I came across this dynamical system on $\mathbb{R}_{>0}$: consider the group homomorphism $$\mathbb{R}_{>0} \longrightarrow \rm{Diff}(\mathbb{R}_{>0})$$ which sends $\alpha$ to $x^\alpha$. Composing with the isomorphism of topological groups $\mathbb{R}_{>0} \sim (\mathbb{R},+)$ (given by the exponential) this gives a continuous dynamical system $$\mathbb{R} \longrightarrow \rm{Diff}(\mathbb{R}_{>0})$$ whose associated vector field (on $\mathbb{R}_{>0}$) is $x \textrm{log} x$.
The question is: has anyone heard about such a dynamical system? Is there any literature about it, has it got a name? What can you tell me about it?
Thank you very much.