## Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems

- Author(s): Mohr, Ryan M.
- Advisor(s): Mezic, Igor
- et al.

## Abstract

The dominating methodology used in the study of dynamical systems is the geometric picture introduced by Poincare. The focus is on the structure of the state space and the asymptotic behavior of trajectories. Special solutions such as fixed points and limit cycles, along with their stable and unstable manifolds, are of interest due to their ability to organize the trajectories in the surrounding state space.

Another viewpoint that is becoming more prevalent is the operator-theoretic / functional-analytic one which describes the system in terms of the evolution of functions or measures defined on the state space. Part I of this doctoral dissertation focuses on the Koopman, or composition, operator that determines how a function on the state space evolves as the state trajectories evolve. Most current studies involving the Koopman operator have dealt with its spectral properties that are induced by dynamical systems that are, in some sense, stationary (in the probabilistic sense). The dynamical systems studied are either measure-preserving or initial conditions for trajectories are restricted to an attractor for the system. In these situations, only the point spectrum on the unit circle is considered; this part of the spectrum is called the unimodular spectrum. This work investigates relaxations of these situations in two different directions. The first is an extension of the spectral analysis of the Koopman operator to dynamical systems possessing either dissipation or expansion in regions of their state space. The second is to consider switched, stochastically-driven dynamical systems and the associated collection of semigroups of Koopman operators.

In the first direction, we develop the Generalized Laplace Analysis (GLA) for both spectral operators of scalar type (in the sense of Dunford) and non spectral operators. The GLA is a method of constructing eigenfunctions of the Koopman operator corresponding to non-unimodular eigenvalues. It represents an extension of the ergodic theorems proven for ergodic, measure-preserving, on-attractor dynamics to the case where we have off-attractor dynamics. We also give a general procedure for constructing an appropriate Banach space of functions on which the Koopman operator is spectral. We explicitly construct these spaces for attracting fixed points and limit cycles. The spaces that we introduce and construct are generalizations of the familiar Hilbert Hardy spaces in the complex unit disc.

In the second direction, we develop the theory of switched semigroups of Koopman operators. Each semigroup is assumed to be spectral of scalar-type with unimodular point spectrum, but possibly non-unimodular continuous spectrum. The functions evolve by applying one semigroup for a period of time, then switching to another semigroup. We develop an approximation of the vector-valued function evolution by a linear approximation in the vector space that the functions map into. A basis for this linear approximation is constructed from the vector-valued modes that are coefficients of the projections of the vector-valued observable onto scalar-valued eigenfunctions of the Koopman operator. The unmodeled modes show up as noisy dynamics in the output space. We apply this methodology to traffic matrices of an Internet Service Provider's (ISP's) network backbone. Traffic matrices measure the traffic volume moving between an ingress and egress router for the network's backbone. It is shown that on each contiguous interval of time in which a single semigroup acts the modal dynamics are deterministic and periodic with Gaussian or nearly-Gaussian noise superimposed.

Part II of the dissertation represents a divergence from the first part in that it does not deal with the Koopman operator explicitly. In the second part, we consider the problem of using exponentially mixing dynamical systems to generate trajectories for an agent to follow in its search for a physical target in a large domain. The domain is a compact subset of the n-dimensional Euclidean space R^{n}. It is assumed that the size of the target is unknown and can take any value in some continuous range. Furthermore, it is assumed that the target can be located anywhere in the domain with equal probability.

We cast this problem as one in the field of quantitative recurrence theory, a relatively new sub-branch of ergodic theory. We give constructive proofs for upper bounds of hitting times of small metric balls in R^{n} for mixing transformations of various speeds. The upper bounds and limit laws we derive say, approximately, that the hitting time is bounded above by some constant multiple of the inverse of the measure of the metric ball. From these results, we derive upper bounds for the expected hitting time, with respect to the range of target sizes [delta, V), to be of order O(-ln delta). First order, continuous time dynamics are constructed from discrete time mixing transformations and upper bounds for these hitting times are shown to be proportional to the discrete time case.