I know the flow for the differential equation $\dot{x} = Ax$, where $A$ has no time dependency, is given by $\phi(x,t) = e^{At}$.
Since we want explicit representations we use Jordan Form and properties of the matrix exponential to get $e^{At} = T^{-1}e^{Jt}T$, we then analyse instead the phase portrait of $e^{Jt}$.
However, do the matrices $T$ and $T^{-1}$ not alter the phase portrait for $e^{At}$?
The two linear autonomous systems $$ \dot{\mathbf{x}} = A \mathbf{x} \tag{1} $$ and $$ \dot{\mathbf{x}} = J \mathbf{x} \tag{2} $$ where $J = P^{-1} A P$ is the Jordan Canonical Form of $A$ are topologically equivalent.
Hence, to understand the qualitative properties of the linear system (1), it is often useful to plot the solutions of (2) and understand its stability behavior.