Boundary-induced multistability and bursting oscillations in frequency switching slow–fast systems

Frequency-switching mechanisms, prevalent in engineering and scientific dynamical systems, introduce non-smooth dynamics that significantly impact system behavior. This study integrates frequency switching with slow–fast analysis in non-smooth dynamical systems, emphasizing the role of switching-induced multistability in shaping bursting dynamics. Using a modified normal-form vector field of a transcritical bifurcation subjected to state-dependent frequency-switching excitation, we show that although individual subsystems exhibit only equilibrium solutions, their interactions across the switching boundary induce non-smooth periodic orbits. These limit cycles coexist with equilibria, yielding switching-induced multistability. This multistability governs frequency-switching dynamics and determines the organization, patterns, and stability of bursting oscillations through a dual regulation mechanism involving the switching threshold and initial conditions: the switching threshold drives bifurcation evolution to control post-switching attractor selection, yielding diverse bursting patterns characterized by monostable–monostable, bistable–bistable, and bistable–monostable switching sequences; meanwhile, initial conditions exploit multistability to provide alternative stable switching pathways, thereby avoiding trajectory divergence during switching transitions and stabilizing bursting solutions. Our findings contribute to the characterizing and regulating bursting dynamics in frequency-switching systems.