Route to mixed-mode oscillations via step-shaped sharp transition of equilibria in a nonlinear gyroscope oscillator

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.