Weak signal detection based on Mathieu-Duffing oscillator with time-delay feedback and multiplicative noise

Abstract This paper presents analytical studies of the stochastic differential equation with Mathieu-Duffing oscillator under velocity feedback control with a time delay. We derive the analytic expressions of the stationary probability density function by the stochastic center manifold and stochastic averaging method to investigate the stochastic bifurcation. We first propose a three-step procedure for weak signals detecting: (1) Ascertain the existence and detect the frequency by the “transient vacancy”(TV) of chaotic motion. (2) Detect the phase based on Melnikov function. (3) After that the frequency and phase are known, we detect the amplitude by the transition between chaotic and large-scale periodic motion. In addition, the effects of the time-delayed feedback on the theoretical chaotic threshold are investigated under Gaussian white noise based on the Langevin and the Melnikov function. The time-delayed feedback τ can reduce the theoretical chaotic threshold, which is beneficial to detect the weak signal with the change of motion state. Subsequently, the “TV” method has obvious advantages of higher accuracy from the perspective of numerical simulation.