This is probably a trivial question but I don't have a clue where to begin.
Suppose $x(t)$ is a T-periodic solution to the differential equation $$\frac{dX}{dt}=F(X)$$ where F(X) is in $C^1$. Show that $x(t+\Delta t)$ is also a solution for any $\Delta t$.
I'm not necessarily looking for a full proof but rather a hint would be amazing.
Let $y(t)=x(t+\Delta t)$. Then
$$\frac{\mathrm dy}{\mathrm dt}(t)=\frac{\mathrm dx}{\mathrm dt}(t+\Delta t)=F(x(t+\Delta t)).$$
This is in fact true even if $x$ wasn't assumed periodic. However, the periodicity condition in this case replaces a boundary (or initial) condition, and in this case the periodicity is clearly preserved by the shift.