Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback
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ORIGINAL PAPER
Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback J Chen1, Y Shen2*
, X Li1, S Yang2 and S Wen3
1
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China
2
Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China 3
Transportation Institute, Shijiazhuang Tiedao University, Shijiazhuang 050043, People’s Republic of China Received: 10 November 2018 / Accepted: 15 July 2019
Abstract: The stability and existence conditions of Hopf bifurcation of a commensurate fractional-order van der Pol oscillator with time-delayed feedback are studied. Firstly, the necessary and sufficient conditions for the asymptotic stability of the equilibrium point of fractional-order van der Pol oscillator with linear displacement feedback are obtained, and it is found that the conditions are not only related to the feedback gain, but also to the fractional order. Secondly, regarding time delay as a bifurcation parameter, the stability of the commensurate fractional-order van der Pol system with time-delayed feedback is investigated based on the characteristic equation. Under some conditions, the critical value of time delay is calculated. The equilibrium point is stable when the parameter is less than the critical value and will be unstable if the parameter is greater than it. Moreover, the conditions for the occurrence of Hopf bifurcation are obtained. Finally, choosing four typical system parameters, some numerical simulations are carried out to verify the correctness of the obtained theoretical results. Keywords: Fractional-order van der Pol oscillator; Time delay; Stability; Hopf bifurcation PACS Nos.: 02.30.Hq; 05.45.-a
1. Introduction Fractional calculus is a branch of applied mathematics that deals with the study on fractional-order integral and derivative operators in real or complex domains. A great number of real-world objects can be generally identified and described by the fractional-order model. The main advantage of the fractional-order model in comparison with the integer-order model is that a fractional-order derivative can provide excellent performance in the description of memory and hereditary properties of various processes. In recent years, the study of oscillatory behaviors in fractional-order systems has received considerable attention in various fields, such as physics, engineering, economics, biology, and materials science [1–24]. Meanwhile, many different types of van der Pol oscillators containing fractional-order derivatives have attracted more and more
*Corresponding author, E-mail: [email protected]
attention [17, 18, 23–30]. For example, Tavazoei et al. [25] found a simple criterion which determined the oscillation range for a fractional-order van der Pol oscillator. Attari et al. [26] established the boundary between oscillatory and non-oscillatory regions using a describing function method for a
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