In "Differential Equations" by Stephen Wiggins (Example F.45) it is said that for the simple dynamical system
$$\dot x=\lambda x$$
$$\dot y=\mu y$$
where $(x, y) \in R^2$ and $\lambda, \mu > 0$, we have that the $x$ axis is the unstable manifold, whereas the $y$ axis is the stable one.
I could not figure how the second assertion could be true
Best Answer
Clearly, these uncoupled system has solution $$x=ke^{\lambda t},$$ and $$y=le^{\mu t}.$$ Since both $\lambda, \mu$ are positive then the system is unstable.
It is necessary that $\mu<0$ to get stability along the $y$-axis.