order Van der Pol equation is investigated. The relations that express the frequency and amplitude of the generated oscillations as functions of the orders and parameters are shown. Moreover, some examples are provided for comparing approximations with numerical solutions of the periodic oscillations. Numerical results reveal that the residue harmonic balance method is very effective for obtaining approximate solutions of fractional oscillations.

Suggested Citation

  • Xiao, Min & Zheng, Wei Xing & Cao, Jinde, 2013. "Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 89(C), pages 1-12.
    • Handle: RePEc:eee:matcom:v:89:y:2013:i:c:p:1-12
    DOI: 10.1016/j.matcom.2013.02.006
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    References listed on IDEAS

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    1. Laskin, Nick, 2000. "Fractional market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 482-492.
    2. Gafiychuk, V. & Datsko, B. & Meleshko, V., 2008. "Analysis of fractional order Bonhoeffer–van der Pol oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(2), pages 418-424.
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    Cited by:

    1. Yonkeu, R. Mbakob, 2023. "Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).

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