Alternate stability switches induced by time delay in nonlinear fractional oscillators

Abstract The van der Pol–Duffing oscillator equipped with fractional time delay feedback is investigated, with emphasis on in-depth stability analysis of equilibrium. An analytical stability criterion is deduced, based on which the parameter space of the feedback gain coefficient and the fractional order is split into time-delay-dependent and -independent regions. In the time-delay-dependent region, it appears that the equilibrium may experience stability losing or gaining repeatedly, named as a stability switch. Moreover, the critical time delay for stability switch happening is determined semi-analytically. An interesting finding consists in a series of alternate stability switches as the fractional order is fixed in a certain range. It implies the fractional time delay feedback would be potentially applicable to design flexible and efficient controllers to suppress or actuate certain motions of nonlinear dynamical systems.