Data-Driven Operator Theoretic Methods for Global Phase Space Learning

In this work, we developed new 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, the bistable toggle switch is considered to illustrate the proposed results. The construction of this global Koopman operator is very helpful in experimental works as multiple experiments cannot be performed at the same time starting from several initial conditions.