论文标题
使用曲折的持久同源性检测动态系统中的HOPF分叉
Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
论文作者
论文摘要
动态系统中的分叉表征了系统行为的定性变化。因此，它们的检测很重要，因为它们可以发出从正常系统操作到迫在眉睫的故障的过渡。尽管在这种情况下已经使用了标准持续的同源性，但通常需要分析持久图的集合，这反过来又大大提高了计算成本。使用曲折的持久性，我们只能在一个持久图中捕获动态系统状态空间中的拓扑变化。在这里，我们提出了使用Zigzag（Buzz）的分叉，这是一种使用锯齿形持久性研究和检测分叉的单步方法。 Buzz方法能够成功地检测两个合成示例和示例动态系统中的这种行为。
Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.
