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 or of any other type.
A dynamical system is often described in terms of differential equations that describe its behavior for a short period of time. To determine the behavior for longer periods 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 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 of minimum height and the point of maximum height. The point is also a limit set, as trajectories converge to it; the point is not a limit set. Because of the dissipation, the point is also an attractor. If there were no dissipation, would not be an attractor.
Let f(t, •) be a function which specifies the dynamics of the system. That is, if s is an element of the phase space, i.e., s totally specifies the state of the system at some instant, then f(0, s) = s and for t>0, f(t, s) evolves s forward t units of time. For example, if our system is an isolated point particle in one dimension, then its position in phase space is given by (x,v) where x is the position of the particle and v is its velocity. If the particle is not acted on by any potential (flies around freely) then dynamics is given by f(t,(x,v)) = (x+t*v,v).
Since the basin of attraction is in a close neighborhood of A, i.e. contains an open set containing A, every state 'close enough' to A is attracted to A. Technically the notion of an attractor depends on the topology placed on the phase space, but normally the standard topology on ℝn is assumed.
Other definitions of attractor are sometimes used. For example, some require that an attractor have positive measure (preventing a point from being an attractor), others relax the requirement that B(A) be a neighborhood.
Two simple attractors are the fixed point and the limit cycle. There can be many other geometrical sets that are attractors. When these sets (or the motions on them), are hard to describe, then the attractor is a strange attractor, as described in the section below.
A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but its power spectrum still consists only of sharp lines.
An attractor is informally described as strange if it has non-integer dimension or if the dynamics on it are chaotic. The term was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions, but some are like a Cantor dust, and therefore not differentiable.
Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg–Landau, the Kuramoto–Sivashinsky, and the two-dimensional, forced Navier–Stokes equations are all known to have global attractors of finite dimension.
For the three-dimensional, incompressible Navier–Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.