koopman算符下的标准映射的动力学
首发时间:2020-10-26
摘要:标准映射虽然是一个相对简单的二维系统,但是在不同的非线性条件下它会产生不同的动力学模式。而Koopman算符提供了有效工具用于帮助人们研究系统的动力学性质,主要描述了可观测量沿相空间轨道的演化规律,可以用于提取系统的关键演化模式。本文利用Koopman算符理论通过考察时间序列来构造相关的数据矩阵,近似得到此算符的有限维表达,在不同非线性参数k下的标准映射中获得算符的本征值和本征函数。选择合适的本征值所对应的本征函数对相空间进行划分,并且分析了标准映射在不同参数条件下的主要动力学模式。
关键词: 标准映射 动力学行为 Koopman算符 周期轨道 混沌
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Dynamics of standard mapping under Koopman operator
Abstract:Although the standard mapping is a relatively simple two-dimensional system, it will produce different dynamic modes under different nonlinear conditions. The Koopman operator provides an effective tool to help people study the dynamics of the system, mainly describing the evolution of the observables along the phase space orbit, and can be used to extract the key evolution mode of the system. This paper uses the Koopman operator theory to construct the relevant data matrix by investigating the time series, and approximately obtains the finite-dimensional expression of this operator, and obtains the eigenvalue and eigenfunction of the operator from the standard mapping under different nonlinear parameters k. The eigenfunction corresponding to the appropriate eigenvalue is selected to divide the phase space, and the main dynamic modes of the standard mapping under different parameter conditions are analyzed.
Keywords: Standard mapping dynamic behavior Koopman operator periodic orbit chaos
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