Bifurcation and stability analysis of commensurate fractional-order van der Pol oscillator with time-delayed feedback

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.