Fixed Point

3 min read#foundations

A state where the dynamics stand still — an equilibrium — classified by how nearby trajectories behave around it.

Contents

Fixed Point

A fixed point (or equilibrium) is a state where the system, left alone, stays put forever. For a flow it is a point x^* where the velocity vanishes,

f(x^*) = 0,

so there is nowhere to go. For a map it is a point that returns to itself, F(x^*) = x^*. Fixed points are the anchors of a Phase Portrait: every other trajectory is organized by how it approaches, avoids, or orbits them.

Finding them is the easy part — just solve f(x)=0. The interesting question is what happens near one.

Classifying a fixed point

Zoom in close enough and a smooth flow looks linear, governed by the Jacobian matrix A = Df(x^*). The eigenvalues of A tell you the local geography:

  • Node — eigenvalues real, same sign. Trajectories dive straight in (stable, both negative) or straight out (unstable, both positive).
  • Saddle — real eigenvalues of opposite sign. Trajectories approach along one direction and flee along another; saddles are always unstable.
  • Spiral (focus) — complex eigenvalues. Trajectories wind in or out, rotating as they go — the hallmark of decaying or growing oscillation.
  • Center — purely imaginary eigenvalues. Trajectories form closed loops, neither approaching nor receding, as in a frictionless oscillator.

Whether each type attracts or repels is the subject of Stability, which reads off the sign of the real parts of those eigenvalues.

See the types side by side

The vector field below is the damped oscillator \dot x = y,\; \dot y = -x - 0.3y, whose only fixed point — the origin — is a stable spiral. Trajectories rotate inward toward it.

A stable spiral at the origin. Crank the friction past 2 and the spiral straightens into a node.

Swap the rule to \dot x = y,\; \dot y = x and the origin becomes a saddle: most trajectories sweep in, bend, and shoot back out along the diagonal escape directions.

One point, many roles

The same equilibrium can change character as a parameter varies — a stable spiral can lose its stability and spawn a Limit Cycle, or two fixed points can collide and annihilate. Those transitions are exactly what Bifurcation catalogs.

See also