I have been studying differential geometry and dynamical systems. I learned, that in smooth geometry, there are all those objects, operations, and morphisms: metric, pushforward, pullback, tangent space, (co)-vectors, Lie brackets. For dynamical systems, we also have a plural vocabulary: flow, eigenvectors, and holonomicity. My question is:
Given a dynamical system $\Sigma$ given by tuple $(\mathbb{R}, \mathcal{X})$ and defined by position vector $x^i \partial_i$ respective to origin point $\mathcal{O}$ such that velocity vector $\dot{x} \in T_\mathcal{O} \mathcal{X}$ is given by smooth map $a(x)$. For a countable covering atlas $(\varphi_i, U_i)$, how can we describe $a(x)$ in local coordinate chat $U_i$ i.e. local coordinates $\{u_i^j\}$?
I hope this is not too complicated. I am new to this, and everything is so "competitive", it is hard to learn it by myself.