FKM Kompass
L3

Fixed points and phase-plane stability

A fixed point is a state at which every rate of change vanishes, so a system left exactly there stays there — and drawing the trajectories in the phase plane (position against velocity, say) turns the question of whether it stays there when nudged into geometry: linearise about the point, and the eigenvalues of the resulting 2×2 matrix decide whether nearby trajectories spiral in, are thrown out along a preferred direction, or circle indefinitely, which is why a pendulum hanging down and the same pendulum balanced upright are one equation with two completely different fixed points.

Practise this concept

Instead of plotting position against time, plot position against velocity. Each state of the system is then a single point, and the equations of motion become a flow: through every point runs one arrow saying where the state goes next. A whole solution is a curve threaded along those arrows, and the picture of all of them together is the phase portrait. Fixed points are where the arrow has zero length — put the system exactly there and it never moves. A pendulum has two of them within one full turn: hanging straight down, and balanced straight up.

Both are fixed points, but nobody would call them equally stable, and the difference is visible in the flow around each one. To get at it quantitatively, write the state as the fixed point plus a small offset and keep only the terms linear in that offset. What is left is a 2×2 matrix, and its eigenvalues say what happens to the offset: a negative real part means it decays, so trajectories come in; a positive real part means it grows, so they leave; an imaginary part means the offset rotates while it does either. That gives the standard catalogue — stable and unstable nodes, spirals, and the saddle, which has one eigenvalue of each sign and so attracts along one direction while repelling along another. The upright pendulum is a saddle, which is exactly why balancing it is possible in principle and hopeless in practice: only a perturbation aimed exactly along the incoming direction ever falls back.

There is one case the linear analysis genuinely cannot settle. When both eigenvalues are purely imaginary, the linearised picture shows closed orbits — a centre — but the terms just discarded are the same size as the effect being predicted, so whether the true trajectories spiral slowly in, slowly out, or close exactly is decided by the nonlinear terms. An undamped pendulum really does have closed orbits; add the slightest damping and the centre becomes an inward spiral. That borderline case is not a defect of the method so much as the place where nonlinear dynamics starts: it is where limit cycles, bifurcations and eventually chaos all live.

Key ideas

Requires: Damped oscillations

Unlocks: Bifurcations and the loss of stability, Limit cycles and Poincaré sections

Common misconceptions

  • Reading "fixed point" as "nothing is happening" — what vanishes is the rate of change of the state variables, not the activity inside the system. A chemical reactor held at a steady concentration sits at a fixed point while reactions run flat out in both directions, and a steadily spinning rotor is a fixed point of the equation for its angular velocity even though it is visibly moving.
  • Assuming a fixed point must be either attracting or repelling, so one negative eigenvalue is enough to call it stable. A saddle does both at once: it pulls trajectories in along one direction and throws them out along another, and since almost no real perturbation lands exactly on the incoming direction, almost every nearby state approaches for a while and then leaves.
  • Treating the linearisation as the whole answer, in both directions: either trusting it always, or dismissing it as "only an approximation" that decides nothing. Eigenvalues with a non-zero real part do settle the local behaviour — that is a theorem, not a guess — but when the real parts are exactly zero, a centre in the linear picture, the discarded nonlinear terms decide whether the real trajectories creep inwards, creep outwards, or genuinely close, and there the linear analysis is silent.

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Part of these UT courses

This concept is mapped as covering part of the following course(s) in the University of Tartu curriculum.