Fast–slow dynamics related to sharp transition behaviors in the Rayleigh oscillator with two slow square wave excitations

This paper aims to report the fast–slow dynamics in the Rayleigh oscillator with two slow square wave excitations. Typically, a square wave excitation is a periodic excitation that has instantaneous alternation between maximum and minimum amplitude levels, which may trigger a transient response in the system. We show that, by introducing two square wave excitations with multiple frequencies, fast–slow oscillations with catastrophic jumps can be observed in the Rayleigh oscillator. Then, we explain the generation mechanisms of these fast–slow oscillations based on the fast–slow analysis, in which the two slow square wave excitations with different frequencies are transformed based on their frequency ratio, and subsequently treated as a single slow variable. Our study shows that square wave excitations play a crucial role in determining the fast–slow dynamics of the system. Under the control of square wave excitations, the equilibrium can exhibit rapid jumps in different levels, and the variation of amplitude of the limit cycle is not gradual, characterized by instantaneous increase and decrease. Such phenomenon can be referred to as the step-shaped sharp transition of attractor, which accounts for the generation of fast–slow oscillations. Furthermore, with the increase of the frequency ratios, the rectangular-pulse-shaped explosion can be created in the solutions branches of the fast subsystem, which thus leads to fast–slow oscillations with multiple catastrophic jumps. Based on this, the fast–slow oscillations of point–point type and fast–slow oscillations of cycle–cycle type are revealed.