On bursting patterns and their mechanisms modulated by the rotation of the discontinuous boundary

Piecewise-smooth dynamical systems are widely utilized in the dynamic description of physical processes, characterized by the state space segmented into multiple distinct regions by discontinuous boundaries. These boundaries frequently give rise to complex dynamical behaviors. This paper aims to investigate the impact of the rotation of the discontinuous boundary (RDB) on the fast–slow dynamics and to elucidate the cross-regional transition mechanisms of RDB-modulated bursting. A theoretical model is developed based on a modified Chua’s circuit. The modulatory effects of RDB on the topological structure of equilibrium branches and the bifurcation characteristics are systematically analyzed. The results indicate that variations in the rotation parameter δ lead to significant adjustments in the bifurcation structures of both the smooth subsystems and the discontinuous boundary, thereby altering the bifurcation sequences encountered during boundary crossing and shaping the resulting bursting patterns. Four types of RDB-modulated bursting oscillations are identified, namely “PB–PB”, “NHB–NHB”, “FB–NFB”, and “Non-switching”. The cross-regional transition mechanisms of each type are further elucidated. Several previously unreported dynamical mechanisms are identified and analyzed in detail. For example, a reordering of the relative sequence of key bifurcation points along the equilibrium branch can lead to the disappearance of spiking oscillations. Moreover, RDB may cause a key bifurcation point to lie outside the modulation range of the slow variable, thereby confining the trajectory to a single subregion. Overall, this work deepens the understanding of multiscale complex dynamics in piecewise-smooth systems. These findings provide new theoretical insights and research perspectives for the control and application of non-smooth dynamical systems.