稳定不动点动力学作用角表示的等稳线、等时线和库普曼谱
Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics
摘要：
对于渐近周期系统，通过计算所谓的等时线（isochrons），即在极限环上收敛到相同轨迹的点集，可以得到动力学的强大（相位）简化。受激励系统的分析，类似的减少已被尝试为非周期系统承认一个稳定的不动点。在这种情况下，仍然可以定义等时线，但它们不能捕获轨迹的渐近行为。相反，我们称之为isostables的兴趣集在文献中被定义为在不动点的稳定慢流形上朝着相同轨迹收敛的点集。然而，事实证明，这个定义的等稳仅适用于系统的慢-快动态。此外，有效的方法来计算的等稳态是失踪。本文基于Koopman算子的谱性质，给出了稳定不动点的等稳线的定义和计算的一般框架.更准确地说，等稳线被定义为库普曼算子的特定本征函数的水平集。通过这种方法，等稳线是与系统的渐近性质相关的唯一且定义明确的对象。此外，该框架揭示了等稳线和等时线是两个不同但互补的概念，它们定义了一组动力学的作用角坐标。此外，一个有效的算法计算的等稳，这依赖于沿轨迹的拉普拉斯平均值的评价。该方法与激发FitzHugh-Nagumo模型和洛伦兹模型说明。最后，我们讨论了这些方法的基础上的Koopman运营商框架涉及到系统的全局线性化和特殊的Lyapunov函数的推导。
英文摘要：
For asymptotically periodic systems, a powerful (phase) reduction of the dynamics is obtained by computing the so-called isochrons, i.e. the sets of points that converge toward the same trajectory on the limit cycle. Motivated by the analysis of excitable systems, a similar reduction has been attempted for non-periodic systems admitting a stable fixed point. In this case, the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories. Instead, the sets of interest-that we call isostables-are defined in literature as the sets of points that converge toward the same trajectory on the stable slow manifold of the fixed point. However, it turns out that this definition of the isostables holds only for systems with slow-fast dynamics. Also, efficient methods for computing the isostables are missing. The present paper provides a general framework for the definition and the computation of the isostables of stable fixed points, which is based on the spectral properties of the so-called Koopman operator. More precisely, the isostables are defined as the level sets of a particular eigenfunction of the Koopman operator. Through this approach, the isostables are unique and well-defined objects related to the asymptotic properties of the system. Also, the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action-angle coordinates for the dynamics. In addition, an efficient algorithm for computing the isostables is obtained, which relies on the evaluation of Laplace averages along the trajectories. The method is illustrated with the excitable FitzHugh-Nagumo model and with the Lorenz model. Finally, we discuss how these methods based on the Koopman operator framework relate to the global linearization of the system and to the derivation of special Lyapunov functions.
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