Bifurcation and Chaos Control of Mixed Rayleigh-LiéNard Oscillator With Two Periodic Excitations and Mixed Delays

Abstract

This paper investigates the bifurcation, chaos, and active control of a mixed Rayleigh-Liénard oscillator with mixed time delays. First, the effects of system parameters on the supercritical pitchfork bifurcations are discussed in detail by applying the fast-slow separation method. Second, it is rigorously proved by the Melnikov method that chaotic vibration exists when the parameters of the uncontrolled system are selected above the threshold of chaos occurrence. By fine-tuning the system parameters, a criterion for designing the control parameters to make the Melnikov function non-zero is derived. In addition, the routes to chaos in controlled system are explored by bifurcation diagrams, largest Lyapunov exponents, phase portraits, Poincaré maps, basins of attraction, frequency spectra, and displacement time series. The results indicate that by properly adjusting the displacement feedback coefficient and the amplitude of parameter excitation, the chaotic motion caused by increasing of the amplitude of external excitation and strength of distributed time delay can be effectively suppressed. This research result can provide theoretical support for exploring the potential chaotic motion of other types of oscillators.