Manifold-based stability assessment in bursting oscillations induced by transcritical bifurcation under frequency switching

This study investigates the mechanisms driving bursting oscillation stability and unconventional bursting dynamics related to transcritical bifurcations under frequency switching. Through analytical and numerical examination of five canonical vector fields spanning one- and two-dimensional systems under dual typical switching schemes, we establish that the boundedness of basins of attraction, which leads to divergent behaviors due to basin of attraction mismatch or closure, originates from the configuration of stable manifolds of the saddle points constituting the transcritical bifurcation. Based on this mechanism, a simulation-free methodology is proposed for prior detection of the divergence potential and tracking corresponding threshold windows by quantifying the intersections between the y-values of switching positions and the invariant manifolds. Based on this framework, two-dimensional vector fields with transcritical bifurcation are classified into two categories according to their invariant manifold structures: the encircling configuration of saddle-point stable manifolds typically yields zero or double intersection points, associated with potential divergence behaviors caused by basin closure or mismatch with threshold variations, and the splayed configuration consistently exhibits a single intersection point, ensuring stable frequency-switching processes.