An attractor is a set towards which a variable, moving according to the dictates of a dynamical system, evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed. The evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate.
If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. If the variable is a scalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.
A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighbourhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).
A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamic system specify its behavior over any given short
eriod of time. To determine the system's behavior for a longer period, it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers.
Dynamical systems in the physical world tend to be dissipative: if it were not for some driving force, the motion would cease. (Dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system into its typical behavior. This one part of the phase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.
Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set is a set of points such that there exists some initial state that ends up arbitrarily close to the limit set (i.e. to each point of the set) as time goes to infinity. Attractors are limit sets, but not all limit sets are attractors: It is possible to have some points of a system converge to a limit set, but different points when perturbed slightly off the limit set may get knocked off and never return to the vicinity of the limit set.
For example, the damped pendulum has two invariant points: the point x0 of minimum height and the point x1 of maximum height. The point x0 is also a limit set, as trajectories converge to it; the point x1 is not a limit set. Because of the dissipation, the point x0 is also an attractor. If there were no dissipation, x0 would not be an attractor.