Attractors

Attractors are geometric forms that characterize long-term behaviour in the phase space. Roughly speaking, an attractor is what the behaviour of a systems settles down to, or is attracted to. They are globally stable in the sense that the system will return if perturbed off the attractor, as long as it remains within the basin of attraction.

The simplest kind of attractor is a fixed point. An example of this is a pendulum subject to friction: no matter how it starts swinging, the pendulum always comes to rest at the same point. (a pendulum bob standing upright and not moving is at an unstable fixed point.)
(catastrophe theory is a theory describing the geometry of discontinuities in systems with fixed point attractors.)

The next most complicated attractor is a limit cycle, forming a closed loop in phase space. A limit cycle describes stable oscillations, such as the motion of a pendulum clock or the beating of a heart.

Compound oscillations, or quasi-periodic behaviour are described by a torus, which resembles the surface of a doughnut. One oscillation is described around the larger perimeter of the doughnut, the other, perpendicular to it, around the smaller section. Higher dimensional tori can be used to describe combinations of more than two oscillations. (a multi-dimensionsal torus is called a hypertorus.) Despite the complexity of these latter examples, all these attractors describe predictable systems.

Chaotic, or strange attractors, on the other hand correspond to unpredictable motions and have a more complex geometric form. (see "Chaos" by Crutchfield, Farmer, Packard, and Shaw, in Scientific American, vol. 255, No. 6, December 1986)



The most famous "strange attractor" is the Lorentz attractor or butterfly. The strange attractor occurs in a bounded space, yet its length is infinite. The attractor can be studied by taking a section through it, called a Poincaré section. The section has a fractal structure. Chaotic attractors are globally stable but contain local instabilities that amplify noise. In a strange attractor periodic orbits are unstable and act as repellors.