mathematics//dynamical systems//equilibrium and stability//attractor

An attractor is the state, or set of states, that a system ends up in from a whole region of starting points: a pendulum hanging at rest, a room settling at its thermostat's setpoint, a heart beating at its own rhythm. It is what a control engineer designs a system to have (the desired state as an attractor, reached from wherever the system starts), and what a scientist looks for to know what a system will do in the long run without solving it.

An attractor is the state, or set of states, that a system ends up in from a whole region of starting points: a pendulum hanging at rest, a room settling at its thermostat's setpoint, a heart beating at its own rhythm. It is what a control engineer designs a system to have (the desired state as an attractor, reached from wherever the system starts), and what a scientist looks for to know what a system will do in the long run without solving it.

Formally it is an invariant set, one that a trajectory never leaves once on it, that draws in every trajectory starting near it; the set of all starting points that end up on it is its basin of attraction. Attractors come in three shapes. A stable equilibrium is a point, such as a damped spring at rest, which every nearby trajectory spirals or creeps into. A limit cycle is a closed orbit that nearby trajectories wind onto, such as a beating heart or an oven cycling under an on-off thermostat. A strange attractor is a fractal set on which the motion is chaotic, bounded and never repeating, the butterfly of the Lorenz system being the famous one.

The basin matters as much as the attractor. A system can have several attractors, each with its own basin, and which one it ends on depends only on where it starts: a balancing controller that recovers from small tilts and falls from large ones has a stable equilibrium with a finite basin. Estimating that basin is what a Lyapunov function is used for.

A linear system has only one possible kind. Its attractor, if any, is the equilibrium, and its basin is the whole space or nothing: either every eigenvalue has a negative real part and everything returns, or something escapes. Limit cycles, strange attractors and several attractors side by side all need nonlinearity (equilibrium and stability).

The opposite is a repeller, a state that every neighbouring trajectory leaves; it is an equilibrium too, but one nobody observes for long.