The Phase Plane

The phase plane is simply a map of a two-quantity system: one quantity goes on each axis, and each solution becomes a path across the map. "Phase" here just means the state of the system at a moment, the pair of values $(x_1, x_2)$. It's good for seeing every solution's long-run behavior at a glance. The formula $\lambda^2 - T\lambda + D = 0$ looks like more algebra, but it's built from two easy numbers: $T$ (the trace, the sum of the diagonal, which is also the sum of the eigenvalues) and $D$ (the determinant, which is the product of the eigenvalues). Those two numbers tell you whether things grow or shrink and whether they turn. It looks like a lot of cases, but it's really just a few simple questions.

The origin is an equilibrium. $A\vec 0 = \vec 0$, so if you start at the origin you stay there. Every other trajectory is a curve showing how $x_1$ and $x_2$ change together, with time running along the curve instead of along an axis.

Why the eigenvalues decide everything. From the last lesson, each eigenvalue gives a solution $e^{\lambda t}\vec v$. If $\lambda$ is negative, that piece shrinks toward the origin; if positive, it grows away. Every solution is a mix of these pieces, so the signs of the eigenvalues tell you which way everything flows. Complex eigenvalues bring in sines and cosines through Euler's formula, which is where the turning comes from. It's the same story as the characteristic roots of $ay'' + by' + cy = 0$: negative real parts die out, imaginary parts oscillate.

Eigenvectors of real eigenvalues show up as straight-line trajectories.

A common mistake. It's tempting to call any picture with curves flowing in a stable node, because "everything goes to the origin" is what you notice first. It isn't that simple: a stable spiral also flows in, but it turns as it goes. The difference is whether the eigenvalues are real or complex, which you can check with $T^2 - 4D$: positive means real, negative means complex. There's an animation further down that shows a spiral turning into a center and then into an outward spiral as the real part changes sign.

Phase portraits are how engineers read stability at a glance, and how ecologists see predator–prey cycles: the predator–prey lab plots exactly these curves.