  In this work, we propose to apply the recently developed Koopman operator
techniques to explore the global phase space of a nonlinear system from
time-series data. In particular, we address the problem of identifying various
invariant subsets of a phase space from the spectral properties of the
associated Koopman operator constructed from time-series data. Moreover, in the
case when the system evolution is known locally in various invariant subspaces,
then a phase space stitching result is proposed that can be applied to identify
a global Koopman operator. A biological system, bistable toggle switch and a
second-order nonlinear system example are considered to illustrate the proposed
results. The construction of this global Koopman operator is very helpful in
experimental works as multiple experiments can't be run at the same time
starting from several initial conditions.
