On the lifting and reconstruction of dynamical systems with multiple attractors

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.