I'm working on the Palis's ''Geometrical Theory of Dynamical Systems''. I have the next problem:
Let $X$ and $Y$ be $C^1$ vector fields on $\mathbb{R}^m$. Suppose that $0$ is an attracting hyperbolic singularity for $X$ and $Y$. Show that there exists a homeomorphism $h$ of a neighbourhood of the origin which conjugates the diffeomorphisms $X_{t = 1}$ and $Y_{t = 1}$ but does not take orbits of $X$ to orbits of $Y$.
I'm a bit lost to find the solution. I would like someone to give me a clue to get started. What I understand is that, the hypothesis of the problem tells us about the behavior of the system near $0$, so intuitively, the $h$ sends orbits close to $0$ in orbits close to $0$. The problem could fail if we take orbits far from 0. Does anyone have an idea on how to proceed?