Poincaré began drawing these portraits in the 1880s: when the three-body problem refused to yield formulas, he mapped the flow of every solution at once instead.
For x′ = Ax, two eigenvalues settle everything, because every solution is a mix of eλt motions along fixed directions.
Real parts run the radius — negative drains the plane, positive erupts. Imaginary parts do one job only: they turn it.
The center is a knife-edge: real part exactly zero, orbits closing forever. One hair left or right, and every loop unravels.
Trajectories never cross — the flow tiles the plane like grain in wood.