Bursting Oscillations Induced by the Variable Discontinuous Boundary in Chua’s Circuit

Bursting oscillations and bifurcation mechanisms in piecewise circuit systems have long been a research focus in the fields of dynamics and control. While most previous studies have focused on systems with a fixed discontinuous boundary, this paper examines systems with a variable discontinuous boundary. We develop a mathematical model based on Chua’s circuit and present numerical simulations of bursting oscillations under varying parameters. The impact of the translation of the discontinuous boundary (TDB) on the system’s topological structure and non-smooth bifurcations is analyzed. By combining the two-parameter bifurcation set of equilibrium points with the superposition diagram of the transformed phase diagram (TPD), we reveal mechanisms behind different bursting modes induced by the TDB. A Multisim-based simulation circuit is designed to validate the research results. It is found that the topology of the equilibrium branches of each subsystem remains the same during the TDB, but the number of smooth bifurcations changes due to the switching between subsystems. The TDB also alters the characteristics of the boundary equilibrium point, leading to the catastrophic disappearance or emergence of non-smooth limit cycles, which consequently changes the number of spiking oscillations. Moreover, the interaction of the slow passage effect (SPE) and the TDB causes the trajectory to remain in a delayed segment, affecting the number of spiking oscillations per period. Additionally, the TDB may lead to the disappearance of bursting oscillations, then the system exhibits a behavior similar to that of simple harmonic motion. Our study expands the scope of research on piecewise-smooth circuit systems.