Chaos theory is a field of study in
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, with applications in several disciplines including
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
,
biologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
, and
philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
. Chaos theory studies the behavior of
dynamical systemA dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
s that are highly sensitive to initial conditions, an effect which is popularly referred to as the
butterfly effectIn chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a small change at one place in a nonlinear system can result in large differences to a later state...
. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are
deterministicIn mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...
, meaning that their future behavior is fully determined by their initial conditions, with no
randomRandomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply
chaos.
Chaotic behavior can be observed in many natural systems, such as the
weatherWeather is the state of the atmosphere, to the degree that it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the troposphere, just below the stratosphere. Weather refers, generally, to day-to-day temperature and precipitation activity, whereas climate...
. Explanation of such behavior may be sought through analysis of a chaotic
mathematical modelA mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
, or through analytical techniques such as
recurrence plotIn descriptive statistics and chaos theory, a recurrence plot is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space...
s and
Poincaré mapIn mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to...
s.
Applications
Chaos theory is applied in many scientific disciplines:
geologyGeology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...
,
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, programming,
microbiologyMicrobiology is the study of microorganisms, which are defined as any microscopic organism that comprises either a single cell , cell clusters or no cell at all . This includes eukaryotes, such as fungi and protists, and prokaryotes...
,
biologyBiology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
,
computer scienceComputer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
,
economicsEconomics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
,
engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
,
finance"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
,
meteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
,
philosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
,
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
,
politicsPolitics is a process by which groups of people make collective decisions. The term is generally applied to the art or science of running governmental or state affairs, including behavior within civil governments, but also applies to institutions, fields, and special interest groups such as the...
,
population dynamicsPopulation dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...
,
psychologyPsychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
, and
roboticsThe word "beam" in BEAM robotics is an acronym for Biology, Electronics, Aesthetics, and Mechanics. This is a term that refers to a style of robotics...
.
Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions,
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, and mechanical and magneto-mechanical devices, as well as computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather, the dynamics of satellites in the
solar systemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
, the time evolution of the magnetic field of celestial bodies, population growth in
ecologyEcology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
, the dynamics of the action potentials in
neuronA neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...
s, and
molecular vibrationA molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion...
s. There is some controversy over the existence of chaotic dynamics in
plate tectonicsPlate tectonics is a scientific theory that describes the large scale motions of Earth's lithosphere...
and in economics.
A successful application of chaos theory is in ecology where dynamical systems such as the
Ricker modelThe Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number a t+1 of individuals in generation t + 1 as a function of the number of individuals in the previous generation,Here r is interpreted as an intrinsic growth rate and k as...
have been used to show how population growth under density dependence can lead to chaotic dynamics.
Chaos theory is also currently being applied to medical studies of
epilepsyEpilepsy is a common chronic neurological disorder characterized by seizures. These seizures are transient signs and/or symptoms of abnormal, excessive or hypersynchronous neuronal activity in the brain.About 50 million people worldwide have epilepsy, and nearly two out of every three new cases...
, specifically to the prediction of seemingly random seizures by observing initial conditions.
Quantum chaosQuantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle...
theory studies how
the correspondenceIn physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers....
between
quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and
classical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
works in the context of chaotic systems. Recently, another field, called relativistic chaos, has emerged to describe systems that follow the laws of
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
The motion of
N stars in response to their self-gravity (the
gravitational N-body problemThe n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
) is generically chaotic.
In electrical engineering, chaotic systems are used in communications, random number generators, and encryption systems.
In
numerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, the Newton-Raphson method of approximating the roots of a function can lead to chaotic iterations if the function has no real roots.
Chaotic dynamics
In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although there is no universally accepted mathematical definition of chaos, a commonly used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:
- it must be sensitive to initial conditions;
- it must be topologically mixing; and
- its periodic orbits must be dense
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
.
The requirement for sensitive dependence on initial conditions implies that there is a set of initial conditions of positive measure which do not converge to a cycle of any length.
Sensitivity to initial conditions
Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour. However, it has been shown that the last two properties in the list above actually imply sensitivity to initial conditions and if attention is restricted to
intervalsIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
, the second property implies the other two (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list). It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) purely
topologicalTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
conditions, which are therefore of greater interest to mathematicians.
Sensitivity to initial conditions is popularly known as the "
butterfly effectIn chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a small change at one place in a nonlinear system can result in large differences to a later state...
", so called because of the title of a paper given by Edward Lorenz in 1972 to the
American Association for the Advancement of ScienceThe American Association for the Advancement of Science is an international non-profit organization with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsibility, and supporting scientific education and science outreach for the...
in Washington, D.C. entitled
Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This is most familiar in the case of weather, which is generally predictable only about a week ahead.
The
Lyapunov exponentIn mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
characterises the extent of the sensitivity to initial conditions. Quantitatively, two
trajectoriesA trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
in
phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
with initial separation
diverge
where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.
There are also measure-theoretic mathematical conditions (discussed in ergodic theory) such as mixing or being a K-system which relate to sensitivity of initial conditions and chaos.
Topological mixing
Topological mixing (or topological
transitivity) means that the system will evolve over time so that any given region or
open setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
of its
phase spaceIn mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
will eventually overlap with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored
dyeA dye is a colored substance that has an affinity to the substrate to which it is being applied. The dye is generally applied in an aqueous solution, and requires a mordant to improve the fastness of the dye on the fiber....
s or fluids is an example of a chaotic system.
Topological mixing is often omitted from popular accounts of chaos, which equate chaos with sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behaviour: all points except 0 tend to infinity.
Density of periodic orbits
DensityIn topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may not be chaotic. For example, an irrational rotation of the circle is topologically transitive, but does not have dense periodic orbits, and hence does not have sensitive dependence on initial conditions. The one-dimensional
logistic mapThe logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
defined by
x → 4
x (1 –
x) is one of the simplest systems with density of periodic orbits. For example,
→
→
(or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).
Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.
Strange attractors
Some dynamical systems, like the one-dimensional
logistic mapThe logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
defined by
x → 4
x (1 –
x), are chaotic everywhere, but in many cases chaotic behaviour is found only in a subset of phase space. The cases of most interest arise when the chaotic behaviour takes place on an
attractorAn attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.
An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly.
Unlike fixed-point attractors and
limit cycles, the attractors which arise from chaotic systems, known as
strange attractors, have great detail and complexity. Strange attractors occur in both
continuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
dynamical systems (such as the Lorenz system) and in some
discreteDiscrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
systems (such as the
Hénon map). Other discrete dynamical systems have a repelling structure called a
Julia setIn the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
which forms at the boundary between basins of attraction of fixed points – Julia sets can be thought of as strange
repellers. Both strange attractors and Julia sets typically have a
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
structure, and a
fractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
can be calculated for them.
Minimum complexity of a chaotic system
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their
dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
ality. However, the Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s) if it has three or more dimensions.
Finite dimensionalIn mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
linear systemA 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 general, nonlinear case....
s are never chaotic; for a dynamical system to display chaotic behaviour it has to be either
nonlinearIn mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
, or infinite-dimensional.
The
Poincaré–Bendixson theoremIn mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane.-Theorem:...
states that a two dimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations with a total of seven terms on the right hand side, five of which are linear terms and two of which are quadratic (and therefore nonlinear). Another well-known chaotic attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is (quadratic) nonlinear. Sprott found a three dimensional system with just five terms on the right hand side, and with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.
While the Poincaré–Bendixson theorem means that a continuous dynamical system on the Euclidean
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
cannot be chaotic, two-dimensional continuous systems with
non-Euclidean geometryNon-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
can exhibit chaotic behaviour. Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional. A theory of linear chaos is being developed in the
functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a branch of mathematical analysis.
History
An early proponent of chaos theory was
Henri PoincaréJules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
. In the 1880s, while studying the
three-body problemThree-body problem has two distinguishable meanings in physics and classical mechanics:# In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then...
, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898
Jacques HadamardJacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, "Hadamard's billiards", Hadamard was able to show that all trajectories are unstable in that all particle trajectories diverge exponentially from one another, with a positive
Lyapunov exponentIn mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
.
Much of the earlier theory was developed almost entirely by mathematicians, under the name of
ergodic theoryErgodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
. Later studies, also on the topic of nonlinear differential equations, were carried out by
G.D. Birkhoff-External links:* − from National Academies Press, by Oswald Veblen....
, , M.L. Cartwright and
J.E. LittlewoodJohn Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
, and
Stephen SmaleSteven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.
Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the
logistic mapThe logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
. What had been beforehand excluded as measure imprecision and simple "
noiseIn common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...
" was considered by chaos theories as a full component of the studied systems.
The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated
iterationIteration means the act of repeating a process usually with the aim of approaching a desired goal or target or result. Each repetition of the process is also called an "iteration," and the results of one iteration are used as the starting point for the next iteration.-Mathematics:Iteration in...
of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems.
An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on
weather predictionMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
in 1961. Lorenz was using a simple digital computer, a
Royal McBeeRoyal McBee was the name of the computer manufacturing and retail division of Royal Typewriter which made the early computers RPC 4000 and RPC 9000...
LGP-30The LGP-30, standing for Librascope General Purpose and then Librascope General Precision, was an early off-the-shelf computer. It was manufactured by the Librascope company of Glendale, California , and sold and serviced by the Royal Precision Electronic Computer Company, a joint venture with the...
, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. Lorenz's discovery, which gave its name to
Lorenz attractorThe Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...
s, showed that even detailed atmospheric modelling cannot in general make long-term weather predictions. Weather is usually predictable only about a week ahead.
The year before,
Benoît MandelbrotBenoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
found recurring patterns at every scale in data on cotton prices. Beforehand, he had studied
information theoryInformation theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
and concluded noise was patterned like a
Cantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). This challenged the idea that changes in price were
normally distributed. In 1967, he published "
How long is the coast of Britain? Statistical self-similarity and fractional dimension"How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoît Mandelbrot, first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2...
", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
ly small measuring device. Arguing that a ball of twine appears to be a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a
fractalA fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
(for example, the Koch curve or "snowflake", which is infinitely long yet encloses a finite space and has
fractal dimensionIn fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...
equal to circa 1.2619, the
Menger spongeIn mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
and the Sierpiński gasket). In 1975 Mandelbrot published
The Fractal Geometry of NatureThe Fractal Geometry of Nature is an influential book written in 1982 by the Franco-American mathematician Benoît Mandelbrot. It is a revised and enlarged version of his 1977 book entitled Fractals: Form, Chance and Dimension.- See also :* Fractal...
, which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.
Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol and in 1958 by R.L. Ives. However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.
In December 1977 the
New York Academy of SciencesThe New York Academy of Sciences is the third oldest scientific society in the United States. An independent, non-profit organization with more than members in 140 countries, the Academy’s mission is to advance understanding of science and technology...
organized the first symposium on Chaos, attended by David Ruelle,
Robert MayRobert McCredie May, Baron May of Oxford, OM, AC, PRS is an Australian scientist who has been Chief Scientific Adviser to the UK Government, President of the Royal Society, and a Professor at Sydney and Princeton. He now holds joint professorships at Oxford, and Imperial College London...
,
James A. YorkeJames A. Yorke is a Distinguished University Professor of Mathematics and Physics and chair of the Mathematics Department at the University of Maryland, College Park. He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology...
(coiner of the term "chaos" as used in mathematics),
Robert ShawRobert Stetson Shaw is an American physicist who was part of Eudaemonic Enterprises in Santa Cruz in the late 1970s and early 1980s. In 1988 he was awarded a MacArthur Fellowship for his work in Chaos theory.-Chaos theory:...
(a physicist, part of the
EudaemonsThe Eudaemons were a small group headed by graduate physics students J. Doyne Farmer and Norman Packard at the University of California Santa Cruz in the late 1970s. The group's immediate objective was to find a way to beat roulette, but a loftier objective was to use the money made from roulette...
group with
J. Doyne FarmerJ. Doyne Farmer is an American physicist and entrepreneur, with interest in chaos theory and complexity. He is a professor at the Santa Fe Institute. He was also a member of Eudaemonic Enterprises.-Biography:...
and
Norman PackardNorman Harry Packard is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. He is an alumnus of Reed College and the University of California, Santa Cruz. Packard is known for his contributions to both chaos theory and cellular automata...
who tried to find a mathematical method to beat
rouletteRoulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....
, and then created with them the Dynamical Systems Collective in
Santa CruzSanta Cruz is the county seat and largest city of Santa Cruz County, California in the US. As of the 2010 U.S. Census, Santa Cruz had a total population of 59,946...
,
CaliforniaCalifornia is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...
), and the meteorologist Edward Lorenz.
The following year,
Mitchell FeigenbaumMitchell Jay Feigenbaum is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.- Biography :...
published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.
In 1979,
Albert J. LibchaberAlbert J. Libchaber is a Detlev W. Bronk Professor at Rockefeller University. He won the Wolf Prize in Physics in 1986.-Education:...
, during a symposium organized in Aspen by
Pierre HohenbergPierre C. Hohenberg is a French-American theoretical physicist, who works primarily on statistical mechanics....
, presented his experimental observation of the
bifurcationBifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
cascade that leads to chaos and turbulence in
convectiveConvection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
Rayleigh–Benard systems. He was awarded the
Wolf Prize in PhysicsThe Wolf Prize in Physics is awarded once a year by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Mathematics, Medicine and Arts. The Prize is often considered the most prestigious...
in 1986 along with Mitchell J. Feigenbaum "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".
Then in 1986 the New York Academy of Sciences co-organized with the
National Institute of Mental HealthThe National Institute of Mental Health is one of 27 institutes and centers that make up the National Institutes of Health...
and the
Office of Naval ResearchThe Office of Naval Research , headquartered in Arlington, Virginia , is the office within the United States Department of the Navy that coordinates, executes, and promotes the science and technology programs of the U.S...
the first important conference on Chaos in biology and medicine. There,
Bernardo HubermanBernardo Huberman is a Senior Fellow at HP Labs, and Director of the Social Computing Lab at HP Labs. He received his Ph.D. in Physics from the University of Pennsylvania, and is currently a Consulting Professor in the Department of Applied Physics and the Symbolic System Program at Stanford...
presented a mathematical model of the eye tracking disorder among schizophrenics. This led to a renewal of
physiologyPhysiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...
in the 1980s through the application of chaos theory, for example in the study of pathological
cardiac cycleThe cardiac cycle is a term referring to all or any of the events related to the flow or blood pressure that occurs from the beginning of one heartbeat to the beginning of the next. The frequency of the cardiac cycle is described by the heart rate. Each beat of the heart involves five major stages...
s.
In 1987,
Per BakPer Bak was a Danish theoretical physicist who coauthored the 1987 academic paper that coined the term "self-organized criticality."- Life and work :...
,
Chao TangChao Tang is a Chinese physicist and professor at the University of California at San Francisco.In 1987, as a post-doctoral research scientist in the Solid State Theory Group of Brookhaven National Laboratory, he and another fellow post-doctoral scientist, Kurt Wiesenfeld, along with their mentor,...
and
Kurt WiesenfeldKurt Wiesenfeld is an American physicist working primarily on non-linear dynamics. His works primarily concern stochastic resonance, spontaneous synchronization of coupled oscillators, and non-linear laser dynamics...
published a paper in
Physical Review LettersPhysical Review Letters , established in 1958, is a peer reviewed, scientific journal that is published 52 times per year by the American Physical Society...
describing for the first time
self-organized criticalityIn physics, self-organized criticality is a property of dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale-invariance characteristic of the critical point of a phase transition, but without the need to tune...
(SOC), considered to be one of the mechanisms by which
complexityIn general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...
arises in nature.
Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display
scale-invariantIn physics and mathematics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor...
behaviour. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behaviour such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the
Omori lawAn aftershock is a smaller earthquake that occurs after a previous large earthquake, in the same area of the main shock. If an aftershock is larger than the main shock, the aftershock is redesignated as the main shock and the original main shock is redesignated as a foreshock...
describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in
econophysicsEconophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes and nonlinear dynamics...
); landscape formation; forest fires; landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by
Niles EldredgeNiles Eldredge is an American paleontologist, who, along with Stephen Jay Gould, proposed the theory of punctuated equilibrium in 1972.-Education:...
and
Stephen Jay GouldStephen Jay Gould was an American paleontologist, evolutionary biologist, and historian of science. He was also one of the most influential and widely read writers of popular science of his generation....
). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. These "applied" investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
The same year,
James GleickJames Gleick is an American author, journalist, and biographer, whose books explore the cultural ramifications of science and technology...
published
Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, (though his history under-emphasized important Soviet contributions). At first the domain of work of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to
Thomas KuhnThomas Samuel Kuhn was an American historian and philosopher of science whose controversial 1962 book The Structure of Scientific Revolutions was deeply influential in both academic and popular circles, introducing the term "paradigm shift," which has since become an English-language staple.Kuhn...
's concept of a
paradigm shiftA Paradigm shift is, according to Thomas Kuhn in his influential book The Structure of Scientific Revolutions , a change in the basic assumptions, or paradigms, within the ruling theory of science...
exposed in
The Structure of Scientific RevolutionsThe Structure of Scientific Revolutions , by Thomas Kuhn, is an analysis of the history of science. Its publication was a landmark event in the history, philosophy, and sociology of scientific knowledge and it triggered an ongoing worldwide assessment and reaction in — and beyond — those scholarly...
(1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by J. Gleick.
The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics,
topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, physics, population biology, biology, meteorology, astrophysics, information theory, etc.).
Distinguishing random from chaotic data
It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.
All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point. Thus, given a time series to test for determinism, one can:
- pick a test state;
- search the time series for a similar or 'nearby' state; and
- compare their respective time evolutions.
Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.
Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (e.g., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.
Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.
When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions. Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback system. In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.
The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in philosophy.
Cultural references
Chaos theory has been mentioned in numerous movies and works of literature. For instance, it was mentioned extensively in Michael Chrichton's novel
Jurassic Park and more briefly in its sequel. Other examples include the film
ChaosChaos is a 2006 crime film directed by Tony Giglio starring Jason Statham, Ryan Phillippe, and Wesley Snipes.-Plot:During a hostage incident on a bridge, Detective York accidentally shoots and kills the hostage, and Detective Conners , his partner, shoots the criminal...
,
The Butterfly EffectThe Butterfly Effect is a 2004 American sci-fi psychological thriller film that is written and directed by Eric Bress and J. Mackye Gruber and starring Ashton Kutcher and Amy Smart...
, the sitcom
The Big Bang TheoryThe Big Bang Theory is an American sitcom created by Chuck Lorre and Bill Prady, both of whom serve as executive producers on the show, along with Steven Molaro. All three also serve as head writers...
,
Tom StoppardSir Tom Stoppard OM, CBE, FRSL is a British playwright, knighted in 1997. He has written prolifically for TV, radio, film and stage, finding prominence with plays such as Arcadia, The Coast of Utopia, Every Good Boy Deserves Favour, Professional Foul, The Real Thing, and Rosencrantz and...
's play
ArcadiaArcadia is a 1993 play by Tom Stoppard concerning the relationship between past and present and between order and disorder and the certainty of knowledge...
and the video game
Tom ClancyThomas Leo "Tom" Clancy, Jr. is an American author, best known for his technically detailed espionage, military science, and techno thriller storylines set during and in the aftermath of the Cold War, along with video games on which he did not work, but which bear his name for licensing and...
's Splinter Cell: Chaos Theory. The influence of chaos theory in shaping the popular understanding of the world we live in was the subject of the BBC documentary
High Anxieties - The Mathematics of Chaos directed by
David MaloneDavid Malone, author of The Debt Generation, is also director of acclaimed documentaries on philosophy, science and religion originally broadcast in the UK by the BBC and Channel 4.-Work:Malone's work includes...
. Chaos theory is also the subject of discussion in the BBC documentary "The Secret Life of Chaos" presented by the physicist
Jim Al-KhaliliJim Al-Khalili OBE is an Iraqi-born British theoretical physicist, author and science communicator. He is Professor of Theoretical Physics and Chair in the Public Engagement in Science at the University of Surrey...
.
See also
Examples of chaotic systems
- Arnold's cat map
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name....
- Bouncing ball dynamics
- Cliodynamics
thumb|Clio—detail from [[The Art of Painting|The Allegory of Painting]] by [[Johannes Vermeer]]Cliodynamics is a new multidisciplinary area of research focused at mathematical modeling of historical dynamics.-Origins:The term was originally coined by Peter...
- Coupled map lattice
A coupled map lattice is a dynamical system that models the behavior of non-linear systems . They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of chaos where the number of effective degrees of freedom diverge as the...
- Chua's circuit
Chua's circuit is a simple electronic circuit that exhibits classic chaos theory behavior. It was introduced in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time...
- Double pendulum
In mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of...
- Dynamical billiards
A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed...
- Economic bubble
An economic bubble is "trade in high volumes at prices that are considerably at variance with intrinsic values"...
- Hénon map
- 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 behavior of the orbits of the van der Pol oscillator...
- List of chaotic maps
- Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
- Rössler attractor
The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor...
- Standard map
- Swinging Atwood's machine
The swinging Atwood's machine is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions.Specifically, it comprises two masses ...
- Tilt A Whirl
Other related topics
- Butterfly effect
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a small change at one place in a nonlinear system can result in large differences to a later state...
- Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of 'expansion' and 'contraction'. Anosov systems are a special case of Axiom A...
- Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
- Chaos theory in organizational development
Chaos theory in organizational development refers to a subset of chaos theory which incorporates principles of quantum mechanics and presents them in a complex systems environment.-Background:...
- Complexity
In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...
- Control of chaos
In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different...
- Edge of chaos
The phrase edge of chaos was coined by mathematician Doyne Farmer to describe the transition phenomenon discovered by computer scientist Christopher Langton. The phrase originally refers to an area in the range of a variable, λ , which was varied while examining the behavior of a cellular automaton...
- Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
- Julia set
In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets defined from a function...
- Mandelbrot set
The Mandelbrot set is a particular mathematical set of points, whose boundary generates a distinctive and easily recognisable two-dimensional fractal shape...
- Predictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively.-Predictability and Causality:...
- Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle...
- Santa Fe Institute
The Santa Fe Institute is an independent, nonprofit theoretical research institute located in Santa Fe and dedicated to the multidisciplinary study of the fundamental principles of complex adaptive systems, including physical, computational, biological, and social systems.The Institute houses a...
- Synchronization of chaos
Synchronization of chaos is a phenomenon that may occur when two, or more, chaotic oscillators are coupled, or when a chaotic oscillator drives another chaotic oscillator...
- Unintended consequence
In the social sciences, unintended consequences are outcomes that are not the outcomes intended by a purposeful action. The concept has long existed but was named and popularised in the 20th century by American sociologist Robert K. Merton...
People
- Ralph Abraham
Ralph H. Abraham is an American mathematician. He has been a member of the mathematics department at the University of California, Santa Cruz since 1968.- Life and work :...
- Michael Berry
- Leon O. Chua
Leon Ong Chua is an IEEE Fellow and a professor in the electrical engineering and computer sciences department at the University of California, Berkeley, which he joined in 1971...
- Ivar Ekeland
Ivar Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well as popular books on mathematics, which have been published in French, English, and...
- Doyne Farmer
- Mitchell Feigenbaum
Mitchell Jay Feigenbaum is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants.- Biography :...
- Martin Gutzwiller
Martin C. Gutzwiller is a physicist. He is known for the invention of the variational wave-function describing electrons with strong localized interactions , for the first investigation of relations between classical and quantum mechanics in chaotic systems , as well as new solutions to...
- Brosl Hasslacher
Brosl Hasslacher was a theoretical physicist.Brosl Hasslacher obtained a bachelor's in physics from Harvard University in 1962. He did his Ph.D. with D.Z. Freeman and C.N. Yang at the State University of New York at Stony Brook...
- Michel Hénon
Michel Hénon is a mathematician and astronomer. He is currently at the Nice Observatory.In astronomy, Hénon is well known for his contributions to stellar dynamics. In late 1960s and early 1970s he was involved in dynamical evolution of star clusters, in particular the globular clusters...
- Edward Lorenz
- Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov was a Russian mathematician, mechanician and physicist. His surname is sometimes romanized as Ljapunov, Liapunov or Ljapunow....
- Ian Malcolm
Ian Malcolm may refer to:*Sir Ian Malcolm , British member of Parliament, clan chieftain*Ian Malcolm , fictional character in Jurassic Park and The Lost World, books written by Michael Crichton...
- Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
- Norman Packard
Norman Harry Packard is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. He is an alumnus of Reed College and the University of California, Santa Cruz. Packard is known for his contributions to both chaos theory and cellular automata...
- Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
- Otto Rössler
Otto E. Rössler is a German biochemist and is notable for his work on chaos theory and his theoretical equation known as the Rössler attractor.-Biography:...
- David Ruelle
David Pierre Ruelle is a Belgian-French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens he coined the term strange attractor, and founded a new theory of turbulence...
- Oleksandr Mikolaiovich Sharkovsky
Oleksandr Mikolaiovich Sharkovsky is a prominent Ukrainian mathematician most famous for developing Sharkovsky's Theorem in 1964. In 2006 he became a member of the National Academy of Sciences of Ukraine.-External links:...
- Robert Shaw
Robert Stetson Shaw is an American physicist who was part of Eudaemonic Enterprises in Santa Cruz in the late 1970s and early 1980s. In 1988 he was awarded a MacArthur Fellowship for his work in Chaos theory.-Chaos theory:...
- Floris Takens
Floris Takens 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 attractor, a term they coined, as opposed to the then-prevailing theory of accretion of modes....
- James A. Yorke
James A. Yorke is a Distinguished University Professor of Mathematics and Physics and chair of the Mathematics Department at the University of Maryland, College Park. He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology...
Articles
- A.N. Sharkovskii, "Co-existence of cycles of a continuous mapping of the line into itself", Ukrainian Math. J., 16:61–71 (1964)
- Li, T. Y.
Tien-Yien Li is a University Distinguished Professor of Mathematics at Michigan State University and a Guggenheim Fellow. Li and co-author James Yorke published a paper in 1975 entitled "Period Three Implies Chaos," in which the mathematical term chaos was coined.-References:* T.Y. Li, and J.A...
and Yorke, J. A.James A. Yorke is a Distinguished University Professor of Mathematics and Physics and chair of the Mathematics Department at the University of Maryland, College Park. He and Benoit Mandelbrot were the recipients of the 2003 Japan Prize in Science and Technology...
"Period Three Implies Chaos." American Mathematical MonthlyThe American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894. It is currently published 10 times each year by the Mathematical Association of America....
82, 985–92, 1975. Online version (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online, but which don't provide article views. The online content is identical to the hardcopy text. Citation variations will be related to country of publication).
- Kolyada, S. F. "Li-Yorke sensitivity and other concepts of chaos", Ukrainian Math. J. 56 (2004), 1242–1257.
- C. Strelioff, A. Hübler (2006). Medium-Term Prediction of Chaos, PRL 96, 044101
- A. Hübler, G. Foster, K. Phelps (2007). Managing Chaos: Thinking out of the Box Complexity, vol. 12, pp. 10–13
Semitechnical and popular works
- Ralph H. Abraham and Yoshisuke Ueda (Ed.), The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory, World Scientific Publishing Company, 2001, 232 pp.
- Michael Barnsley
Michael Fielding Barnsley is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. He received his Ph.D in Theoretical Chemistry from University of Wisconsin–Madison in 1972...
, Fractals Everywhere, Academic Press 1988, 394 pp.
- Richard J Bird, Chaos and Life: Complexity and Order in Evolution and Thought, Columbia University Press 2003, 352 pp.
- John Briggs
John Briggs, Distinguished Professor and Co-Chair of the Department of English Language, Comparative Literature, and Writing at Western Connecticut State University, is co-author of Seven Life Lessons of Chaos with the physicist David Peat...
and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper Perennial 1990, 224 pp.
- John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper Perennial 2000, 224 pp.
- Lawrence A. Cunningham, From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis, George Washington Law Review, Vol. 62, 1994, 546 pp.
- Predrag Cvitanović, Universality in Chaos, Adam Hilger 1989, 648 pp.
- Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 1988, 272 pp.
- James Gleick
James Gleick is an American author, journalist, and biographer, whose books explore the cultural ramifications of science and technology...
, Chaos: Making a New ScienceChaos: Making A New Science is the best-selling book by James Gleick that first introduced the principles and early development of chaos theory to the public...
, New York: Penguin, 1988. 368 pp.
- John Gribbin, Deep Simplicity,
- L Douglas Kiel, Euel W Elliott (ed.), Chaos Theory in the Social Sciences: Foundations and Applications, University of Michigan Press, 1997, 360 pp.
- Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National Book Trust, 2003.
- Hans Lauwerier, Fractals, Princeton University Press, 1991.
- Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996.
- Chapter 5 of Alan Marshall (2002) The Unity of nature, Imperial College Press: London
- Heinz-Otto Peitgen
thumb|180px|right|Heinz-Otto PeitgenHeinz-Otto Peitgen is a German mathematician. Peitgen is one of the most prominent researchers in the study of fractals.- Life :...
and Dietmar SaupeDietmar Saupe is a fractal researcher and professor of computer science, Department of Computer and Information Science, University of Konstanz, Germany....
(Eds.), The Science of Fractal Images, Springer 1988, 312 pp.
- Clifford A. Pickover
Clifford A. Pickover is an American author, editor, and columnist in the fields of science, mathematics, and science fiction, and is employed at the IBM Thomas J. Watson Research Center in Yorktown, New York.- Biography :He received his Ph.D...
, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
- Ilya Prigogine
Ilya, Viscount Prigogine was a Russian-born naturalized Belgian physical chemist and Nobel Laureate noted for his work on dissipative structures, complex systems, and irreversibility.-Biography :...
and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
- Heinz-Otto Peitgen and P. H. Richter, The Beauty of Fractals : Images of Complex Dynamical Systems, Springer 1986, 211 pp.
- David Ruelle
David Pierre Ruelle is a Belgian-French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens he coined the term strange attractor, and founded a new theory of turbulence...
, Chance and Chaos, Princeton University Press 1993.
- Ivars Peterson
Ivars Peterson is an award-winning mathematics writer.He is currently Director of Publications for Journals and Communications at the Mathematical Association of America....
, Newton's Clock: Chaos in the Solar System, Freeman, 1993.
- David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.
- Peter Smith, Explaining Chaos, Cambridge University Press, 1998.
- Ian Stewart
Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...
, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990.
- Steven Strogatz
Steven Henry Strogatz is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University...
, Sync: The emerging science of spontaneous order, Hyperion, 2003.
- Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993.
- M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, 1992.
External links
- Nonlinear Dynamics Research Group with Animations in Flash
- The Chaos group at the University of Maryland
- The Chaos Hypertextbook. An introductory primer on chaos and fractals
- ChaosBook.org An advanced graduate textbook on chaos (no fractals)
- Society for Chaos Theory in Psychology & Life Sciences
- Nonlinear Dynamics Research Group at CSDC, Florence
Florence is the capital city of the Italian region of Tuscany and of the province of Florence. It is the most populous city in Tuscany, with approximately 370,000 inhabitants, expanding to over 1.5 million in the metropolitan area....
ItalyItaly , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...
- Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum
- Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.
- Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens
- Gleick's Chaos (excerpt)
- Systems Analysis, Modelling and Prediction Group at the University of Oxford
- A page about the Mackey-Glass equation
- High Anxieties - The Mathematics of Chaos (2008) BBC documentary directed by David Malone
David Malone, author of The Debt Generation, is also director of acclaimed documentaries on philosophy, science and religion originally broadcast in the UK by the BBC and Channel 4.-Work:Malone's work includes...