  This manuscript presents a novel approach to nonlinear system identification
leveraging densely defined Liouville operators and a new "kernel" function that
represents an integration functional over a reproducing kernel Hilbert space
(RKHS) dubbed an occupation kernel. The manuscript thoroughly explores the
concept of occupation kernels in the contexts of RKHSs of continuous functions,
and establishes Liouville operators over RKHS where several dense domains are
found for specific examples of this unbounded operator. The combination of
these two concepts allow for the embedding of a dynamical system into a RKHS,
where function theoretic tools may be leveraged for the examination of such
systems. This framework allows for trajectories of a nonlinear dynamical system
to be treated as a fundamental unit of data for nonlinear system identification
routine. The approach to nonlinear system identification is demonstrated to
identify parameters of a dynamical system accurately, while also exhibiting a
certain robustness to noise.
