The motivation of our research is to establish a Laplace-domain theory that provides principles and methodology to analyze and synthesize systems with nonlinear dynamics. A semigroup of composition operators defined for nonlinear autonomous dynamical systems---the Koopman semigroup and its associated Koopman generator---plays a central role in this study. We introduce the resolvent of the Koopman generator, which we call the Koopman resolvent, and provide its spectral characterization for three types of nonlinear dynamics: ergodic evolution on an attractor, convergence to a stable equilibrium point, and convergence to a (quasi-)stable limit cycle. This shows that the Koopman resolvent provides the Laplace-domain representation of such nonlinear autonomous dynamics. A computational aspect of the Laplace-domain representation is also discussed with emphasis on non-stationary Koopman modes.

中文翻译：

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Koopman Resolvent：非线性自治动力系统的拉普拉斯域分析

我们研究的动机是建立一个拉普拉斯域理论，为分析和合成具有非线性动力学的系统提供原理和方法。为非线性自主动力系统定义的组合算子半群——Koopman 半群及其相关的 Koopman 生成器——在本研究中发挥着核心作用。我们介绍了 Koopman 生成器的求解器，我们称之为 Koopman 求解器，并提供了三种非线性动力学的谱表征：吸引子上的遍历演化、收敛到稳定的平衡点和收敛到（准）稳定的极限循环。这表明 Koopman 解析器提供了这种非线性自主动力学的拉普拉斯域表示。