Chaos as an intermittently forced linear system.
Chaos as an intermittently forced linear system.
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DOI：
10.1038/s41467-017-00030-8
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发表时间：
2017-05-30
影响因子：
16.6
通讯作者：
Kutz JN
中科院分区：
文献类型：
--
作者：
Brunton SL;Brunton BW;Proctor JL;Kaiser E;Kutz JN
理解混沌中有序和无序的相互作用是现代定量科学的核心挑战。非线性动力学的近似线性表示长期以来一直在寻找，这使得人们对库普曼理论产生了极大的兴趣。我们提出了一个普遍的，数据驱动的分解混沌作为一个间歇性强迫线性系统。这项工作结合了延迟嵌入和Koopman理论，将混沌动力学分解成一个线性模型，在领先的延迟坐标与强迫低能量延迟坐标，这被称为汉克尔替代观点的Koopman（HAVOK）分析。这种分析适用于洛伦兹系统和现实世界的例子，包括地球的磁场逆转和麻疹疫情。在每种情况下，强迫统计是非高斯的，长尾对应于罕见的间歇性强迫之前切换和突发现象。强迫活动划分相干相空间区域的动态近似线性的那些是强非线性的。神经科学或金融等领域产生的大量数据需要有效的策略来挖掘数据，以揭示潜在的动态。在这里，Brunton等人开发了一种数据驱动技术来分析混沌系统并根据强迫线性模型预测其动态。
Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear. The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.