Different vibration-switching modes induced by pulse-shaped explosion in a hybrid van der Pol-Rayleigh-Duffing system

Abstract

Mixed-mode vibrations belong to a typical multiscale dynamic phenomenon, which are manifested as a combination of limit cycle vibrations (defined as excited states) and equilibrium point vibrations (defined as silent states). Revealing the switching mechanism between the two different vibration states is an important topic in nonlinear dynamics. This work proposes a novel switching mechanism based on a hybrid van der Pol-Rayleigh-Duffing oscillator with slow-varying harmonic excitations, known as pulse-shaped explosion (PSE for short). PSE behavior is manifested as the system roots tending to the positive or negative infinities near some certain critical values, which can formally transform a smooth system into a combination of several non-smooth systems. To explore the occurrence of the PSE, we calculate the equilibrium state characteristics of the model and obtain the variation law of the system real solution with the cosine excitation function by the function monotonicity theory. From the general analyses of the relaxation oscillations and mixed-mode vibrations, we show that three different vibration-switching modes namely the switching between the excited and silent states induced by the PSEs, the switching between the two excited states induced by the PSEs and the switching between the two silent states induced by the PSEs can be exhibited. In addition, we show that the bifurcation-delay phenomenon caused by the cosine function slowly varying through the supercritical Hopf bifurcations can significant prolong the silent state process and the slow passage effect can terminate the trajectories’ tendency to the positive and negative infinities with the PSE to form a complete periodic structure. Our findings contribute to a better analysis of the mixed-mode vibrations and provide valuable insights for the mechanism research and encourage or inhibit strategic planning of the mixed-mode vibrations.