a spiral point

A spiral point (also called a focus) is an equilibrium that trajectories wind around as they approach or leave — they corkscrew rather than coming in straight. Think of water swirling down a drain, or a coin spinning down a funnel: it both circles AND moves inward, tracing a spiral. A stable spiral pulls every nearby state into a tightening inward whirl; an unstable spiral sends them whirling outward.

It is the case of COMPLEX eigenvalues, a pair alpha plus or minus i beta with a nonzero imaginary part. The imaginary part beta supplies the rotation — the state turns at angular rate beta, which is why trajectories curl. The real part alpha supplies the growth or decay of the radius — the distance from the point scales like e^(alpha t). So if alpha is negative the spiral tightens inward forever (a stable spiral, a damped oscillation winding to rest); if alpha is positive it loosens outward (an unstable spiral, a growing oscillation). The two ingredients, rotate-and-shrink or rotate-and-grow, are read straight off the real and imaginary parts of the eigenvalues.

Spirals are the phase-plane fingerprint of damped vibration — a struck bell, a settling suspension, an RLC circuit ringing down all show inward spirals. The key honest point is the role of the real part: a spiral is stable only when alpha is strictly negative. Push alpha to exactly zero and the spiral stops shrinking and becomes a closed loop — a center — the delicate borderline case between a stable and an unstable focus.