Dynamical systems and phase portraits

Write a system like x′ = f(x, y), y′ = g(x, y) and every point in the (x, y) plane gets an arrow: the direction and speed a particle sitting there would move next. Following those arrows traces out trajectories, and the whole picture — arrows everywhere, trajectories threading through them — is the phase portrait. A fixed point is any point where both f and g vanish, so the arrow has zero length and nothing moves; it's the system's equilibrium.

Near a fixed point, replacing the nonlinear f and g with their linear (Jacobian) approximation turns the messy original system into a linear one, whose behaviour is fully controlled by that matrix's eigenvalues. Two real eigenvalues of the same sign give a node — everything flows straight in (both negative, stable) or straight out (both positive, unstable). Two real eigenvalues of opposite sign give a saddle: trajectories approach along one direction and flee along another, so almost every nearby starting point eventually gets flung away even though a special line of points flows straight in. A complex-conjugate pair gives a spiral, winding in or out depending on the sign of the real part, with the imaginary part controlling how fast it winds.

This linear picture is only a local approximation — it describes the phase portrait in a small neighbourhood of the fixed point, and can fail to say anything definite at all when an eigenvalue's real part is exactly zero, because then the nonlinear terms that the linearisation threw away are precisely what decides whether the point is stable, unstable, or something stranger (a centre, a limit cycle). This gap is why the classification is called qualitative: it tells you the shape of the flow, not a closed-form trajectory, and it earns that shape by discarding exactly the information that would let you solve the system exactly.