The Phase Plane

In one dimension a flow can only slide left or right, so its whole repertoire is "drift onto a fixed point." Add a second variable and the world cracks open. A trajectory in the plane can circle, spiral, and loop — motions impossible on the line. The phase plane is where we watch a two-variable system

\dot{x} = f(x,y), \qquad \dot{y} = g(x,y)

by plotting (x,y) and hiding time inside the curve. At every point the vector (\dot{x},\dot{y}) is an arrow saying "go this way, this fast"; stitch the arrows into streamlines and you get the phase portrait — a single picture of every possible destiny at once. Near a fixed point the Jacobian's eigenvalues dictate which of a small gallery of shapes the portrait takes.

The gallery of a linear fixed point

Linearizing gives \dot{\boldsymbol{\eta}} = J\boldsymbol{\eta}, and the eigenvalues \lambda_{1,2} of the 2\times2 matrix J sort the origin into one of five archetypes:

Node — \lambda_{1,2} real, same sign. Trajectories run along straight eigen-directions; stable if both negative, unstable if both positive.
Saddle — \lambda_{1,2} real, opposite signs. Always unstable: flow comes in along the stable eigenvector, leaves along the unstable one.
Spiral (focus) — \lambda = \alpha \pm i\beta, \alpha \ne 0. Trajectories rotate while decaying (\alpha < 0) or growing (\alpha > 0).
Centre — \lambda = \pm i\beta purely imaginary. Closed orbits: the system cycles forever, neither winding in nor out.
Star / degenerate node — repeated real eigenvalue: a limiting case between node and spiral.

Explore the portrait

Below is the phase plane of \dot{\mathbf{x}} = A\mathbf{x}. The slider swaps A between six canonical matrices; each trajectory is drawn both forwards and backwards in time from a ring of starting points, with an arrowhead marking the flow direction. Watch the eigenvalue signature reshape the whole picture — a saddle's hyperbolic sweep, a spiral's inward winding, a centre's closed loops.

The trace–determinant map

Astonishingly, the whole gallery is charted by just two numbers. The eigenvalues of a 2\times2 matrix A obey

\lambda^2 - \tau\lambda + \Delta = 0, \qquad \tau = \operatorname{tr}A,\ \ \Delta = \det A, \qquad \lambda = \tfrac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}.

So the type of the fixed point is read straight from (\tau, \Delta):

\Delta < 0 ⟹ saddle (real, opposite signs);
\Delta > 0 and \tau^2 > 4\Delta ⟹ node (real, same sign — stable if \tau < 0);
\Delta > 0 and \tau^2 < 4\Delta ⟹ spiral (complex — stable if \tau < 0);
\tau = 0, \Delta > 0 ⟹ centre.

The parabola \tau^2 = 4\Delta is the borderline between spiralling and non-spiralling motion; the whole right half-plane \Delta > 0,\ \tau < 0 is "stable." One glance at two numbers and you have named the fixed point — no eigenvector, no integration.

A phase portrait deliberately throws time away. Two systems that trace the same loop at wildly different speeds share one portrait, because we plot the track, not the timetable. That is the source of the phase plane's power and its one blind spot: it shows you the geometry of every destiny at a glance, but to recover how fast you must put the arrows' lengths back in. For qualitative questions — will it settle? will it oscillate? — the track is all you need, which is why Poincaré built an entire theory of differential equations out of pictures with the clock removed.

Worked example: classify from the matrix

Classify the origin for \dot{x} = -x - 3y,\ \dot{y} = 3x - y.

Step 1 — the matrix. A = \begin{pmatrix} -1 & -3 \\ 3 & -1 \end{pmatrix}.

Step 2 — trace and determinant. \tau = -1 + (-1) = -2 and \Delta = (-1)(-1) - (-3)(3) = 1 + 9 = 10.

Step 3 — discriminant. \tau^2 - 4\Delta = 4 - 40 = -36 < 0, so the eigenvalues are complex: \lambda = -1 \pm 3i.

Step 4 — verdict. Complex eigenvalues with negative real part (\alpha = -1): the origin is a stable spiral. Trajectories wind inward, completing a turn about every 2\pi/\beta = 2\pi/3 units of time while their radius shrinks like e^{-t}. Set the slider above to "Stable spiral" to see exactly this motion.

A linear centre is fragile. Purely imaginary eigenvalues put the fixed point exactly on the borderline \tau = 0, where the linearization is non-hyperbolic and Hartman–Grobman gives no guarantee. In the true nonlinear system the neglected higher-order terms can tip those closed loops into a slow inward or outward spiral — so a picture-perfect centre in the linear analysis may not survive. The other archetypes (node, saddle, spiral) are hyperbolic and robust; only the centre, sitting on the knife-edge, demands you go back and check the nonlinear terms before trusting it.