Show that if exist $\mu>0$, $k\geq 1$ such that $\lVert e^{At} x\rVert\leq ke^{-\mu t}\lVert x\rVert$, for all $t\in\mathbb{R}$ (condition 1), then $x'=Ax'$ is topologycal conjugated to $x'=-x$, for all $x\in\mathbb{R}^{n}$ (condition 2).
This a proposition about equivalence, if $x'=Ax$ is an "atractor" (condition 0) then implies condition 1 and 2. I proved that $\mbox{ condition 0}\implies\mbox{ condition 1}$ and $\mbox{ condition 2}\implies\mbox{ condition 0}$, but I can not see $1)\implies 2)$, any hint. Thanks!