![Poisson saturne revisited](https://upload.wikimedia.org/wikipedia/commons/7/70/Poisson_saturne_revisited.jpg)
In the
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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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 a ...
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 referen ...
.
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 (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
, a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, or even a complicated set with a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
structure known as a ''strange attractor'' (see
strange attractor
In the mathematical field of dynamical systems, 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 ...
below). If the variable is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, 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
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
.
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 traj ...
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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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
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 t ...
,
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 the ...
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 the mathematical field of dynamical systems, 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 ...
), 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 R
2 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
![Julia immediate basin 1 3](https://upload.wikimedia.org/wikipedia/commons/2/25/Julia_immediate_basin_1_3.png)
:
An attractor is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
''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
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 suf ...
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
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
(preventing a point from being an attractor), others relax the requirement that ''B''(''A'') be a neighborhood.
Types of attractors
Attractors are portions or
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of the
phase space of a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
. 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 (geometry), position of an element (i.e., Point (m ...
. 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
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavi ...
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
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.
Thr ...
.
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 ...
. 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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
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,
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s), then the attractor is called a ''
strange attractor
In the mathematical field of dynamical systems, 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 ...
''.
Fixed point
![Critical orbit 3d](https://upload.wikimedia.org/wikipedia/commons/0/07/Critical_orbit_3d.png)
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 the ...
, 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 t ...
,
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 chemistry, ...
.
In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly
hemispherical
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
, 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 ...
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 it ...
, 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 to ...
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:
![torus](https://upload.wikimedia.org/wikipedia/commons/1/17/Torus.png)
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
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
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
![Lorenz attractor yb](https://upload.wikimedia.org/wikipedia/commons/5/5b/Lorenz_attractor_yb.svg)
An attractor is called strange if it has a
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
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 attra ...
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
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lo ...
.
Attractors characterize the evolution of a system
![Logistic Map Bifurcation Diagram, Matplotlib](https://upload.wikimedia.org/wikipedia/commons/c/c8/Logistic_Map_Bifurcation_Diagram%2C_Matplotlib.svg)
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
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
,
, whose basins of attraction for various values of the parameter ''r'' are shown in the figure. If
, all starting ''x'' values of
will rapidly lead to function values that go to negative infinity; starting ''x'' values of
will also go to negative infinity. But for