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.