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Title: Connecting border collision with saddle-node bifurcation in switched dynamical systems
Authors: Ma, Y
Tse, CKM 
Kousaka, T
Kawakami, H
Issue Date: Sep-2005
Source: IEEE transactions on circuits and systems. II, Express briefs, Sept. 2005, v. 52, no. 9, p. 581-585
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. 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.
Keywords: Border collision
Saddle-node bifurcation
Switched dynamical systems
Unstable solutions
Publisher: Institute of Electrical and Electronics Engineers
Journal: IEEE transactions on circuits and systems. II, Express briefs 
ISSN: 1549-7747
EISSN: 1558-3791
DOI: 10.1109/TCSII.2005.850488
Rights: © 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
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