 Solution

The flow of the vector field is obtained by solvingDefine $g^{\epsilon }(x,u)$ to be this solution at parameter $\epsilon$. Uniqueness implies $g^0=\mathrm{id}$ and $g^{\epsilon +\delta }=g^{\epsilon }\circ g^{\delta }$ wherever both sides are defined: shifting the solution parameter gives the same initial-value problem. In particular $g^{-\epsilon }$ is the local inverse. This constructs the one-parameter group generated by the vector field, generally as a local group if the flow is not complete.