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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s, an attractor is a set of states toward which a system tends to evolve, 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 represented
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ically as an ''n''-dimensional
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
. The attractor is a region in ''n''-dimensional space. In
physical systems A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
, the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in
economic systems An economic system, or economic order, is a system of production, resource allocation and distribution of goods and services within a society or a given geographic area. It includes the combination of the various institutions, agencies, entitie ...
, they may be separate variables such as the
inflation rate In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
and the
unemployment rate Unemployment, according to the OECD (Organisation for Economic Co-operation and Development), is people above a specified age (usually 15) not being in paid employment or self-employment but currently available for work during the refere ...
. 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 Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
, a finite set of points, a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, a manifold, or even a complicated set with a fractal structure known as a ''strange attractor'' (see strange attractor below). 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 A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...
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 be periodic or
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
. 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 a repeller (or ''repellor'').


Motivation of attractors

A
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
is generally described by one or more differential or
difference equations In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
. 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 to integrate the equations, either through analytical means or through
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
, often with the aid of computers. Dynamical systems in the physical world tend to arise from
dissipative system A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dis ...
s: 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 balance, killing off initial transients and settle the system into its typical behavior. The subset 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 set In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...
s 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 A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
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 due to air resistance, the point is also an attractor. If there was no dissipation, 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 (see strange attractor), in which case the evolution of any two distinct points of the attractor result in exponentially diverging trajectories, which complicates prediction when even the smallest noise is present in the system.


Mathematical definition

Let ''t'' represent time and let ''f''(''t'', •) be a function which specifies the dynamics of the system. That is, if ''a'' is a point in an ''n''-dimensional phase space, representing the initial state of the system, then ''f''(0, ''a'') = ''a'' and, for a positive value of ''t'', ''f''(''t'', ''a'') is the result of the evolution of this state after ''t'' 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 R2 with coordinates (''x'',''v''), where ''x'' is the position of the particle, ''v'' is its velocity, ''a'' = (''x'',''v''), and the evolution is given by : f(t,(x,v))=(x+tv,v).\ An attractor is a subset ''A'' of the phase space characterized by the following three conditions: * ''A'' is ''forward invariant'' under ''f'': if ''a'' is an element of ''A'' then so is ''f''(''t'',''a''), for all ''t'' > 0. * There exists a neighborhood of ''A'', called the basin of attraction for ''A'' and denoted ''B''(''A''), which consists of all points ''b'' that "enter ''A'' in the limit ''t'' → ∞". More formally, ''B''(''A'') is the set of all points ''b'' in the phase space with the following property: :: For any open neighborhood ''N'' of ''A'', there is a positive constant ''T'' such that ''f''(''t'',''b'') ∈ ''N'' for all real ''t'' > ''T''. * There is no proper (non-empty) subset of ''A'' having the first two properties. Since the basin of attraction contains an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
containing ''A'', every point that is sufficiently close to ''A'' is attracted to ''A''. The definition of an attractor uses a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
on the phase space, but the resulting notion usually depends only on the topology of the phase space. In the case of R''n'', the Euclidean norm is typically used. Many other definitions of attractor occur in the literature. For example, some authors 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 portions or subsets of the phase space of a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
. Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points,
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
,
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s, and simple regions of
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
. More complex attractors that cannot be categorized as simple geometric subsets, such as topologically wild sets, were known of at the time but were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was
robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
and that its attractor had the structure of a Cantor set. Two simple attractors are a fixed point and the
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, 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 o ...
. 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 and
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
) of fundamental geometric objects (e.g.
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
,
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s,
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
s,
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
s, manifolds), then the attractor is called a '' strange attractor''.


Fixed point

A fixed 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 a damped
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, the level and flat water line of sloshing water in a glass, or the bottom center of a bowl contain 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 between stable 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 the
nonlinear dynamics In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
of
stiction Stiction is the static friction that needs to be overcome to enable relative motion of stationary objects in contact. The term is a portmanteau of the words ''static'' and ''friction'', and is perhaps also influenced by the verb '' to stick''. Any ...
,
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
, surface roughness,
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
(both
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
and
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adaptab ...
ity), and even
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
. In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
shape, are both much more complex surfaces when examined under a microscope, and their shapes change or
deform Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * De ...
during contact. Any physical surface can be seen to have a rough terrain of multiple peaks, valleys, saddle points, ridges, ravines, and plains. 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 considered stationary or fixed points, some of which are categorized as attractors.


Finite number of points

In a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
system, an attractor can take the form of a finite number of points that are visited in sequence. Each of these points is called a
periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2''n'' points, 3 points, 3×2''n'' points, 4 points, 5 points, or any given positive integer number of points.


Limit cycle

A
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, 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 o ...
is a periodic orbit of a continuous dynamical system that is isolated. It concerns a cyclic attractor. Examples include the swings of a
pendulum clock A pendulum clock is a clock that uses a pendulum, a swinging weight, as its timekeeping element. The advantage of a pendulum for timekeeping is that it is a harmonic oscillator: It swings back and forth in a precise time interval dependent on i ...
, 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 the
escapement An escapement is a mechanical linkage in mechanical watches and clocks that gives impulses to the timekeeping element and periodically releases the gear train to move forward, advancing the clock's hands. The impulse action transfers energy ...
mechanism to maintain the cycle.


Limit torus

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 an
irrational fraction In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions. A ration ...
(i.e. they are incommensurate), the trajectory is no longer closed, and the limit cycle becomes a limit
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
. This kind of attractor is called an -torus if there are incommensurate frequencies. For example, here is a 2-torus: A time series corresponding to this attractor is a
quasiperiodic Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpred ...
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 The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
still consists only of sharp lines.


Strange attractor

An attractor is called strange if it has a fractal structure. This is often the case when the dynamics on it are
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
, but strange nonchaotic attractors also exist. If a strange attractor is chaotic, exhibiting sensitive 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. The term strange attractor was coined by
David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
and
Floris Takens Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attr ...
to describe the attractor resulting from a series of bifurcations of a system describing fluid flow. Strange attractors are often
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
in a few directions, but some are
like In English, the word ''like'' has a very flexible range of uses, ranging from conventional to non-standard. It can be used as a noun, verb, adverb, adjective, preposition, particle, conjunction, hedge, filler, and quotative. Uses Comparisons ' ...
a
Cantor dust In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. ...
, 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. Examples of strange attractors include the double-scroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor.


Attractors characterize the evolution of a system

The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters. An example is the well-studied logistic map, x_=rx_n(1-x_n), whose basins of attraction for various values of the parameter ''r'' are shown in the figure. If r=2.6, all starting ''x'' values of x<0 will rapidly lead to function values that go to negative infinity; starting ''x'' values of x>1 will also go to negative infinity. But for 0 the ''x'' values rapidly converge to x\approx0.615, i.e. at this value of ''r'', a single value of ''x'' is an attractor for the function's behaviour. For other values of ''r'', more than one value of x may be visited: if ''r'' is 3.2, starting values of 0 will lead to function values that alternate between x\approx0.513 and x\approx0.799. At some values of ''r'', the attractor is a single point (a "fixed point"), at other values of ''r'' two values of ''x'' are visited in turn (a
period-doubling bifurcation In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. W ...
), or, as a result of further doubling, any number ''k'' × 2''n'' values of ''x''; at yet other values of r, any given number of values of ''x'' are visited in turn; finally, for some values of ''r'', an infinitude of points are visited. Thus one and the same dynamic equation can have various types of attractors, depending on its starting parameters.


Basins of attraction

An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor. For a stable
linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...
, every point in the phase space is in the basin of attraction. However, in nonlinear 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.


Linear equation or system

A single-variable (univariate) linear difference equation of the
homogeneous form In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
x_t=ax_ diverges to infinity if , ''a'', > 1 from all initial points except 0; there is no attractor and therefore no basin of attraction. But if , ''a'', < 1 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 linear
matrix difference equation A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the eq ...
in a dynamic
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
''X'', of the homogeneous form X_t=AX_ in terms of square matrix ''A'' will have all elements of the dynamic vector diverge to infinity if the largest
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of ''A'' 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 ''n''-dimensional space of potential initial vectors is the basin of attraction. Similar features apply to linear
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. The scalar equation dx/dt =ax causes all initial values of ''x'' except zero to diverge to infinity if ''a'' > 0 but to converge to an attractor at the value 0 if ''a'' < 0, making the entire number line the basin of attraction for 0. And the matrix system dX/dt=AX gives divergence from all initial points except the vector of zeroes if any eigenvalue of the matrix ''A'' 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.


Nonlinear equation or system

Equations or systems that are
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
can give rise to a richer variety of behavior than can linear systems. One example is Newton's method of iterating to a root of a nonlinear expression. If the expression has more than one
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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, for the function f(x)=x^3-2x^2-11x+12, the following initial conditions are in successive basins of attraction: :2.35287527 converges to 4; :2.35284172 converges to −3; :2.35283735 converges to 4; :2.352836327 converges to −3; :2.352836323 converges to 1. Newton's method can also be applied to
complex functions Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
to find their roots. Each root has a basin of attraction in the complex 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 are fractals.


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 condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s, if it has a global attractor, then this attractor will be of finite dimensions.


See also

*
Cycle detection In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function that maps a finite set to itself, and any initial value in , the sequence of itera ...
*
Hyperbolic set In dynamical systems theory, a subset Λ of a smooth manifold ''M'' is said to have a hyperbolic structure with respect to a smooth map ''f'' if its tangent bundle may be split into two invariant subbundles, one of which is contracting and th ...
*
Stable manifold 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 repello ...
*
Steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
*
Wada basin In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of ''one'' of the lakes ...
*
Hidden oscillation In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bound ...
* Rössler attractor *
Stable distribution In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be sta ...
*
Convergent evolution Convergent evolution is the independent evolution of similar features in species of different periods or epochs in time. Convergent evolution creates analogous structures that have similar form or function but were not present in the last com ...


References


Further reading

* * * * * * * * Edward N. Lorenz (1996) ''The Essence of Chaos'' *
James Gleick James Gleick (; born August 1, 1954) is an American author and historian of science whose work has chronicled the cultural impact of modern technology. Recognized for his writing about complex subjects through the techniques of narrative nonficti ...
(1988) ''Chaos: Making a New Science''


External links


Basin of attraction on Scholarpedia

A gallery of trigonometric strange attractors

Double scroll attractor
Chua's circuit simulation


Chaoscope, a 3D Strange Attractor rendering freeware


an
software laboratory

Online strange attractors generator



Economic attractor
{{Chaos theory Limit sets