In dynamical systems, an attractor is a set to which the system evolves after a long enough time. For the set to be an attractor, trajectories that get close enough to the attractor must remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with fractal structures known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...

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 trajectory is an imagined trace of positions followed by an object moving through space. ...

Motivation and definition

Dynamical systems are often described in terms of differential equations. These equations describe the behavior of the system 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. (The 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. The part of the phase space of the dynamical system corresponding to the typical behavior is the attracting set or attractor. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In calculus, the integral of a function is an extension of the concept of a sum. ... Friction is the force that opposes the relative motion or tendency of such motion of two surfaces in contact. ... Thermodynamics (from the Greek thermos meaning heat and dynamics meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Phase space of a dynamical system with focal stability. ...

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 are the states a system goes to after an infinite amount of time. Attractors are limit sets, but not all limit sets are attractors. It is possible to have a system converge to a limit set, but if placed in the limit set, have small perturbations that knock it off to never return. Perturbation is a term used in astronomy to describe alterations to an objects orbit caused by gravitational interactions with other bodies. ...

As an 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. Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ...

Mathematical definition

We fix a function f(t, •) 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 isn't acted on by any potential (flies around freely) then dynamics is given by f(t,(x,v))=x+t*v.

The attractor A is a subset of the phase space such that: A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

• A is invariant under f, i.e., if s is an element of A then f(t,s) is in A for all t.
• There is a neighborhood of A, B(A) called the basin of attraction for A, with B(A)={s| for all neighborhoods N of A there is a T so that for all t>T f(t,s) in N}. In other words B(A) is the set of points who 'enter A in the limit'.
• There is no subset of A with the first two properties.

Note that requiring the basin of attraction to be a neighborhood of A, i.e. contain an open set containing A, requires every state 'close enough' to A be attracted to A. Technically the notion of an attractor depends on the topology placed on the phase space but normally the standard topology on R^n is assumed. A neighbourhood or neighborhood (see spelling differences) is a geographically localised community located within a larger city or suburb. ...

Note that other definitions of attractor will occasionally be used. For instance some sources require that an attractor have positive measure (preventing a point from being an attractor) or relax the requirement than 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. The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Three lines — the red and blue lines have same slope, while the red and green ones have same y-intercept. ... An open surface with X-, Y-, and Z-contours shown. ... Volume is how much space a thing has. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ... In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. ... Structural stability is a mathematical concept concerning whether a given function is sensitive to a small perturbation. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

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. In mathematics, in the area of dynamical systems, a limit-cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches minus-infinity. ...

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. Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ...

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. In mathematics, in the area of dynamical systems, a limit-cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches minus-infinity. ... In mathematics, in the area of dynamical systems, a limit-cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches minus-infinity. ... In topology, a point x of a set S is called an isolated point, if there exists a neighborhood of x not containing other points of S. In particular, in an Euclidean space (or in a metric space), x is an isolated point of S, if one can find an... A pendulum clock uses a pendulum as its time base. ...

Image File history File links Download high-resolution version (1024x668, 29 KB) Van de Pol phase portrait. ...

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 will no longer be closed, and the limit cycle becomes a limit torus. We call this kind of attractor N_t-torus if there are N_t incommensurate frequencies. For example it is a 2-torus: This article is about the meaning of commensurable and derived words in mathematics. ...

Image File history File links Torus. ...

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of N_t periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does no longer have a strict periodicity, but its power spectrum still consists only of sharp lines. The power spectrum is a plot of the portion of a signals power (energy per unit time) falling within given frequency bins. ...

Strange attractor

A plot of Lorenz's strange attractor for values ρ=28, σ = 10, β = 8/3

An attractor is informally described as strange if it has non-integer dimension or if the dynamics on the attractor 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 and like a Cantor dust (and therefore not differentiable) in others. Image File history File links Lorenz_attractor_yb. ... Image File history File links Lorenz_attractor_yb. ... In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ... :For other senses of this word, see dimension (disambiguation). ... A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... (Born August 20, 1935) Belgian-French physicist. ... Floris Takens (born 12 November 1940) is a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. ... In mathematics, specifically in the study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in the systems long-term dynamical behaviour. ... This word should not be confused with homomorphism. ... Cantor dust, named after the mathematician Georg Cantor, is the two-dimensional version of the Cantor set. ...

The Hénon attractor, Rössler attractor, and the Lorenz attractor are examples of strange attractors. The Hénon map is a discrete-time dynamical system. ... Rössler attractor with , , Rössler attractor as a stereogram with , , The Rössler attractor is a set of ordinary differential equations that define a continuously chaotic function consisting of several surfaces. ... A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ...

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. A parabola A parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone, and a plane tangent to the cone or parallel to some plane tangent to the cone. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In fluid dynamics, the Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. ...

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 dimension. In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...

Further reading

• Edward N. Lorenz (1996) The Essence of Chaos ISBN 0-295-97514-8
• James Gleick (1988) Chaos: Making a New Science ISBN 0-295-97514-8

Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... 1996 (MCMXCVI) was a leap year starting on Monday of the Gregorian calendar, and was designated the International Year for the Eradication of Poverty. ... James Gleick (August 1, 1954– ) is an author, journalist, and biographer, whose books explore the cultural ramifications of science and technology. ... 1988 (MCMLXXXVIII) was a leap year starting on Friday of the Gregorian calendar. ...

See also

• Wada basin
• Stable manifold
• Hyperbolic set

In mathematics, the lakes of Wada are three disjoint connected open sets of the plane with the counterintuitive property that they all have the same boundary. ... In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. ... In mathematics, a subset of a manifold is said to have hyperbolic structure when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding. ...

External links

• A gallery of polynomial strange attractors
• Animated Pickover Strange Attractors
• Chaoscope, a 3D Strange Attractor rendering freeware

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.
• David Ruelle (1989). Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press. ISBN 0-12-601710-7.
• R. Temam (1997). Infinite dimensional dynamical systems in mechanics and physics, 2nd edition. Springer-Verlag. ISBN 0-387-94866-X.
• Manfred Schroeder (1991). Fractals, Chaos, Power Laws. W.H. Freeman and Company. ISBN 0-7167-2136-8.
• http://www.research.ibm.com/journal/rd/471/martens.html
• http://mathworld.wolfram.com/Attractor.html