Connecting border collision with saddle-node bifurcation in switched dynamical systems
Connecting border collision with saddle-node bifurcation in switched dynamical systems
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DOI：
10.1109/tcsii.2005.850488
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发表时间：
2005-09
期刊：
影响因子：
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通讯作者：
Yue Ma;C. Tse;T. Kousaka;H. Kawakami
中科院分区：
文献类型：
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作者：
Yue Ma;C. Tse;T. Kousaka;H. Kawakami
已知的切换动力系统表现出边界冲突，其中当一个或多个参数变化时，特定操作被终止，而新操作被假定。在这篇简短的文章中，我们报告了这类系统中边界碰撞和鞍结点分叉之间的微妙关系。我们的主要发现是，边界碰撞和鞍结点分叉实际上是由同一鞍结点分叉产生的不稳定解联系在一起的。由于不稳定的解不能直接观察到，所以这种微妙的联系还不为人所知。这也解释了为什么边界碰撞表现为从原来的稳定运行到新的稳定运行的“跳跃”。此外，随着鞍结点分叉和边界碰撞切向合并，连接缩短并最终消失，导致在边界碰撞处出现明显的连续过渡，而不是“跳跃”。在这篇简短的文章中，我们描述了一种有效的方法来跟踪解决方案，无论它们的稳定性如何，允许发现微妙的现象。
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.