用局部动力学解释混沌系统的模拟预测
Using local dynamics to explain analog forecasting of chaotic systems
摘要：
类比是系统状态的最近邻居。通过使用时间上的类似物及其后继物，人们能够产生经验预测。几种模拟预报方法已用于大气应用，并在著名的动力系统上进行了测试。虽然在实践中有效，模拟方法和动力系统之间的理论联系被忽视了。模拟预测可以与感兴趣的系统的真实动力学方程相关。本研究探讨不同的模拟预测策略的性质，采取本地近似系统的动态。我们发现，模拟预测性能与流图的局部雅可比矩阵高度相关，并且模拟预测与线性回归相结合可以捕获该雅可比矩阵的投影。所提出的方法允许估计模拟预测误差，并比较不同的模拟方法。这些结果是来自解析和测试两个简单的混沌动力系统的数值。
英文摘要：
Analogs are nearest neighbors of the state of a system. By using analogs and their successors in time, one is able to produce empirical forecasts. Several analog forecasting methods have been used in atmospheric applications and tested on well-known dynamical systems. Although efficient in practice, theoretical connections between analog methods and dynamical systems have been overlooked. Analog forecasting can be related to the real dynamical equations of the system of interest. This study investigates the properties of different analog forecasting strategies by taking local approximations of the system's dynamics. We find that analog forecasting performances are highly linked to the local Jacobian matrix of the flow map, and that analog forecasting combined with linear regression allows to capture projections of this Jacobian matrix. The proposed methodology allows to estimate analog forecasting errors, and to compare different analog methods. These results are derived analytically and tested numerically on two simple chaotic dynamical systems.
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