An analytical criterion for alternate stability switches in nonlinear oscillators with varying time delay

Abstract

Varying time delay feedback has found wide applications in stabilization and controlling of nonlinear dynamic systems. It has been a challenging topic to analyze nonlinear systems with time delay feedback, especially when the delay is varying. Here in this paper an analytical criterion is deduced, according to which alternate stability switches can be determined for nonlinear oscillators equipped with fast varying periodic time delay feedback. The criterion renders an analytical boundary being given by a concise and explicit function in the parameter space of interest. The controlled Duffing and the Duffing–van der Pol oscillators are investigated by the presented criterion. Analytical results are in well consistence with the numerical ones obtained by the Runge–Kutta method. It reveals that the stability region depends on the amplitude of varying time delay, but not on the frequency as long as it is large enough.