Connecting Border Collision with Saddle-Node Bifurcation in Switched Dynamical Systems

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. In this brief, 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 connection has not been known. This also explains why border collision manifests itself as a “jump” from an original stable operation to a new stable operation. Furthermore, as the saddle-node bifurcation and the border collision merge tangentially, the connection shortens and eventually vanishes, resulting in an apparently continuous transition at border collision in lieu of a “jump.” In this brief, we describe an effective method to track solutions regardless of their stability, allowing the subtle phenomenon to be uncovered.