The Koopman operator provides a linear perspective on non-linear dynamics by focusing on the evolution of observables in an invariant subspace. Observables of interest are typically linearly reconstructed from the Koopman eigenfunctions. Despite the broad use of Koopman operators over the past few years, there exist some misconceptions about the applicability of Koopman operators to dynamical systems with more than one fixed point. In this work, an explanation is provided for the mechanism of lifting for the Koopman operator of nonlinear systems with multiple attractors. Considering the example of the Duffing oscillator, we show that by exploiting the inherent symmetry between the basins of attraction, a linear reconstruction with three degrees of freedom in the Koopman observable space is sufficient to globally linearize the system.

翻译：Koopman算子通过关注不变量子空间中观测量的演化，提供了一种理解非线性动力学的线性视角。通常，我们通过Koopman本征函数对感兴趣的观测量进行线性重构。尽管近年来Koopman算子得到了广泛应用，但关于其对具有多个不动点的动力系统的适用性仍存在一些误解。本文针对多吸引子非线性系统的Koopman算子提升机制给出了解释。通过考虑Duffing振荡器实例，我们证明利用吸引域之间的固有对称性，在Koopman观测量空间中仅需三个自由度的线性重构即可实现系统的全局线性化。