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 a dynamical system 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算子通过关注不变子空间中可观测量的演化，为非线性动力学提供了线性视角。通常，我们利用Koopman本征函数对感兴趣的可观测量进行线性重构。尽管Koopman算子在过去几年中得到了广泛应用，但关于其应用于具有多个不动点的动力系统时，仍存在一些误解。本文针对具有多个吸引子的动力系统，解释了Koopman算子提升机制的原理。以Duffing振子为例，我们证明通过利用吸引域之间的固有对称性，在Koopman可观测空间中使用三个自由度的线性重构足以实现系统的全局线性化。