Delay-coordinate maps and the spectra of Koopman operators

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the discrete spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions from high-dimensional data in systems with pure point or mixed spectra.