Wilcox Formula and Two Pathways Lemma

In this talk we consider an analogue of Wilcox formula for vector fields on Banach manifolds. We use this result to show that trajectories of dynamic system on manifold can be approximated by trajectories of another structured system which is obtained in terms of commutators of vector fields of the original system. This result is stated as Two Pathways Lemma.
We demonstrate applications of this Lemma for approximation of trajectories of the original system for small and large times.