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.Comment: 8 page