Near-Threshold Bursting Is Delayed by a Slow Passage near a Limit Point

In a general model for square-wave bursting oscillations, we examine the fast transition between the slowly varying quiescent and active phases. In this type of bursting, the transition occurs at a saddle-node (SN) bifurcation point of the fast-variable subsystem when the slow variable is taken to be the bifurcation parameter. A critical case occurs when the SN bifurcation point is also a steady solution of the full bursting system. In this case near the bursting threshold, the transition suffers a large delay. We propose a first investigation of this critical case that has been noted accidentally but never explored. We present an asymptotic analysis local to the SN point of the fast subsystem and quantitatively describe the slow passage near the SN point underlying the transition delay. Our analysis reveals that bursting solutions showing the longest delays and, correspondingly, the bursting threshold appear near but not exactly at the SN point, as is commonly assumed.