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An attractor is a set to which a dynamical system evolves after a long enough time. That is, points that get close enough to the attractor remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with a fractal structure known as a strange attractor. 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 or of any other type.

Motivation

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 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.

Mathematical definition

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).

An attractor is a subset A of the phase space such that:

* A is invariant under f; i.e., if s is an element of A then so is f(t,s), for all t.

* There is a neighborhood of A, B(A) called the basin of attraction for A, such that B(A) = { s | for all neighborhoods N of A there is a T such that for all t > T, f(t,s) ∈ N }. In other words, B(A) is the set of points that 'enter A in the limit'.

* There is no proper subset of A with the first two properties.

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.

Types of attractors

Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by textbooks of that era, attractors were thought of as being geometrical subsets of the phase space: points, lines, surfaces, volumes. The (topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

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.

Fixed point

A fixed point is a point that a system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass. It corresponds to a fixed point of the evolution function that is also attracting.

Limit cycle

See main article limit cycle

A limit cycle is a periodic orbit of the system that is isolated. Examples include the swings of a pendulum clock, the tuning circuit of a radio, and the heartbeat while resting. The ideal pendulum is not an example because its orbits are not isolated. In phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit.

Limit tori

There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. If two of these frequencies form an irrational fraction (i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit torus. We call this kind of attractor Nt-torus if there are Nt incommensurate frequencies. For example here is a 2-torus:

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of Nt 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.

Strange attractor

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.

Lorenz Attractor, ρ = 28, σ = 10, β = 8/3 Mathematica:

ρ = 28;
σ = 10;
β = 8/3;

NDSolve[{x'[t] == σ(y[t] - x[t]), y'[t] == x[t](ρ - z[t]) - y[t], z'[t] == x[t] y[t] - β z[t], x[0] == z[0] == 0, y[0] == 1},
{x, y, z}, {t, 0, 200}, MaxSteps -> Infinity];

Examples of strange attractors include the Hénon attractor, Rössler attractor, Lorenz attractor, Tamari attractor.

Partial differential equations

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.

Further reading

* Edward N. Lorenz (1996) The Essence of Chaos ISBN 0-295-97514-8

* James Gleick (1988) Chaos: Making a New Science ISBN 0-140-09250-1

See also

* Hyperbolic set

* Stable manifold

* Steady state

* Wada basin

* Great Attractor

Links

• A gallery of polynomial strange attractors
• Animated Pickover Strange Attractors
• Chaoscope, a 3D Strange Attractor rendering freeware
• Strange Attractor a DVD featuring Terence McKenna
• 1D, 2D and 3D of strange attractors, include Tamari Attractor
• Research abstract and software laboratory

References

• David Ruelle and Floris Takens (1971). "On the nature of turbulence". Communications of Mathematical Physics 20: 167-192.
• D. Ruelle (1981). "Small random perturbations of dynamical systems and the definition of attractors". Communications of Mathematical Physics 82: 137-151.
• John Milnor (1985). "On the concept of attractor". Communications of Mathematical Physics 99: 177-195.
• J. Milnor (main author) Attractor on scholarpedia.
• David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN 0-12-601710-7.
• Ruelle, David (August 2006). "What is...a Strange Attractor?" (PDF). Notices of the American Mathematical Society 53 (7): pp.764–765. Retrieved on 2008-01-16.
• Ben Tamari (1997). Conservation and Symmetry Laws and Stabilization Programs in Economics. Ecometry ltd. ISBN 965-222-838-9.

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