Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations. Consider a systems of linear differential equations x = ax. Similar to a direction field, a phase portrait is a graphical tool to visualize how the solutions of a given system of differential equations would behave in the long run. In this context, the cartesian plane where the phase portrait resides is called the phase plane. The parametric curves traced by the solutions are sometimes also called their trajectories. Remark: it is quite labor-intensive, but it is possible to sketch the phase portrait by hand without first having to solve the system of equations that it represents. Just like a direction field, a phase portrait can be a tool to predict the behaviors of a system"s solutions. To do so, we draw a grid on the phase plane.
