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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. • Rayleigh oscillator with two slow square wave excitations. • Step-shaped sharp transition induced by slow square wave excitations. • Rectangular-pulse-shaped explosion induced by slow square wave excitations. • Fast–slow dynamics related to the slow square wave excitations. [ABSTRACT FROM AUTHOR]

Sharp transitions in relation to the variation of system parameters are frequently encountered in many multiple-timescale systems, and they have been found to be an important factor related to the generation of mixed-mode oscillations (MMOs). The present paper aims to report a novel type of sharp transition, referred to as step-shaped sharp transition, in a nonlinear gyroscope oscillator with multiple-frequency excitations, and investigate the resulting MMOs. We show that step-shaped sharp quantitative changes in relation to the variation of system parameters can be observed in the equilibrium branch, which yields the step-shaped sharp transition. In particular, with the increase of the frequency ratio between the parametric and external excitations, more step-shaped sharp transitions appear in the equilibrium branches, which evolve into the ones displaying different structures. Based on this, the rectangular-pulse-shaped explosion of equilibria is created. Furthermore, these sharp transitions can form active areas for the MMOs, leading to the alternations between large-amplitude and small-amplitude oscillations, and finally the route to MMOs is created. Our findings have significant implications for understanding the fast-slow dynamics of the nonlinear gyroscope oscillator, contributing to the exploration of new routes to MMOs. Thus, the results could provide theoretical support for the potential application of gyroscopes. • The nonlinear gyroscope oscillator with multiple-frequency excitations. • A novel sharp transition type: step-shaped sharp transition of equilibria. • Rectangular-pulse-shaped explosion induced by step-shaped sharp transitions. • Mixed-mode oscillations induced by two novel sharp transitions. [ABSTRACT FROM AUTHOR]