Local and global holomorphic extensions of time-varying real analytic vector fields

In this paper, we consider time-varying real analytic vector fields as curves on the space of real analytic vector fields. Using a suitable topology on the space of real analytic vector fields, we study and characterize different properties of time-varying real analytic vector fields. We study holomorphic extensions of time-varying real analytic vector fields and show that under suitable integrability conditions, a time-varying real analytic vector field on a manifold can be extended to a time-varying holomorphic vector field on a neighbourhood of that manifold. Moreover, we develop an operator setting, where the nonlinear differential equation governing the flow of a time-varying real analytic vector field can be considered as a linear differential equation on an infinite dimensional locally convex vector space. Using the holomorphic extension results, we show that the integrability of the time-varying vector field ensures the convergence of the sequence of Picard iterations for this linear differential equation. This gives us a series representation for the flow of an integrable time-varying real analytic vector field. We also define the exponential map between integrable time-varying real analytic vector fields and their flows. Using the holomorphic extensions of time-varying real analytic vector fields, we show that the exponential map is sequentially continuous.