Local coordinates for numerical integrators on manifolds

In constructing numerical integrators for ordinary differential equations on manifolds a common strategy is to introduce local coordinates on the manifold $\mathcal{M}$ in a neighbourho od of the point $p\in\mathcal{M}$ where the time stepping procedure is performed. If we denote with $T\mathcal{M}$ the tangent bundle of $\mathcal{M}$, typically in a neighbourhood of $p$ we are looking for a smooth mapping $\mathcal{R} :T\mathcal{M}\rightarrow \mathcal{M}$ that is reasonably easy to compute and such that the solution $y(t)$ of the differential equation around $p$ can be expressed in the form $$y(t)=\mathcal{R}_p(\sigma (t)), \hskip0.5cm \sigma (t)\in T_p\mathcal{M}.$$ If $\mathcal{R}_p$ is such that it possible to derive a differential equation for $\sigma (t)