Department of Physics, University of California, Davis, California 95616, USA.
Complexity Sciences Center, University of California, Davis, California 95616, USA.
Chaos. 2019 Sep;29(9):093128. doi: 10.1063/1.5099091.
Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, often described by complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces a block diagonal structure in operator approximations, revealing hidden organization. Practically, if the symmetries are known, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.
具有对称性的非线性动力系统表现出丰富多样的行为，这些行为通常通过复杂的吸引子盆地图谱和增强与抑制的分岔来描述。对称论证提供了一种研究这些集体行为并简化其分析的方法。Koopman 算子是一个无限维的线性算子，它通过对状态空间中函数的线性演化来完全捕捉系统的非线性动力学。重要的是，与局部线性化不同，它保留了系统的全局非线性特征。我们展示了对称性如何影响 Koopman 算子的结构及其谱性质。事实上，我们表明，对称考虑还可以简化使用扩展和核动态模态分解方法（EDMD 和核 DMD）寻找 Koopman 算子近似值。具体来说，表示论允许我们证明同构分量基在算子近似中诱导出一个块对角结构，揭示了隐藏的组织。实际上，如果知道对称性，EDMD 和核 DMD 方法可以被修改，以更有效地计算 Koopman 算子近似值及其特征值、特征函数和特征模式。最后，我们讨论了测量噪声的影响。
