Attractor

An attractor is a set of states that a Dynamical System tends toward as time goes on, and stays near once it arrives. Release the system from many different starting conditions and, after the initial transient fades, the trajectories all end up doing the same thing — orbiting the same shape in State Space. That shape is the attractor, and the collection of starting points that flow to it is its basin of attraction.

A hierarchy of shapes

Attractors come in a small zoo of types, in order of increasing richness:

• A fixed point — the system runs down to rest, like a Damped Oscillator coming to a stop. The attractor is a single point.
• A limit cycle — the system settles into a steady repeating oscillation. The attractor is a closed loop; see Limit Cycle.
• A torus — two incommensurate frequencies combine, and the trajectory winds forever over the surface of a doughnut without closing.
• A strange attractor — the trajectory is bounded and never repeats, folded into a Fractal. This is the signature of Chaos; see Strange Attractor.

What makes a set an attractor

Three conditions: it is invariant (start on it, stay on it), it attracts an open neighborhood of nearby states, and it is minimal — no smaller piece of it does the same job. The attracting property is why attractors are what you actually observe in nature: unstable behaviors are washed out, and the system is left riding its attractor.

Basins and competition

A system can have several attractors at once, each with its own basin. The boundaries between basins decide the system's fate from a given start, and those boundaries can themselves be fractal — making the long-term outcome practically unpredictable even when each attractor is simple. A change in a parameter can make an attractor appear, vanish, or swap stability with another: a Bifurcation.

See also

• Strange Attractor
• Limit Cycle
• Lorenz System