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Encyclopedia > Chaos theory
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3
Systems science Portal

In mathematics, chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos. Chaos theory is a scientific theory describing erratic behaviour in certain nonlinear dynamical systems. ... Image File history File links Lorenz_attractor_yb. ... Image File history File links Lorenz_attractor_yb. ... A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ... Download high resolution version (2100x1500, 380 KB)  Acadac was inspired to create this graphic after reading: Roger Lewin (1992) Complexity: Life and the Edge of Chaos Steven Johnson (2001) Emergence: The Connected Lives of Ants, Brains, Cities, and Software   File links The following pages link to this file: Emergence Complex... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... The dynamical system concept is a mathematical formalization for any fixed rule which describes the time dependence of a points position in its ambient space. ... Point attractors in 2D phase space. ... Random redirects here. ... A deterministic system is a conceptual model of the philosophical doctrine of determinism applied to a system for understanding everything that has and will occur in the system, based on the physical outcomes of causality. ... For other uses, see Chaos (disambiguation). ...


Chaotic behaviour is also observed in natural systems, such as the weather. This may be explained by a chaos-theoretical analysis of a mathematical model of such a system, embodying the laws of physics that are relevant for the natural system. A mathematical model uses mathematical language to describe a system. ... A physical law or a law of nature is a scientific generalization based on empirical observations. ...

Contents

Overview

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices. Observations of chaotic behaviour in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. Everyday examples of chaotic systems include weather and climate.[1] There is some controversy over the existence of chaotic dynamics in the plate tectonics and in economics.[2][3][4] An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... For alternative meanings see laser (disambiguation). ... Chemical reactions are also known as chemical changes. ... --68. ... This article is about the Solar System. ... See also Earths magnetic field The magnetic fieldof a rotating body of conductive gas or liquid develops self-amplifying electric currents, and thus a self-generated magnetic field, due to a combination of differential rotation (different angular velocity of different parts of body), Coriolis forces and induction. ... Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... For the journal, see Ecology (journal). ... Schematic of an electrophysiological recording of an action potential showing the various phases which occur as the wave passes a point on a cell membrane. ... A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. ... // Meteorology (from Greek: μετέωρον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... The tectonic plates of the world were mapped in the second half of the 20th century. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ...


Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. A related field of physics called quantum chaos theory studies systems that follow the laws of quantum mechanics. Recently, another field, called relativistic chaos,[5] has emerged to describe systems that follow the laws of general relativity. Quantum chaos is an interdisciplinary branch of physics, arising from so-called semi-classical models. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ...


As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics.[citation needed] For example, the Lorenz system pictured is chaotic, but has a clearly defined structure. Bounded chaos is a useful term for describing models of disorder. A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ...


History

Fractal fern created using chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an Iterated function system (IFS).
Fractal fern created using chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an Iterated function system (IFS).

The first discoverer of chaos was Henri Poincaré. In 1890, while studying the three-body problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point.[6] In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature.[7] 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 exponent. The boundary of the Mandelbrot set is a famous example of a fractal. ... This article is about the group of pteridophyte plants. ... The chaos game or chaosgame is a means of creating a fractal, using a polygon and a random point inside it. ... Menger sponge, created by using IFS. Iterated function systems or IFSs, are a kind of fractal which were conceived in their present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... This page is a candidate for speedy deletion. ... In physics and mathematics, the Hadamard dynamical system or Hadamards billiards is a chaotic dynamical system, a type of dynamical billiards. ... In 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 theory. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff,[8] A. N. Kolmogorov,[9][10][11] M.L. Cartwright and J.E. Littlewood,[12] and Stephen Smale.[13] 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. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... George David Birkhoff George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944, Cambridge, Massachusetts) was an American mathematician, best known for what is now called the ergodic theorem. ... Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Dame Mary Cartwright was a leading British mathematician of the 20th century. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan, and winner of the Fields Medal in 1966. ...


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 map. What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied systems. A linear system is a model based on some kind of linear operator. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... Measure can mean: To perform a measurement. ... This article is about noise as in sound. ...


The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration 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. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models. ENIAC ENIAC, short for Electronic Numerical Integrator And Computer,[1] was the first large-scale, electronic, digital computer capable of being reprogrammed to solve a full range of computing problems,[2] although earlier computers had been built with some of these properties. ...

Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence was important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961.[14] Lorenz was using a simple digital computer, a Royal McBee LGP-30, 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. Image File history File links Download high resolution version (2976x2420, 995 KB) Date 05. ... Image File history File links Download high resolution version (2976x2420, 995 KB) Date 05. ... In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. ... Wingtip vortices stream from an F-15E as it disengages from a KC-10 Extender following midair refueling. ... Fixed-wing aircraft is a term used to refer to what are more commonly known as aeroplanes in Commonwealth English (excluding Canada) or airplanes in North American English. ... The term critical point can mean any of: critical point (thermodynamics) critical point (mathematics) critical loops (topology) critical point (set theory) This is a disambiguation page: a list of articles associated with the same title. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ле́в Дави́дович Ланда́у) (January 22, 1908 – April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... In physics, the Landau-Hopf theory of turbulence was until the mid 1970s the accepted theory of how a fluid flow becomes turbulent. ... (Born August 20, 1935) Belgian-French physicist. ... Floris Takens (born 12 November 1940) is a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. ... Turbulence is part of daily experience: no microscopes or telescopes are needed to notice the meanders of cigarette smoke, the gracious arabesques of cream poured into coffee or turbulent whirls in a mountain torrent. ... In the study of dynamical systems, an attractor is a set, curve, or space to which a system irreversibly evolves, if left undisturbed. ... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... // Meteorology (from Greek: μετέωρον, meteoron, high in the sky; and λόγος, logos, knowledge) is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting. ... Royal McBee is the name of the comupter manufacturing and retail division of Royal Typewriter wich made the early computers RPC 4000 and RPC 9000 [[1]]. Other sources also credit it as the manufacturer of the LGP-21 and the LGP-30 [2] [3] [4]. Note that the Wikipedia article... The LGP-30, standing for Librascope General Precision, was an early off the shelf computer manufactured and sold by Royal Precision Electronic Computer Company. ...


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.[15] Lorenz's discovery, which gave its name to Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most). A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ...


The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices.[16] Beforehand, he had studied information theory and concluded noise was patterned like a Cantor set: 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.[17] Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur, e.g., in a stock's prices after bad news, thus challenging normal distribution theory in statistics, aka Bell Curve) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).[18][19] In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional dimension," 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 infinitesimally small measuring device.[20] 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 fractal (for example, the Koch curve or "snowflake", which is infinitely long yet encloses a finite space and has fractal dimension equal to circa 1.2619, the Menger sponge and the Sierpiński gasket). In 1975 Mandelbrot published The Fractal Geometry of Nature, 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. Beno t Mandelbrot was the first to use a computer to plot the Mandelbrot set. ... Not to be confused with information technology, information science, or informatics. ... This article is about noise as in sound. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ... The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... This article is about the field of statistics. ... 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. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... The first four iterations of the Koch snowflake. ... In 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. ... For other uses, see Sponge (disambiguation). ...


Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol[21] and in 1958 by R.L. Ives.[22][23] However, Yoshisuke Ueda seems to have been the first experimenter to have identified a chaotic phenomenon as such by using an analog computer on November 27, 1961. The chaos exhibited by an analog computer is a real phenomenon, in contrast with those that digital computers calculate, which has a different kind of limit on precision. Ueda's supervising professor, Hayashi, did not believe in chaos, and thus he prohibited Ueda from publishing his findings until 1970.[24] A page from the Bombardiers Information File (BIF) that describes the components and controls of the Norden bombsight. ... Year 1970 (MCMLXX) was a common year starting on Thursday (link shows full calendar) of the Gregorian calendar. ...


In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist, part of the Eudaemons group with J. Doyne Farmer and Norman Packard who tried to find a mathematical method to beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz), and the meteorologist Edward Lorenz. New York Academy of Sciences is a society of some 20,000 scientists of all disciplines from 150 countries. ... (Born August 20, 1935) Belgian-French physicist. ... Robert McCredie Bob May, Baron May of Oxford OM AC Kt (born 8 January 1936 in Australia) is a cross-bench member of the British House of Lords and President of the Royal Society. ... Robert Stetson Shaw is an American physicist who was part of Eudaemonic Enterprises in Santa Cruz in the late 1970s and early 1980s. ... For Eudaemons in mythology, see Daemon. ... J. Doyne Farmer is an American physicist and one of the founding fathers of chaos theory. ... Norman Packard Norman Packard (born 1954 in Silver City, New Mexico) is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. ... Roulette is a casino and gambling game named after the French word meaning small wheel. In the game a croupier spins a wheel in one direction, then spins a ball in the opposite direction around a tilted circular surface running around the circumference of the wheel. ... Santa Cruz, Spanish and Portuguese for Holy Cross, is the name of several cities, regions, and other geographical features around the world: Argentina Puerto Santa Cruz, Santa Cruz province Santa Cruz Province, Argentina Bolivia Santa Cruz de la Sierra Santa Cruz Department Brazil Santa Cruz, Espírito Santo Santa Cruz... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ...


The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps.[25] Feigenbaum had applied fractal geometry to the study of natural forms such as coastlines. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena. Mitchell Jay Feigenbaum (born December 19, 1944; Philadelphia, USA) is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constant. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ...


In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective Rayleigh–Benard systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems".[26] Albert J. Libchaber (1934-) is a Detlev W. Bronk Professor at Rockefeller University. ... Look up Bifurcation in Wiktionary, the free dictionary. ... Convection in the most general terms refers to the movement of currents within fluids (i. ... Past winners of the Wolf Prize in Physics: 1978 Chien-Shiung Wu 1979 George Eugene Uhlenbeck, Giuseppe Occhialini 1980 Michael E. Fisher, Leo P. Kadanoff, Kenneth G. Wilson 1981 Freeman J. Dyson, Gerard t Hooft, Victor F. Weisskopf 1982 Leon M. Lederman, Martin M. Perl 1983/4 Erwin L. Hahn... Mitchell Jay Feigenbaum (born December 19, 1944; Philadelphia, USA) is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constant. ...


The New York Academy of Sciences then co-organized, in 1986, with the National Institute of Mental Health and the Office of Naval Research the first important conference on Chaos in biology and medicine. Bernardo Huberman thereby presented a mathematical model of the eye tracking disorder among schizophrenics.[27] Chaos theory thereafter renewed physiology in the 1980s, for example in the study of pathological cardiac cycles. The National Institute of Mental Health (NIMH) is part of the federal government of the United States and the largest research organization in the world specializing in mental illness. ... ONR Logo The Office of Naval Research (ONR), headquartered in Arlington, Virginia (Ballston), is an office of the U.S. Navy that carries out scientific research to support the Navy and Marine Corps in the interest of national security. ... This article or section does not cite any references or sources. ... Cardiac events occuring in a single cardiac cycle Cardiac cycle is the term referring to all or any of the events related to the flow of blood that occur from the beginning of one heartbeat to the beginning of the next. ...


In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[28] describing for the first time self-organized criticality (SOC), considered to be one of the mechanisms by which complexity arises in nature. Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have centred around large-scale natural or social systems that are known (or suspected) to display scale-invariant 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 law[29] describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); 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 Eldredge and Stephen Jay Gould). Worryingly, 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. Per Bak (December 8, 1948 in Denmark † October 16, 2002 in Copenhagen) was a danish theoretical physicist attributed with the development of the concept of self-organized criticality. ... Chao Tang is physicist, Professor at the university of California at San Francisco. ... Physical Review Letters is one of the most prestigious journals in physics. ... The theory of self-organized criticality (SOC) claims that whenever a self-organizing dynamical system is open or dissipative, it exhibits critical (scale-invariant) behaviour similar to that displayed by static systems undergoing a second-order phase transition. ... Complexity in general usage is the opposite of simplicity. ... In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. ... Global earthquake epicenters, 1963–1998. ... In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. ... A solar flare is a violent explosion in the Suns atmosphere with an energy equivalent to tens of millions of hydrogen bombs. ... In finance, financial markets facilitate: The raising of capital (in the capital markets); The transfer of risk (in the derivatives markets); and International trade (in the currency markets). ... Econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in Economics, usually those including uncertainties or stochastic elements and nonlinear dynamics. ... Fire in San Bernardino, California Mountains (image taken from the International Space Station) A wildfire, also known as a forest fire, vegetation fire, grass fire, or bushfire (in Australasia), is an uncontrolled fire in wildland often caused by lightning; other common causes are human carelessness and arson. ... This entry refers to the geological term landslide. ... An epidemic is generally a widespread disease that affects many individuals in a population. ... This article is about biological evolution. ... Punctuated equilibrium, or punctuated equilibria, is a theory of evolution which states that changes such as speciation can occur relatively quickly, with long periods of little change—equilibria—in between. ... Dr. Niles Eldredge (born August 25, 1943) is an American paleontologist, who, along with Stephen Jay Gould, proposed the theory of punctuated equilibrium in 1972. ... Stephen Jay Gould (September 10, 1941 – May 20, 2002) was an American paleontologist, evolutionary biologist, and historian of science. ... In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a common factor. ... For other uses of War, see War (disambiguation). ...


The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced general principles of chaos theory as well as its history to the broad public. At first the domains 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 Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimed that this new theory was an example of such as shift, a thesis upheld by J. Gleick. James Gleick (August 1, 1954– ) is an author, journalist, and biographer, whose books explore the cultural ramifications of science and technology. ... In mathematics, a nonlinear system is a system which is not linear i. ... Thomas Samuel Kuhn (July 18, 1922 – June 17, 1996) was an American intellectual who wrote extensively on the history of science and developed several important notions in the philosophy of science. ... Paradigm shift is the term first used by Thomas Kuhn in his 1962 book The Structure of Scientific Revolutions to describe a change in basic assumptions within the ruling theory of science. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


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, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc.). For other uses, see Topology (disambiguation). ... Not to be confused with information technology, information science, or informatics. ...


Chaotic dynamics

For a dynamical system to be classified as chaotic, it must have the following properties:[30]

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. In mathematics, mixing is a concept applied in ergodic theory, that is, the study of stochastic processes and measure-preserving dynamical systems. ... In mathematics, iterated functions are the objects of study in fractals and dynamical systems. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of...


Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science 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. Point attractors in 2D phase space. ... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... The American Association for the Advancement of Science (AAAS) is an organization that promotes cooperation between scientists, defends scientific freedom, encourages scientific responsibility and supports scientific education for the betterment of all humanity. ...


Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by iterating the mapping on the real line that maps x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behaviour, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems. Phase space of a dynamical system with focal stability. ...


Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimensional torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above. An irrational rotation is the map r:[0,1]->[0,1], r(x)=x + θ (mod 1) where θ is an irrational number. ...


Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Look up dye in Wiktionary, the free dictionary. ...


Linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be nonlinear. Also, by the Poincaré–Bendixson theorem, a continuous dynamical system on the plane cannot be chaotic; among continuous systems only those whose phase space is non-planar (having dimension at least three, or with a non-Euclidean geometry) can exhibit chaotic behaviour. However, a discrete dynamical system (such as the logistic map) can exhibit chaotic behaviour in a one-dimensional or two-dimensional phase space. A linear system is a mathematical model of a system based on the use of a linear operator. ... In mathematics, a nonlinear system is one whose behavior cant be expressed as a sum of the behaviors of its parts (or of their multiples. ... In mathematics, the Poincaré–Bendixson theorem is a statement about the behaviour of trajectories of continuous dynamical systems on the plane. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... This article is about the mathematical construct. ... 2-dimensional renderings (ie. ... Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ... Discrete time is non-continuous time. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ...


Attractors

Some dynamical systems are chaotic everywhere (see e.g. Anosov diffeomorphisms) 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 attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region. 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 diffeomorphisms were introduced by D. V. Anosov, who proved... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...


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. In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...

Phase diagram for a damped driven pendulum, with double period motion
Phase diagram for a damped driven pendulum, with double period motion

For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin. Image File history File links Download high resolution version (2280x1504, 612 KB)Damped driven TEL-Atomic Multipurpose Chaotic Pendulum, phase diagram of velocity versus position, displaying double period behavior. ... Image File history File links Download high resolution version (2280x1504, 612 KB)Damped driven TEL-Atomic Multipurpose Chaotic Pendulum, phase diagram of velocity versus position, displaying double period behavior. ... For other uses, see Pendulum (disambiguation). ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ... This article needs to be wikified. ...


Strange attractors

While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler map, which experiences period-two doubling route to chaos, like the logistic map. In mathematics, in the area of dynamical systems, a limit-cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches minus-infinity. ... In the study of dynamical systems, an attractor is a set, curve, or space to which a system irreversibly evolves, if left undisturbed. ... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3 A trajectory of Lorenzs equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, noted for its butterfly shape. ... In 1979 Otto Rössler found the inspiration from a Taffy-pulling machine for his Non-linear three-dimensional deterministic dynamical system. ...


Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set 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 fractal structure. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ... Hénon attractor for a = 1. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...


The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems. The Poincaré-Bendixson theorem is a statement about the behaviour of trajectories in two-dimensional continuous dynamical systems. ...


The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion. This article is about the problem in classical mechanics. ...


Minimum complexity of a chaotic system

Bifurcation diagram of a logistic map, displaying chaotic behaviour past a threshold

Simple systems can also produce chaos without relying on differential equations. An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over time. Another example is the Ricker model of population dynamics. Image File history File links Download high resolution version (1838x1300, 570 KB) Summary A bifurcation diagram for the Logistic map: The horizontal axis is the r parameter, the vertical axis is the x variable. ... Image File history File links Download high resolution version (1838x1300, 570 KB) Summary A bifurcation diagram for the Logistic map: The horizontal axis is the r parameter, the vertical axis is the x variable. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... The Ricker model is a classic discrete population model which gives the expected number (or density) of individuals in generation as a function of the number of individuals in the previous generation, Here is interpreted as an intrinsic growth rate and as the carrying capacity of the environment. ...


Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30. A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. ... Stephen Wolfram (born August 29, 1959 in London) is a physicist known for his work in theoretical particle physics, cellular automata, complexity theory, and computer algebra, and is the creator of the computer program Mathematica. ... Rule 30 is an elementary Cellular Automaton Rule. ...


A minimal model for conservative (reversible) chaotic behavior is provided by Arnold's cat map. From order to the chaos and return. ...


Mathematical theory

Sarkovskii'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. In mathematics, Sarkovskiis theorem (or Sharkovskys theorem) is a result, named for Oleksandr Mikolaiovich Sharkovsky, about discrete dynamical systems on the real line. ...


Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, and transfer operator. Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... In 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. ... In 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. ... In descriptive statistics and chaos theory, a recurrence plot (RP) 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. ... In 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 the flow of the system. ... In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. ... In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. ...


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.[31]


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.[32][31] Thus, given a time series to test for determinism, one can:

  1. pick a test state;
  2. search the time series for a similar or 'nearby' state; and
  3. 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.[33]


Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (i.e., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.[34] 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. Practically, anything approaching about 10 dimensions is considered so large that a stochastic description is probably more suitable and convenient anyway.[citation needed]


Applications

Chaos theory is applied in many scientific disciplines: mathematics, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics.[35] For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Biology (disambiguation). ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... The field of finance refers to the concepts of time, money and risk and how they are interelated. ... For other uses, see Philosophy (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... For other uses, see Politics (disambiguation). ... Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... Psychological science redirects here. ... BEAM robotics (acronym for Biology, Electronics, Aesthetics, and Mechanics) is a style of robotics that primarily uses simple analog circuits (instead of a microprocessor; though some mutants exist that do). ...


One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population growth under density dependence can lead to chaotic dynamics. The Ricker model is a classic discrete population model which gives the expected number (or density) of individuals in generation as a function of the number of individuals in the previous generation, Here is interpreted as an intrinsic growth rate and as the carrying capacity of the environment. ...


Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.[36]


Chaos theory in the media

Movies

Jurassic Park is a 1993 science fiction film directed by Steven Spielberg, based on the novel of the same name by Michael Crichton. ...

Books

Michael Crichton, pronounced [1], (born October 23, 1942) is an American author, film producer, film director, and television producer. ... For the feature film based on this book, see Jurassic Park (film). ... The Lost World is a novel by Michael Crichton, published in 1995 by Ballantine Books. ... Ray Douglas Bradbury (born August 22, 1920) is an American literary, fantasy, horror, science fiction, and mystery writer best known for The Martian Chronicles, a 1950 book which has been described both as a short story collection and a novel, and his 1953 dystopian novel Fahrenheit 451. ... “A Sound of Thunder” is a science fiction short story by Ray Bradbury, first published in Collier’s magazine in 1952. ... James Gleick (August 1, 1954– ) is an author, journalist, and biographer, whose books explore the cultural ramifications of science and technology. ...

See also

From order to the chaos and return. ... The Bouncing Ball Simulation System is a program for the Mac OS that provides a physically accurate rendering of the motions of a ball impacting with a sinusoidally vibrating table. ... Chuas circuit is a simple electronic circuit that exhibits classic chaos theory behavior. ... An example of a double pendulum. ... The Bunimovich stadium is a chaotic dynamical billiard A billiard is a dynamical system where a particle alternates between motion in a straight line and specular reflections with a boundary. ... bubbles are things that you make out of soap. ... The Hénon map is a discrete-time dynamical system. ... In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. ... The logistic map is a polynomial mapping, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. ... In 1979 Otto Rössler found the inspiration from a Taffy-pulling machine for his Non-linear three-dimensional deterministic dynamical system. ... Example of the mapping of ten orbits of the Standard map for . ... The Swinging Atwoods Machine (SAM) is a mechanism that resembles a simple Atwoods Machine except that one of the masses is allowed to swing in a two-dimensional plane. ... 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 diffeomorphisms were introduced by D. V. Anosov, who proved... In mathematics, specifically in the study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden qualitative or topological change in the systems long-term dynamical behaviour. ... 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. ... Complexity in general usage is the opposite of simplicity. ... In chaos theory, control of chaos is based on the fact that any chaotic attractor contains an infinite number of unstable periodic orbits. ... The phrase edge of chaos was coined by computer scientist Christopher Langton in 1990. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ... Prediction of future events is an ancient human wish. ... The Santa Fe Institute (or SFI) is a non-profit research institute dedicated to the study of complex systems in Santa Fe, New Mexico founded by George Cowan, David Pines, Stirling Colgate, Murray Gell-Mann, Nick Metropolis, Herb Anderson, Peter A. Carruthers, and Richard Slansky in 1984 to study complex... 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. ... Mitchell Jay Feigenbaum (born December 19, 1944; Philadelphia, USA) is a mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constant. ... There are very few or no other articles that link to this one. ... Michel Hénon (born 1931 in Paris, France) is a mathematician and astronomer. ... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 – November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ... Benoît B. Mandelbrot, PhD, (born November 20, 1924) is a Franco-American mathematician, best known as the father of fractal geometry. Benoît Mandelbrot was born in Poland, but his family moved to France when he was a child; he is a dual French and American citizen and was... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Otto E. Rössler (born 20 May 1940) is a German biochemist. ... (Born August 20, 1935) Belgian-French physicist. ... Oleksandr Mikolaiovich Sharkovsky (born December 7, 1936 in Kiev, Ukraine) is a Ukrainian mathematician who is best known for an important theorem on continuous functions that he proved in 1964. ... Floris Takens (born 12 November 1940) is a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. ... James A. Yorke (born August 3, 1941) is a Distinguished University Professor of Mathematics and Physics at the University of Maryland, and a recipient of the 2003 Japan Prize for his work in chaotic systems. ...

References

  1. ^ Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background", Environmetrics, vol. 8, no. 5, pages 517-532.
  2. ^ Apostolos Serletis and Periklis Gogas,Purchasing Power Parity Nonlinearity and Chaos, in: Applied Financial Economics, 10, 615-622, 2000.
  3. ^ Apostolos Serletis and Periklis Gogas The North American Gas Markets are ChaoticPDF (918 KiB), in: The Energy Journal, 20, 83-103, 1999.
  4. ^ Apostolos Serletis and Periklis Gogas, Chaos in East European Black Market Exchange Rates, in: Research in Economics, 51, 359-385, 1997.
  5. ^ A. E. Motter, Relativistic chaos is coordinate invariant, in: Phys. Rev. Lett. 91, 231101 (2003).
  6. ^ Jules Henri Poincaré (1890) "Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt," Acta Mathematica, vol. 13, pages 1–270.
  7. ^ Jacques Hadamard (1898) "Les surfaces à courbures opposées et leurs lignes géodesiques," Journal de Mathématiques Pures et Appliquées, vol. 4, pages 27–73.
  8. ^ George D. Birkhoff, Dynamical Systems, vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)
  9. ^ Kolmogorov, A. N. (1941a). “Local structure of turbulence in an incompressible fluid for very large Reynolds numbers,” Doklady Akademii Nauk SSSR, vol. 30, no. 4, pages 301–305. Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical Sciences (Series A), vol. 434, pages 9–13 (1991).
  10. ^ Kolmogorov, A. N. (1941b) “On degeneration of isotropic turbulence in an incompressible viscous liquid,” Doklady Akademii Nauk SSSR, vol. 31, no. 6, pages 538–540. Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical Sciences (Series A), vol. 434, pages 15–17 (1991).
  11. ^ Andrey N. Kolmogorov (1954) "Preservation of conditionally periodic movements with small change in the Hamiltonian function," Doklady Akademii Nauk SSSR, vol. 98, pages 527–530. See also Kolmogorov–Arnold–Moser theorem
  12. ^ Mary L. Cartwright and John E. Littlewood (1945) "On non-linear differential equations of the second order,I: The equation y" + k(1−y2)y' + y = bλkcos(λt + a), k large," Journal of the London Mathematical Society, vol. 20, pages 180–189. See also: Van der Pol oscillator
  13. ^ Stephen Smale (January 1960) "Morse inequalities for a dynamical system," Bulletin of the American Mathematical Society, vol. 66, pages 43–49.
  14. ^ Edward N. Lorenz, "Deterministic non-periodic flow," Journal of the Atmospheric Sciences, vol. 20, pages 130–141 (1963).
  15. ^ Gleick, James (1987). Chaos: Making a New Science. London: Cardinal, 17. 
  16. ^ Benoit Mandelbrot (1963) "The variation of certain speculative prices," Journal of Business, vol. 36, pages 394–419.
  17. ^ J.M. Berger and B. Mandelbrot (July 1963) "A new model for error clustering in telephone circuits," I.B.M. Journal of Research and Development, vol 7, pages 224–236.
  18. ^ B. Mandelbrot, The Fractal Geometry of Nature (N.Y., N.Y.: Freeman, 1977), page 248.
  19. ^ See also: Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (N.Y., N.Y.: Basic Books, 2004), page 201.
  20. ^ Benoît Mandelbrot (5 May 1967) "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," Science, Vol. 156, No. 3775, pages 636–638.
  21. ^ B. van der Pol and J. van der Mark (1927) "Frequency demultiplication," Nature, vol. 120, pages 363–364. See also: Van der Pol oscillator
  22. ^ R.L. Ives (10 October 1958) "Neon oscillator rings," Electronics, vol. 31, pages 108–115.
  23. ^ See p. 83 of Lee W. Casperson, "Gas laser instabilities and their interpretation," pages 83–98 in: N. B. Abraham, F. T. Arecchi, and L. A. Lugiato, eds., Instabilities and Chaos in Quantum Optics II: Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, June 28–July 7, 1987 (N.Y., N.Y.: Springer Verlag, 1988).
  24. ^ Ralph H. Abraham and Yoshisuke Ueda, eds., The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory (Singapore: World Scientific Publishing Co., 2001). See Chapters 3 and 4.
  25. ^ Mitchell Feigenbaum (July 1978) "Quantitative universality for a class of nonlinear transformations," Journal of Statistical Physics, vol. 19, no. 1, pages 25–52.
  26. ^ The Wolf Prize in Physics in 1986.
  27. ^ Bernardo Huberman, "A Model for Dysfunctions in Smooth Pursuit Eye Movement" Annals of the New York Academy of Sciences, Vol. 504 Page 260 July 1987, Perspectives in Biological Dynamics and Theoretical Medicine
  28. ^ Per Bak, Chao Tang, and Kurt Wiesenfeld, "Self-organized criticality: An explanation of the 1/f noise," Physical Review Letters, vol. 59, no. 4, pages 381–384 (27 July 1987). However, the conclusions of this article have been subject to dispute. See: http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html . See especially: Lasse Laurson, Mikko J. Alava, and Stefano Zapperi, "Letter: Power spectra of self-organized critical sand piles," Journal of Statistical Mechanics: Theory and Experiment, 0511, L001 (15 September 2005).
  29. ^ F. Omori (1894) "On the aftershocks of earthquakes," Journal of the College of Science, Imperial University of Tokyo, vol. 7, pages 111-200.
  30. ^ Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press. ISBN 0521587506. 
  31. ^ a b Provenzale A. et al.: "Distinguishing between low-dimensional dynamics and randomness in measured time-series", in: Physica D, 58:31-49, 1992
  32. ^ Sugihara G. and May R.: "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series", in: Nature, 344:734-41, 1990
  33. ^ Casdagli, Martin. "Chaos and Deterministic versus Stochastic Non-linear Modelling", in: Journal Royal Statistics Society: Series B, 54, nr. 2 (1991), 303-28
  34. ^ Broomhead D. S. and King G. P.: "Extracting Qualitative Dynamics from Experimental Data", in: Physica 20D, 217-36, 1986
  35. ^ Metaculture.net, metalinks: Applied Chaos, 2007.
  36. ^ Comdig.org, Complexity Digest 199.06

“PDF” redirects here. ... A kibibyte (a contraction of kilo binary byte) is a unit of information or computer storage, commonly abbreviated KiB (never kiB). 1 kibibyte = 210 bytes = 1,024 bytes The kibibyte is closely related to the kilobyte, which can be used either as a synonym for kibibyte or to refer to... The Kolmogorov–Arnold–Moser theorem is a result in dynamical systems about the persistence of quasi-periodic motions under small perturbations. ... Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle. ... Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle. ...

Literature

Articles

Tien-Yien Li is a University Distinguished Professor of Mathematics at Michigan State University and a Guggenheim Fellow. ... James A. Yorke (born August 3, 1941) is a Distinguished University Professor of Mathematics and Physics at the University of Maryland, and a recipient of the 2003 Japan Prize for his work in chaotic systems. ... The American Mathematical Monthly is a mathematical journal published 10 times each year by the Mathematical Association of America since 1894. ... Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... The article entitled A Mathematical Theory of Communication, published in 1948 by mathematician Claude E. Shannon, was one of the founding works of the field of information theory. ... Bell System Technical Journal was the in-house journal of Bell Laboratories. ...

Textbooks

  • Alligood, K. T. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag New York, LLC. ISBN 0-387-94677-2. 
  • Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 0-521-39511-9. 
  • Badii, R.; Politi A. (1997). "Complexity: hierarchical structures and scaling in physics". Cambridge University Press. ISBN 0521663857. 
  • Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed,. Westview Press. ISBN 0-8133-4085-3. 
  • Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0-521-47685-2. 
  • Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer-Verlag New York, LLC. ISBN 0-387-97173-4. 
  • Hoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 981-02-4073-2. 
  • Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 0-472-08472-0. 
  • Moon, Francis (1990). Chaotic and Fractal Dynamics. Springer-Verlag New York, LLC. ISBN 0-471-54571-6. 
  • Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press New, York. ISBN 0-521-01084-5. 
  • Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6. 
  • Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9. 
  • Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 0-521-83912-2. 
  • Tufillaro, Abbott, Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley New York. ISBN 0-201-55441-0. 
  • Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 0-198-52604-0. 

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, 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 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.
  • Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 1988, 272 pp.
  • James Gleick, Chaos: Making a New Science, 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.
  • Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp.
  • Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
  • Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
  • H.-O. Peitgen and P.H. Richter, The Beauty of Fractals : Images of Complex Dynamical Systems, Springer 1986, 211 pp.
  • David Ruelle, Chance and Chaos, Princeton University Press 1993.
  • David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.
  • Peter Smith, Explaining Chaos, Cambridge University Press, 1998.
  • Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990.
  • Steven Strogatz, 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.

Michael Barnsley is the researcher and entrepreneur who has worked on fractal compression; he holds several patents on the technology. ... Johnny Briggs is also the name of the actor who plays Mike Baldwin in the soap opera Coronation Street. ... Johnny Briggs is also the name of the actor who plays Mike Baldwin in the soap opera Coronation Street. ... James Gleick (August 1, 1954– ) is an author, journalist, and biographer, whose books explore the cultural ramifications of science and technology. ... Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ... Clifford A. Pickover is an author, editor, and columnist in the fields of science, mathematics, and science fiction. ... Ilya Prigogine (January 25, 1917 – May 28, 2003) was a Belgian physicist and chemist noted for his work on dissipative structures, complex systems, and irreversibility. ... (Born August 20, 1935) Belgian-French physicist. ... (Born August 20, 1935) Belgian-French physicist. ... Ian Stewart, FRS (b. ... Steven H. Strogatz is Professor of theoretical and applied mechanics at Cornell University. ...

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For other uses, see System (disambiguation). ... Systems science is the science of complex systems. ... For other uses, see System (disambiguation). ... An example of a system: The nervous system. ... There are many definitions of complexity, therefore many natural, artificial and abstract objects or networks can be considered to be complex systems, and their study (complexity science) is highly interdisciplinary. ... Complex adaptive systems are special cases of complex systems. ... A conceptual system is a system that is comprised of non-physical objects, i. ... Cultural system refers to the functional interaction between the different elements of culture in a particular manner. ... The Lorenz attractor is an example of a non-linear dynamical system. ... An economic system is a particular set of social institutions which deals with the production, distribution and consumption of goods and services in a particular society. ... A coral reef near the Hawaiian islands is an example of a complex marine ecosystem. ... In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... GPS redirects here. ... List of bones of the human skeleton Human anatomy is primarily the scientific study of the morphology of the adult human body. ... It has been suggested that this article or section be merged with Information systems. ... World distribution of major legal traditions The three major legal systems of the world today consist of civil law, common law and religious law. ... A system of measurement is a set of units which can be used to specify anything which can be measured and were historically important, regulated and defined because of trade and internal commerce. ... The International System of Units (symbol: SI) (for the French phrase Syst me International dUnit s) is the most widely used system of units. ... A multi-agent system (MAS) is a system composed of several software agents, collectively capable of reaching goals that are difficult to achieve by an individual agent or monolithic system. ... The nervous system is a highly specialized network whose principal components are nerves called neurons. ... In mathematics, a nonlinear system is one whose behavior cant be expressed as a sum of the behaviors of its parts (or of their multiples. ... An operating system (OS) is a software that manages computer resources and provides programmers with an interface used to access those resources. ... A physical system is a system that is comprised of matter and energy. ... A political system is a system of politics and government. ... The human eye is the first element of a sensory system: in this case, vision, for the visual system. ... See Social structure of the United States for an explanation of concepts exsistance within US society. ... This article is about the Solar System. ... Complex systems have a number of properties, some of which are listed below. ... For control theory in psychology and sociology, see control theory (sociology). ... For other uses, see Cybernetics (disambiguation). ... Holism in science, or Holistic science, is an approach to research that emphasizes the study of complex systems. ... Sociotechnical systems theory is theory about the social aspects of people and society and technical aspects of machines and technology. ... Systems biology is a term used very widely in the biosciences, particularly from the year 2000 onwards, and in a variety of contexts. ... System dynamics is an approach to understanding the behaviour of complex systems over time. ... Systems Ecology is a transdiscipline which studies ecological systems, or ecosystems. ... Systems engineering techniques are used in complex projects: from spacecrafts to chip design, from robotics to creating large software products to building bridges, Systems engineering uses a host of tools that include modeling & simulation, requirements analysis, and scheduling to manage complexity Systems Engineering (SE) is an interdisciplinary approach and means... Systems science is the science of complex systems. ... Systems theory is an interdisciplinary field of science. ... Russell Lincoln Ackoff (born 12 February 1919) is a Professor Emeritus of the Wharton School in operations research and systems theory. ... William Ross Ashby (September 6, 1903, London, England - November 15, 1972) was a British psychiatrist and a pioneer in the study of complex systems. ... Gregory Bateson (9 May 1904–4 July 1980) was a British anthropologist, social scientist, linguist and cyberneticist whose work intersected that of many other fields. ... Anthony Stafford Beer (September 25, 1926 - August 23, 2002) was a theorist in operational research and management cybernetics. ... Karl Ludwig von Bertalanffy (September 19, 1901, Vienna, Austria - June 12, 1972, New York, USA) was a biologist who was a founder of general systems theory--which he literally translated from the mathematization of Nicolai Hartmanns Ontology as stated by himself in his seminal work-- .An Austrian citizen, he... Kenneth E. Boulding Kenneth Ewart Boulding (January 18, 1910 - March 18, 1993) was an economist, educator, peace activist, poet, religious mystic, devoted Quaker, systems scientist, and interdisciplinary philosopher. ... British academic Peter Checkland is the developer of soft-systems methodology (SSM) in the field of systems thinking. ... 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George Jiri Klir (1932 Prague, Czechoslovakia) is an Czech-American computer scientist and professor of systems sciences at the Center for Intelligent Systems at the Binghamton University in New York. ... Dr. Lorenz at work Edward Norton Lorenz is an American mathematician and meteorologist, and a contributor to the chaos theory and inventor of the strange attractor notion. ... Niklas Luhmann (December 8, 1927 - November 6, 1998) was a German sociologist, administration expert, and social systems theorist, as well as one the most prominent modern day thinkers in the sociological systems theory. ... Humberto Maturana (born September 14, 1928 in Santiago) is a Chilean biologist whose work crosses over into philosophy and cognitive science. ... Donella Dana Meadows (March 13, 1941 Elgin, Illinois, USA - February 20, 2001, New Hampshire) was a pioneering environmental scientist, a teacher and writer. ... Mihajlo D. Mesarovic (1928) is a Yugoslavian scientist, who was professor of Systems Engineering and Mathematics at Case Western Reserve University. ... Howard Thomas Odum (1924-2002), commonly known as H.T. Odum or Tom Odum, was an eminent American ecosystem ecologist and a professor at the University of Florida. ... Talcott Parsons Talcott Edgar Frederick Parsons (December 13, 1902–May 8, 1979) was for many years the best-known sociologist in the United States, and indeed one of the best-known in the world. ... Ilya Prigogine (January 25, 1917 – May 28, 2003) was a Belgian physicist and chemist noted for his work on dissipative structures, complex systems, and irreversibility. ... Anatol Rapoport (born May 22, 1911) is a Russian-born American Jewish, mathematical psychologist. ... Claude Shannon Claude Elwood Shannon (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called the father of information theory,[1] and was the founder of practical digital circuit design theory. ... Francisco Varela (Santiago, September 7, 1946 – May 28, 2001, Paris) was a Chilean biologist and philosopher who, together with his teacher Humberto Maturana, is best known for introducing the concept of autopoiesis to biology. ... Norbert Wiener Norbert Wiener (November 26, 1894, Columbia, Missouri – March 18, 1964, Stockholm Sweden) was an American theoretical and applied mathematician. ...

  Results from FactBites:
 
Chaos Theory- Crystalinks (2064 words)
Thus chaos theory is, very generally, the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a physical system.
Chaos theory attempts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions.
Chaos theory progressed more rapidly 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 behavior of certain experiments like that of the logistic map.
Chaos Homepage (967 words)
Andrew Ho Chaos theory is among the youngest of the sciences, and has rocketed from its obscure roots in the seventies to become one of the most fascinating fields in existence.
Chaos theory predicts that complex nonlinear systems are inherently unpredictable--but, at the same time, chaos theory also insures that often, the way to express such an unpredictable system lies not in exact equations, but in representations of the behavior of a system--in plots of strange attractors or in fractals.
Chaos theory is great as a way of looking at events which happen in the world differently from the more traditional strictly deterministic view which has dominated science from Newtonian times.
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