A Subtle Link in Switched Dynamical Systems: Saddle-Node Bifurcation Meets Border Collision

Abstract

Switched dynamical systems are known to exhibit border collision, in which a particular operation is terminated and a new operation is assumed as one or more parameters are varied. We report a subtle relation between border collision and saddle-node bifurcation in such systems. Our main finding is that the border collision and the saddle-node bifurcation are actually linked together by unstable solutions which have been generated from the same saddle-node bifurcation. Since unstable solutions are not observable directly, such a subtle relation has not been known. We describe an effective method to track solutions regardless of their stability, allowing the subtle phenomenon to be uncovered. Two typical DC-DC converters are observed to verify our finding.