Classification of equilibria of planar linear systems

Summary

Let $A$ be a real $2 \times 2$ matrix with $\det A \neq 0$, let $\tau = \operatorname{tr} A$ and $\Delta = \det A$, and consider $x' = A x$ in the phase plane. The origin is a saddle if $\Delta < 0$. It is a node if $\Delta > 0$ and $\tau^2 - 4 \Delta \ge 0$, and a spiral, or focus, if $\tau \neq 0$ and $\tau^2 - 4 \Delta < 0$. It is a centre if $\tau = 0$ and $\Delta > 0$. A node or a spiral is stable for $\tau < 0$ and unstable for $\tau > 0$.

Remarks

The eigenvalues are the roots of $\lambda^2 - \tau \lambda + \Delta = 0$. A saddle has real eigenvalues of opposite signs. A node has real eigenvalues of the same sign. A spiral or a centre has complex conjugate eigenvalues. The case $\tau^2 = 4 \Delta$ gives a repeated eigenvalue. The origin is then a star node or a degenerate node. The trace and the determinant do not change under a change of basis $A \mapsto C^{-1} A C$. So the classification depends only on the system, not on the coordinates. The damped harmonic oscillator gives a stable spiral when it is underdamped and a stable node when it is overdamped.

Explanations

L1 human

For a linear system in the plane, two numbers, the trace and the determinant, tell the shape of the phase portrait near the origin. For a node, trajectories run into or out of the origin along straight directions. They turn around it for a spiral, and they go around it in closed loops for a centre.