Summary
Koopman operator theory has attracted a lot of attention in recent years, since it allows for linear representations of quite general (autonomous) nonlinear systems. Two questions that frequently arise in this context are the dimension of the linear Koopman system, and the question how a linear system may represent a nonlinear system with multiple attractors/repellors. We use a bistable duffing oscillator as a simple example to discuss these questions which are both intertwined with the concept of the so-called principal Koopman eigenfunctions. We numerically calculate the five principal eigenfunctions of the bistable Duffing oscillator and show how the nonlinearity, discontinuity and complicated range of the principal Koopman eigenfunctions conceal the effort that is required to obtain a seemingly simple linear system in the Koopman framework.
