UC Santa Barbara

Modeling complex dynamical systems, with an eye towards accurate reconstruction of individual trajectories, is an inherently difficult proposition. This difficulty, which arises from exponential sensitivity to initial conditions, also makes it nearly impossible to simulate long-term trajectories. Fortunately, in many applications, it is sufficient if a reduced-order model is able to just capture the global invariant and quasiperiodic structures which fall within the resolution of observation.

Koopman operator theory is a mathematical formalism which allows one to extract these global structures. Many geometric notions in the state-space, as well as statistical properties on the attractor, are characterizable in terms of the spectral decomposition of this linear operator. The linearization of a system through the Koopman operator is achieved by lifting the dynamics to the space of observables, making the technique applicable to any system for which the flow map is well defined.

The generality of the Koopman formalism is specifically demonstrated here on a nonsmooth system in which a pendulum is subjected to downward kicks at fixed angles. In the absence of damping, it is shown that the fixed space of the operator yields a characterization of the ergodic partition, whereas the set of limit cycles on the attractor gives rise to a continuous spectrum. Under marginal damping, the continuous spectrum is shown to disintegrate and the appearance of dissipative eigenvalues associated with the transient dynamics of the stable fixed point can be related to a linear system by means of a semi-conjugacy.

Critical to the wide-scale application of Koopman operator theory are numerical methods which are able to approximate the spectral decomposition of observables. Given that any computational method must reduce the infinite-dimensional Koopman operator to something finite, this dissertation specifically examines how the Koopman operators of finite-state dynamical systems are related to their infinite dimensional counterparts.

A convergence result is proven here in which the class of continuous, measure-preserving automorphisms on compact metric spaces is shown to be approximable by periodic systems on a finite state-space. The Koopman operators of these so-called ``periodic approximations'' are shown to converge spectrally to that of the original operators in a weak sense. Herein, it is demonstrated that even though the individual rank-one spectral projectors are spurious, smooth weighted sums of these projectors applied to a fixed observable are meaningful and converge to the quantities one would expect for their infinite-dimensional counterparts.

The results are generalizable to handle measure-preserving flows through an intermediate time-discretization. For convergence of the spectra to occur, a sufficient condition is derived requiring the spatial refinements in the periodic approximation to happen at a faster pace than the temporal refinements. Peculiarly, this requirement is somewhat opposite to what the CLF condition dictates for stability of finite difference schemes. Albeit, the periodic approximation may be interpreted as a specific semi-Lagrangian scheme where additional ``global'' efforts are made to prevent two grid points from collapsing into one.

Given the aforementioned approximation results, numerically convergent methods for computing spectra of observables are formulated. The crucial part of the numerical method is the construction of the periodic map, which in the general case can be done by solving a bipartite matching problem. It is shown that for Lipschitz continuous maps, this process can be executed in roughly $O(n^{3m/2})$ complexity, where $m$ is the dimension of the state-space and $n$ the size of the partition in each dimension. Since the remainder of the numerical method exploits the structure of permutation matrices and only requires computing Discrete Fourier transforms, the entire scheme is numerically stable and ``fast'', hence yielding a viable method to compute the spectral decomposition of Koopman operators for the measure-preserving case.