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

Yue Ma, Chi Kong Tse, Takuji Kousaka, Hiroshi Kawakami

Research output: Journal article publicationJournal articleAcademic researchpeer-review

28 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)581-585
Number of pages5
JournalIEEE Transactions on Circuits and Systems II: Express Briefs
Issue number9
Publication statusPublished - 1 Jan 2005


  • Border collision
  • saddle-node bifurcation
  • switched dynamical systems
  • unstable solutions

ASJC Scopus subject areas

  • Electrical and Electronic Engineering


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