  This paper uses data-driven operator theoretic approaches to explore the
global phase space of a dynamical system. We defined conditions for discovering
new invariant subspaces in the state space of a dynamical system starting from
an invariant subspace based on the spectral properties of the Koopman operator.
When the system evolution is known locally in several invariant subspaces in
the state space of a dynamical system, a phase space stitching result is
derived that yields the global Koopman operator. Additionally, in the case of
equivariant systems, a phase space stitching result is developed to identify
the global Koopman operator using the symmetry properties between the invariant
subspaces of the dynamical system and time-series data from any one of the
invariant subspaces. Finally, these results are extended to topologically
conjugate dynamical systems; in particular, the relation between the Koopman
tuple of topologically conjugate systems is established. The proposed results
are demonstrated on several second-order nonlinear dynamical systems including
a bistable toggle switch. Our method elucidates a strategy for designing
discovery experiments: experiment execution can be done in many steps, and
models from different invariant subspaces can be combined to approximate the
global Koopman operator.
