information:extraction, structural:scholar_edition
Time-dependent ordinary differential equations

In the case of time-dependent vector fields F: R n × R → R n {\displaystyle {\boldsymbol {F}}:\mathbb {R} ^{n}\times \mathbb {R} \to \mathbb {R} ^{n}}, one denotes φ t, t 0 (x 0) = x (t + t 0), {\displaystyle \varphi ^{t,t_{0}}({\boldsymbol {x}}_{0})={\boldsymbol {x}}(t+t_{0}),} where x: R → R n {\displaystyle {\boldsymbol {x}}:\mathbb {R} \to \mathbb {R} ^{n}} is the solution of

Then φ t, t 0 (x 0) {\displaystyle \varphi ^{t,t_{0}}({\boldsymbol {x}}_{0})} is the time-dependent flow of F. It is not a "flow" by the definition above, but it can easily be seen as one by rearranging its arguments. Namely, the mapping

indeed satisfies the group law for the last variable:

One can see time-dependent flows of vector fields as special cases of time-independent ones by the following trick. Define

Then y (t) is the solution of the "time-independent" initial value problem

if and only if x (t) is the solution of the original time-dependent initial value problem. Furthermore, then the mapping φ is exactly the flow of the "time-independent" vector field G.