I am working on an example of 1d flow conjugation from class. For a vector field on $\mathbb{R}$, we have a flow $\phi_{t}(x)=xe^t$, with vector field $F(x)=x$ and another flow $\psi_{t}(x)=xe^{2t}$ for a vector field $G(x)=2x$.
We want to show these flows are conjugate. I was given that the conjugation homeomorphism looks like: $h(x)=|x|^{\theta}$ if $x>0$ and $h(x)=-|x|^{\theta}$ if $x\leq 0$.
Finding the theta for this to be a conjugation is an easy exercise ($\theta=2$, but I am confused about where the form of $h$ came from. Is this just an often good candidate as $exp$ is smooth and invertible or is there something deeper going on here? Also, if the former is true, is there a way to extend this to higher dimensions? In general flowing conjugacy functions seems difficult.
The usual procedure is to try to find a conjugacy that is $C^\infty$ outside the origin. This allows you to take derivatives. Namely, assume that $$ h(e^tx)=e^{2t}h(x). $$ Taking derivatives with respect to $t$ we get $h'(e^tx)e^tx=2e^{2t}h(x)$, and letting $t=0$ yields the identity $h'(x)x=2h(x)$. Solving for $h$ gives you the conjugacy (which is $0$ at $0$).
A similar procedure can be used in higher-dimensions but it is much more complicated.