Complex dynamics of compound bursting driven by saturation effect

Abstract The saturation effect, characterized by inherent non-smoothness, plays a pivotal role in shaping the qualitative behavior of nonlinear dynamical systems. To investigate the impact of saturation nonlinearity on compound bursting, a modified Lorenz-like system incorporating a saturation function and slow-varying excitation is considered. Numerical simulations demonstrate that variations in the saturation threshold govern an ordered evolution of compound bursting patterns, manifested as a stepwise increase and reorganization of bursting clusters. A systematic bifurcation analysis of the fast subsystem reveals that saturation fundamentally restructures the equilibrium configuration, stability properties, and bifurcation topology. In addition to classical fold and Hopf bifurcations, various non-smooth bifurcations about equilibrium points and limit cycles are identified. Based on slow–fast analysis, the underlying mechanisms of compound bursting are systematically elucidated. In particular, it is found that the non-smooth homoclinic bifurcation serves as the organizing center, leading to the generation and rearrangement of multiple bursting clusters. Furthermore, the slow passage effect suppresses further transitions near critical bifurcations, resulting in the saturation of the number of bursting clusters. These findings provide new insight into compound bursting in piecewise-smooth slow–fast systems and highlight the essential role of saturation-induced non-smoothness in bursting dynamics.