UC Santa Barbara

The use of the Koopman operator framework in dynamical systems has greatly expanded in recent years. Instead of considering the evolution of the state of a system, the Koopman semigroup tracks the evolution of observables on the state. Since the Koopman operator defined for an arbitrary dynamical system is linear, it allows us to use linear system theory and spectral methods to analyze nonlinear systems. This framework has also been extended to stochastic systems. Since the evolution of observables can only be defined probabilistically for random systems, stochastic Koopman operators are defined by taking the expectation of the future value of observables.

In the first part of this thesis, we review the basic theory of random dynamical systems and stochastic Koopman operators. We can use these operators to represent a nonlinear RDS as an infinite dimensional linear operator. The basic theorems and definitions are given in this section, which will help form the foundation for the algorithms discussed in the second and third sections. Further, some simple examples are given for which the stochastic Koopman operator is well understood. These examples will recur as we use them to test the algorithms in the second section.

The second section is devoted to the analysis of Dynamic Mode Decomposition (DMD) algorithms. DMD algorithms approximate a finite section of the (stochastic) Koopman operator using data from a trajectory. However, these methods are sensitive to noise, and will give a biased approximation if the observables contains randomness. To combat this, we introduce an new DMD variant which can approximate a finite section of the stochastic Koopman operator even when the data contains measurement noise. Further, we extend this algorithm for use with time delayed observables to create a variant of Hankel DMD which will converge for stochastic systems. We then demonstrate these algorithms on numerical examples.

In the final section, we will discuss the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm for stochastic differential equations. The SINDy algorithm allows one to generate a representation of an ODE using a dictionary of functions and data from a trajectory. This algorithm has been extended to SDEs, but the accuracy is limited by the numerical approximations of the drift and diffusion functions. We demonstrate how we can use higher order approximations to these functions to generate a far more accurate representation of the SDE. We then test these approximations on several examples.