Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems
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DOI：
10.3390/a13110278
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发表时间：
2020-09
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通讯作者：
Sarah Tymochko;E. Munch;Firas A. Khasawneh
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作者：
Sarah Tymochko;E. Munch;Firas A. Khasawneh
动力学系统中的分叉刻画了系统行为的质变。因此，它们的检测很重要，因为它们可以发出信号，指示从正常系统运行到即将发生故障的转变。在实验环境中，这种转换可能会导致不正确的数据或对整个实验造成损害。虽然在这种设置中使用了标准的持久同调，但它通常需要分析一组持久图，这反过来又大大增加了计算成本。利用之字形持久化，我们只需在一个持久化图中就可以捕捉到动力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，这是一种利用Zigzag持久性研究和检测分叉的一步方法。该方法能够在两个合成实例和一个实例动力系统中成功地检测到这种行为。
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. In an experimental setting, this transition could lead to incorrect data or damage to the entire experiment. 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.