Q:

How to show $\phi(t,x)$ is a flow of a vector field?

Consider the function $\phi(t,x)$ where $x=(x_1,x_2) \in \mathbb{R}^2$. How can we check if $\phi(t,x)$ is a flow for a vector field?
$$
\phi(t,x)=(x_1\cos(r^2t)+x_2\sin(r^2t),-x_1\sin(r^2t)+x_2\cos(r^2t))
$$
where $r^2=x_1^2+x_2^2$.
My try:
We need to check if $\phi(0,x)=\phi(t,\phi(-t,x))$. I am wondering if there is a shorter way?

A:

If you write $\zeta(t,z)=ϕ_1(t,x)+iϕ_2(t,x)$ where $z=x_1+ix_2$, you will find that
$$
\zeta(t,z)=e^{-i|z|^2t}z
$$
and as a consequence $|\zeta(t,z)|=|z|$ constant along the trajectories, so that the group conditions for the flow are easy to prove.

