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In themathematical field ofdynamical systems, anattractor is a set of states toward which a system tends to evolve,[2] for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.
In finite-dimensional systems, the evolving variable may be representedalgebraically as ann-dimensionalvector. The attractor is a region inn-dimensional space. Inphysical systems, then dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; ineconomic systems, they may be separate variables such as theinflation rate and theunemployment rate.[not verified in body]
If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be representedgeometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be apoint, a finite set of points, acurve, amanifold, or even a complicated set with afractal structure known as astrange attractor (seestrange attractor below). If the variable is ascalar, the attractor is a subset of the real number line. Describing the attractors of chaotic dynamical systems has been one of the achievements ofchaos theory.
Atrajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may beperiodic orchaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called arepeller (orrepellor).
Adynamical system is generally described by one or moredifferential ordifference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary tointegrate the equations, either through analytical means or throughiteration, often with the aid of computers.
Dynamical systems in the physical world tend to arise fromdissipative systems: if it were not for some driving force, the motion would cease. (Dissipation may come frominternal friction,thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to balance, killing off initial transients and settle the system into its typical behavior. The subset of thephase space of the dynamical system corresponding to the typical behavior is the attractor, also known as the attracting section or attractee.
Invariant sets andlimit sets are similar to the attractor concept. Aninvariant set is a set that evolves to itself under the dynamics.[3] Attractors may contain invariant sets. Alimit 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, thedampedpendulum has two invariant points: the pointx0 of minimum height and the pointx1 of maximum height. The pointx0 is also a limit set, as trajectories converge to it; the pointx1 is not a limit set. Because of the dissipation due to air resistance, the pointx0 is also an attractor. If there was no dissipation,x0 would not be an attractor. Aristotle believed that objects moved only as long as they were pushed, which is an early formulation of a dissipative attractor.
Some attractors are known to be chaotic (seestrange attractor), in which case the evolution of any two distinct points of the attractor result in exponentiallydiverging trajectories, which complicates prediction when even the smallest noise is present in the system.[4]
Let represent time and let be a function which specifies the dynamics of the system. That is, if is a point in an-dimensional phase space, representing the initial state of the system, then and, for a positive value of, is the result of the evolution of this state after units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane with coordinates, where is the position of the particle, is its velocity,, and the evolution is given by
An attractor is asubset of thephase space characterized by the following three conditions:
Since the basin of attraction contains anopen set containing, every point that is sufficiently close to is attracted to. The definition of an attractor uses ametric on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of, the Euclidean norm is typically used.
Many other definitions of attractor occur in the literature. For example, some authors require that an attractor have positivemeasure (preventing a point from being an attractor), others relax the requirement that be a neighborhood.[5]
Attractors are portions orsubsets of thephase space of adynamical system. Until the 1960s, attractors were thought of as beingsimple geometric subsets of the phase space, likepoints,lines,surfaces, and simple regions ofthree-dimensional space. More complex attractors that cannot be categorized as simple geometric subsets, such astopologically wild sets, were known of at the time but were thought to be fragile anomalies.Stephen Smale was able to show that hishorseshoe map wasrobust and that its attractor had the structure of aCantor set.
Two simple attractors are afixed point and thelimit cycle. Attractors can take on many other geometric shapes (phase space subsets). But when these sets (or the motions within them) cannot be easily described as simple combinations (e.g.intersection andunion) offundamental geometric objects (e.g.lines,surfaces,spheres,toroids,manifolds), then the attractor is called astrange attractor.
Afixed point of a function or transformation is a point that is mapped to itself by the function or transformation. If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation. The final state that a dynamical system evolves towards corresponds to an attracting fixed point of the evolution function for that system, such as the center bottom position of adampedpendulum, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl containing a rolling marble. But the fixed point(s) of a dynamic system is not necessarily an attractor of the system. For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference betweenstable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).
In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including thenonlinear dynamics ofstiction,friction,surface roughness,deformation (bothelastic andplasticity), and evenquantum mechanics.[6] In the case of a marble on top of an inverted bowl, even if the bowl seems perfectlyhemispherical, and the marble'sspherical shape, are both much more complex surfaces when examined under a microscope, and theirshapes change ordeform during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains.[7] There are many points in this surface terrain (and the dynamic system of a similarly rough marble rolling around on this microscopic terrain) that are consideredstationary or fixed points, some of which are categorized as attractors.
In adiscrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called aperiodic point. This is illustrated by thelogistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2n points, 3 points, 3×2n points, 4 points, 5 points, or any given positive integer number of points.
Alimit cycle is a periodic orbit of a continuous dynamical system that isisolated. It concerns acyclic attractor. Examples include the swings of apendulum clock, and the heartbeat while resting. The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting. For a physical pendulum under friction, the resting state will be a fixed-point attractor. The difference with the clock pendulum is that there, energy is injected by theescapement mechanism to maintain the cycle.
There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. If two of these frequencies form anirrational fraction (i.e. they areincommensurate), the trajectory is no longer closed, and the limit cycle becomes a limittorus. This kind of attractor is called anNt -torus if there areNt incommensurate frequencies. For example, here is a 2-torus:
A time series corresponding to this attractor is aquasiperiodic series: A discretely sampled sum ofNt periodic functions (not necessarilysine waves) with incommensurate frequencies. Such a time series does not have a strict periodicity, but itspower spectrum still consists only of sharp lines.[citation needed]
An attractor is calledstrange if it has afractal structure, that is if it has non-integerHausdorff dimension. This is often the case when the dynamics on it arechaotic, butstrange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibitingsensitive dependence on initial conditions, then any two arbitrarily close alternative initial points on the attractor, after any of various numbers of iterations, will lead to points that are arbitrarily far apart (subject to the confines of the attractor), and after any of various other numbers of iterations will lead to points that are arbitrarily close together. Thus a dynamic system with a chaotic attractor is locally unstable yet globally stable: once some sequences have entered the attractor, nearby points diverge from one another but never depart from the attractor.[8]
The termstrange attractor was coined byDavid Ruelle andFloris Takens to describe the attractor resulting from a series ofbifurcations of a system describing fluid flow.[9] Strange attractors are oftendifferentiable in a few directions, but some arelike aCantor dust, and therefore not differentiable. Strange attractors may also be found in the presence of noise, where they may be shown to support invariant random probability measures of Sinai–Ruelle–Bowen type.[10]
Examples of strange attractors include thedouble-scroll attractor,Hénon attractor,Rössler attractor, andLorenz attractor.
The behavior of a dynamical system may be influenced by its parameters or the choice of initial conditions. Thelogistic map, defined as, is a well-studied example of a system dependent on one parameter. Its attractors for various values of are shown in the figure. For small the attractor is a singlefixed point, which is shown on the bifurcation diagram as one line. For other choices of, more than one value of may become attracting: at the fixed point splits in two creating a period 2 cycle during aperiod-doubling bifurcation. As increases,chaos emerges thorough a period-doubling cascade, meaning the attractor consists of an infinte number of points. At a period 3 orbit can be found. It follows from theSharkovskii's theorem that orbits of any natural period are present in the system. Thus one dynamic equation can have vastly different attractors depending on the choice of parameters.
An attractor's basin of attraction is the region of thephase space, over which iterations are defined, such that any point (anyinitial condition) in that region willasymptotically be iterated into the attractor. For astablelinear system, every point in the phase space is in the basin of attraction. However, innonlinear systems, some points may map directly or asymptotically to infinity, while other points may lie in a different basin of attraction and map asymptotically into a different attractor; other initial conditions may be in or map directly into a non-attracting point or cycle.[11]
A univariate linear homogeneous difference equation diverges to infinity if from all initial points except 0; there is no attractor and therefore no basin of attraction. But if all points on the number line map asymptotically (or directly in the case of 0) to 0; 0 is the attractor, and the entire number line is the basin of attraction.
Likewise, a linearmatrix difference equation in a dynamicvector, of the homogeneous form in terms ofsquare matrix will have all elements of the dynamic vector diverge to infinity if the largesteigenvalues of is greater than 1 in absolute value; there is no attractor and no basin of attraction. But if the largest eigenvalue is less than 1 in magnitude, all initial vectors will asymptotically converge to the zero vector, which is the attractor; the entire-dimensional space of potential initial vectors is the basin of attraction.
Similar features apply to lineardifferential equations. The scalar equation causes all initial values of except zero to diverge to infinity if but to converge to an attractor at the value 0 if, making the entire number line the basin of attraction for 0. And the matrix system gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix is positive; but if all the eigenvalues are negative the vector of zeroes is an attractor whose basin of attraction is the entire phase space.
Equations or systems that arenonlinear can give rise to a richer variety of behavior than can linear systems. One example isNewton's method of iterating to a root of a nonlinear expression. If the expression has more than onereal root, some starting points for the iterative algorithm will lead to one of the roots asymptotically, and other starting points will lead to another. The basins of attraction for the expression's roots are generally not simple—it is not simply that the points nearest one root all map there, giving a basin of attraction consisting of nearby points. The basins of attraction can be infinite in number and arbitrarily small. For example,[12] for the function, the following initial conditions are in successive basins of attraction:
Newton's method can also be applied tocomplex functions to find their roots. Each root has a basin of attraction in thecomplex plane; these basins can be mapped as in the image shown. As can be seen, the combined basin of attraction for a particular root can have many disconnected regions. For many complex functions, the boundaries of the basins of attraction arefractals.
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. TheGinzburg–Landau, theKuramoto–Sivashinsky, and the two-dimensional, forcedNavier–Stokes equations are all known to have global attractors of finite dimension.
For the three-dimensional, incompressible Navier–Stokes equation with periodicboundary conditions, if it has a global attractor, then this attractor will be of finite dimensions.[13]
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