A dynamical system ẋ = A x assigns a little arrow to every point — the direction the state moves next. Follow the arrows and you get a trajectory; the whole picture is the phase portrait. The eigenvalues of A decide the character of the fixed point at the origin: a stable spiral, an outward spiral, a saddle, or an orbiting center. Click anywhere to release a particle and watch it flow.
flow arrows — direction of ẋ at each pointtrajectory — click to release a particleeigen-directions — invariant lines (real case)The fixed point's fate is read straight off the trace and determinant. Real eigenvalues of the same sign give a node (both negative → everything decays in, stable; both positive → blows out); opposite signs give a saddle — stable along one eigen-line, unstable along the other. Complex eigenvalues give a spiral whose winding decays or grows with the sign of the real part ½ tr A, and a pure imaginary pair (tr A = 0, det A > 0) gives closed orbits — a center. This is the phase-plane analysis behind pendulum and oscillator dynamics, the stability check for any linearized controller or robot balance loop, and the geometric heart of how nonlinear systems behave near their equilibria.