phase-plane classification

Instead of plotting each variable against time, plot the variables against each other and watch the state of the system trace a curve in a plane — the phase plane. For a two-dimensional linear system every trajectory has one of a small number of shapes, and classifying that shape tells you, at a glance and without solving, how the system behaves near equilibrium.

For x' = A x in the plane, the geometry of trajectories near the origin is decided entirely by the eigenvalues of the 2-by-2 matrix A. Two real eigenvalues of the same sign give a node — all trajectories flow into the origin (stable, both negative) or out of it (unstable, both positive). Two real eigenvalues of opposite sign give a saddle — trajectories approach along one direction and flee along another, always unstable. A complex pair alpha plus-or-minus i beta gives a spiral, winding inward if alpha is negative (stable) and outward if alpha is positive (unstable); when alpha is exactly zero the trajectories are closed loops, a center, with neutral, non-decaying oscillation. A compact summary uses the trace and determinant of A to locate the case.

This classification is the entry point to the whole qualitative theory of dynamical systems. For nonlinear systems, you linearize near each equilibrium and read off the local type from the Jacobian's eigenvalues — so saddles, nodes, spirals, and centers organize the behavior of pendulums, predator-prey models, chemical oscillators, and electrical circuits. The one honest caveat: a borderline case like a center can be destroyed by the nonlinear terms the linearization ignored, so a linear 'center' may actually spiral slowly in the full system.