A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3

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). The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in Chaos theory. 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. Randomness is a lack of order Purpose, cause, or predictability 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. A deterministic system is a conceptual model of the philosophical Doctrine of Determinism applied to a System for understanding everything that This behavior is known as deterministic chaos, or simply chaos. Chaos (derived from the Ancient Greek, Chaos) typically refers to Unpredictability, and is the antithesis of Cosmos.

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. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i

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. An electrical network is an interconnection of Electrical elements such as Resistors Inductors Capacitors Transmission lines Voltage A laser is a device that emits Light ( Electromagnetic radiation) through a process called Stimulated emission. A chemical reaction is a process that always results in the interconversion of Chemical substances The substance or substances initially involved in a chemical reaction are called Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion 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. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. See also Earth's magnetic field The magnetic field of a rotating body of conductive gas or liquid develops self-amplifying Electric currents and thus 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 Ecology (from Greek grc οἶκος oikos, "house(hold" and grc -λογία -logia) is the scientific study of In Neurophysiology, the action potential is a self-regenerating Wave of Electrochemical activity that allows Nerve cells to carry a signal A molecular vibration occurs when Atoms in a Molecule are in periodic motion while the molecule as a whole has constant translational Everyday examples of chaotic systems include weather and climate. Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary ^[1] There is some controversy over the existence of chaotic dynamics in the plate tectonics and in economics. Plate tectonics (from Greek τέκτων tektōn "builder" or "mason" describes the large scale motions of Earth 's Lithosphere Economics is the social science that studies the production distribution, and consumption of goods and services. ^[2][3][4]

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. Quantum chaos is a branch of Physics which studies how chaotic classical systems (see Dynamical systems and Chaos theory) can be shown to be limits of Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Recently, another field, called relativistic chaos,^[5] has emerged to describe systems that follow the laws of general relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics. For example, the Lorenz system pictured is chaotic, but has a clearly defined structure. The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly Bounded chaos is a useful term for describing models of disorder.

History

Fractal fern created using chaos game. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" A fern is any one of a group of about 20000 Species of Plants classified in the phylum or division Pteridophyta, also known as Filicophyta In Mathematics, the term chaos game, as coined by Michael Barnsley, originally referred to a method of creating a Fractal, using a Polygon Natural forms (ferns, clouds, mountains, etc. ) may be recreated through an Iterated function system (IFS). In Mathematics, iterated function systems or IFS s are a method of constructing Fractals the resulting constructions are always Self-similar.

The first discoverer of chaos was Henri Poincaré. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician 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. 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 ^[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. Jacques Salomon Hadamard ( December 8, 1865 – October 17, 1963) was a French Mathematician best known for his proof of ^[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. In Physics and Mathematics, the Hadamard dynamical system or Hadamard's billiards is a chaotic Dynamical system, a type of Dynamical In Mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a Dynamical system is a quantity that characterizes the rate of separation of

Much of the earlier theory was developed almost entirely by mathematicians, under the name of ergodic theory. Ergodic theory is a branch of Mathematics that studies Dynamical systems with an Invariant measure and related problems 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. George David Birkhoff ( 21 March 1884, Overisel Michigan - 12 November 1944, Cambridge Massachusetts) was an American Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet Dame Mary Lucy Cartwright DBE ( December 17, 1900 &ndash April 3, 1998) was a leading 20th-century British Mathematician John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration Stephen Smale (born July 15, 1930) is an American Mathematician from Flint Michigan. ^[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.

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. A linear system is a mathematical model of a System based on the use of a Linear operator. 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 What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied systems. is a one volume manga created by Tsutomu Nihei as a prequel to his ten-volume work Blame!.

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, short for Electronic Numerical Integrator And Computer, was the first general-purpose electronic Computer.

Turbulence in the tip vortex from an airplane wing. In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes Wingtip vortices are tubes of circulating air which are left behind by the Wing as it generates lift. Overview Fixed-wing aircraft range from small training and recreational aircraft to Wide-body aircraft and military cargo aircraft. 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. Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 In Physics, the Landau-Hopf theory of turbulence, named for Lev Landau and Eberhard Hopf, was until the mid 1970s the accepted theory of how a Fluid David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory. David Pierre Ruelle (b August 20, 1935 Ghent, Belgium) is a Belgian-French Mathematical physicist. Floris Takens (born November 12, 1940) is a Dutch mathematician known for contributions to the theory of chaotic dynamical systems In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes An attractor is a set to which a Dynamical system evolves after a long enough time

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Edward Norton Lorenz ( May 23, 1917) was an American Mathematician and meteorologist, and a pioneer of Chaos theory. Meteorology (from Greek grc μετέωρος metéōros, "high in the sky" and grc -λογία -logia) is the Interdisciplinary ^[14] Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather simulation. Royal 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-30, standing for Librascope General Purpose and then Librascope General Precision, was an early "off the shelf" computer manufactured by the Librascope 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. ^[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). The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly

The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Benoît B Mandelbrot (born 20 November 1924 is a French mathematician, best known as the father of fractal geometry. ^[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. Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. is a one volume manga created by Tsutomu Nihei as a prequel to his ten-volume work Blame!. In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith ^[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). The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. ^[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. How Long Is the Coast of Britain ? Statistical Self-Similarity and Fractional Dimension is a paper by Mathematician Benoît Mandelbrot Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have ^[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. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" The Koch snowflake (or Koch star) is a mathematical Curve and one of the earliest Fractal curves to have been described 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 2619, the Menger sponge and the Sierpiński gasket). In Mathematics, the Menger sponge is a Fractal curve It is the universal curve, in that it has Topological dimension one and any other The Sierpiński triangle, also called the Sierpiński gasket or the Sierpiński Sieve, is a Fractal named after Wacław Sierpiński who described 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.

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. An analog computer (spelt analogue in British English is a form of Computer that uses continuous physical phenomena such as electrical mechanical 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. Year 1970 ( MCMLXX) was a Common year starting on Thursday (link shows full calendar of the Gregorian calendar. ^[24]

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. The New York Academy of Sciences is the third oldest scientific society in the United States David Pierre Ruelle (b August 20, 1935 Ghent, Belgium) is a Belgian-French Mathematical physicist. Robert May or Bob May may refer to Bob May (actor (born 1939 Bob May (golfer (born 1968 Robert L James Yorke may refer to James A Yorke, mathematician James Yorke (clergy 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. The Eudaemons were a small group headed by graduate physics students J J Doyne Farmer (1952 is an American Physicist and Entrepreneur, with interest in Chaos theory Norman Packard (born 1954 in Silver City New Mexico) is a Chaos theory Physicist and one of the founders of the Prediction Company and Roulette is a Casino and Gambling game named after the French word meaning "small wheel" Edward Norton Lorenz ( May 23, 1917) was an American Mathematician and meteorologist, and a pioneer of Chaos theory.

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps. Mitchell Jay Feigenbaum (born December 19 1944) is a mathematical physicist whose pioneering studies in Chaos theory led to the discovery 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 ^[25] Feigenbaum had applied fractal geometry to the study of natural forms such as coastlines. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.

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. Albert J Libchaber (1934- is a Detlev W Bronk Professor at Rockefeller University Convection in the most general terms refers to the movement of molecules within Fluids (i 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". Mitchell Jay Feigenbaum (born December 19 1944) is a mathematical physicist whose pioneering studies in Chaos theory led to the discovery ^[26]

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. 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 The Office of Naval Research ( ONR) headquartered in Arlington Virginia ( Ballston) is the office within the United States Department of the Bernardo Huberman thereby presented a mathematical model of the eye tracking disorder among schizophrenics. Schizophrenia ( from the Greek roots schizein (σχίζειν "to split" and phrēn ^[27] Chaos theory thereafter renewed physiology in the 1980s, for example in the study of pathological cardiac cycles. Physiology (from Greek grc φύσις physis, "nature origin" and grc -λογία -logia) is the study of the mechanical physical 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. Per Bak ( December 8 1948 - October 16 2002) was a Danish theoretical physicist, attributed with the development of Chao Tang is a Chinese Physicist and professor at the University of California at San Francisco. Kurt Wiesenfeld is an American Physicist working primarily on Non-linear dynamics. Physical Review Letters is one of the most prestigious journals in Physics. In Physics, self-organized criticality (SOC is a property of (classes of Dynamical systems which have a critical point as an Attractor. In general usage complexity often tends to be used to characterize something with many parts in intricate arrangement 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. In Physics, the Bak–Tang–Wiesenfeld sandpile model is the first discovered example of a Dynamical system displaying Self-organized criticality 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 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). An earthquake is the result of a sudden release of energy in the Earth 's crust that creates Seismic waves Earthquakes are recorded with a Seismometer 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 In Seismology, the Gutenberg–Richter law expresses the relationship between the magnitude and total number of Earthquakes in any given region and time A solar flare is a violent explosion in a star's (like the Sun 's atmosphere releasing as much Energy as 6 × 1025 Joules Solar flares In Economics, a financial market is a mechanism that allows people to easily buy and sell ( Trade) financial Securities (such as stocks and bonds Econophysics is an interdisciplinary research field applying theories and methods originally developed by physicists in order to solve problems in Economics, usually A wildfire, also known as a wildland fire, forest fire, brush fire, vegetation fire, grass fire, Peat fire, A landslide is a geological phenomenon which includes a wide range of ground movement such as rock falls deep failure of slopes and shallow debris flows which can occur In Epidemiology, an epidemic (from Greek epi- upon + demos people is a classification of a disease that appears as new cases in a eVolution is the third Album by eLDee, it was due to be released in 2008 Punctuated equilibrium is a theory of evolutionary biology which states that most sexually reproducing populations experience little change for most of their geological Niles Eldredge (born August 25 1943 is an American paleontologist, who along with Stephen Jay Gould, proposed the theory of Punctuated equilibrium Stephen Jay Gould (September 10 1941 &ndash May 20 2002 was a prominent American paleontologist, evolutionary biologist, and historian of science 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. 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 War is an international relations Dispute, characterized by organized Violence between National Military units 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 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. James Gleick (born August 1, 1954) is an author journalist and biographer whose books explore the cultural ramifications of science and technology Chaos 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 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. This article describes the use of the term nonlinearity in mathematics 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. Thomas Samuel Kuhn (surname ˈkuːn July 18, 1922  &ndash June 17, 1996) was an American intellectual who wrote extensively Paradigm shift, sometimes known as extraordinary science or revolutionary science, is the term first used by Thomas Kuhn in his influential The Structure of Scientific Revolutions ( 1962) by Thomas Kuhn, is an analysis of the History of science. 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, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. ).

Chaotic dynamics

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

• it must be sensitive to initial conditions,
• it must be topologically mixing, and
• its periodic orbits must be dense. In Mathematics, mixing is an abstract concept originating from Physics: the attempt to describe the irreversible Thermodynamic process of mixing In Mathematics, in the study of Dynamical systems an orbit is a collection of points related by the Evolution function of the dynamical system In Topology and related areas of Mathematics, a Subset A of a Topological space X is called dense (in X) if

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.

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. The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in Chaos theory. Edward Norton Lorenz ( May 23, 1917) was an American Mathematician and meteorologist, and a pioneer of Chaos theory. The American Association for the Advancement of Science (or AAAS) is an organization that promotes cooperation between Scientists defends scientific freedom encourages 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.

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. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a 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.

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. In Mathematics, an irrational rotation is a map r: \rightarrow given by r(x = x + \theta \mod 1

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. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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. A dye can generally be described as a Colored substance that has an affinity to the substrate to which it is being applied

Linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be nonlinear. A linear system is a mathematical model of a System based on the use of a Linear operator. This article describes the use of the term nonlinearity in mathematics 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. In Mathematics, the Poincaré–Bendixson theorem is a statement about the long term behaviour of orbits of Continuous dynamical systems on the plane Continuity may refer to In mathematics: Continuous probability distribution or random variable in probability and statistics For In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry However, a discrete dynamical system (such as the logistic map) can exhibit chaotic behaviour in a one-dimensional or two-dimensional phase space. Discrete time is non-continuous time Sampling at non-continuous times results in discrete-time samples 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

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. In Mathematics, more particularly in the fields of Dynamical systems and Geometric topology, an Anosov map on a Manifold M is a certain 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. An attractor is a set to which a Dynamical 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. In Mathematics, stability theory deals with the stability of solutions (or sets of solutions for Differential equations and Dynamical systems Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor.

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 is a mass that is attached to a pivot from which it can swing freely 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. A pencil is a family of geometric objects such as lines, that have a common property such as passage through a given line in a given plane.

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. In Mathematics, in the area of Dynamical systems, a limit-cycle on a plane or a Two-dimensional manifold An attractor is a set to which a Dynamical system evolves after a long enough time For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. Edward Norton Lorenz ( May 23, 1917) was an American Mathematician and meteorologist, and a pioneer of Chaos theory. The Lorenz attractor, named for Edward N Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a chaotic flow, noted for its Butterfly 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. RosslerStereopng|thumb|right|344px|Rössler attractor as a Stereogram with a=0

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). 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 The Hénon map is a discrete-time Dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. 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. In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under Both strange attractors and Julia sets typically have a fractal structure. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole"

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

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. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over time. 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 "Difference equation" redirects here It should not be confused with a Differential equation. Another example is the Ricker model of population dynamics. The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number a   t +1 (or density

Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions. A cellular automaton (plural cellular automata) is a discrete model studied in computability theory, Mathematics, Theoretical biology Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30. Stephen Wolfram (born August 29, 1959 in London) is a British Physicist, Mathematician and Businessman known for his Rule 30 is a one-dimensional binary Cellular automaton rule introduced by Stephen Wolfram in 1983

A minimal model for conservative (reversible) chaotic behavior is provided by 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

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, Sharkovskii 's theorem is a result about Discrete dynamical systems One of the implications of the theorem is that if a continuous discrete

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. A mathematician is a person whose primary area of study and research is the field of Mathematics. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, and transfer operator. 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 In Mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a Dynamical system is a quantity that characterizes the rate of separation of 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 In Mathematics, particularly in Dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection In Mathematics, particularly in Dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits of a system The transfer operator is different from the transfer homomorphism.

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.

Applications

Chaos theory is applied in many scientific disciplines: mathematics, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Foundations of modern biology There are five unifying principles Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Economics is the social science that studies the production distribution, and consumption of goods and services. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and The field of finance refers to the concepts of Time, Money and Risk and how they are interrelated Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Politics Politics is the process by which groups of people make decisions 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 Psychology (from Greek grc ψῡχή psȳkhē, "breath life soul" and grc -λογία -logia) is an Academic and The word "beam" in BEAM robotics is an acronym for '''B'''iology, '''E'''lectronics, '''A'''esthetics, and '''M'''echanics ^[35]

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, named after Bill Ricker, is a classic discrete population model which gives the expected number a   t +1 (or density

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions. Epilepsy is a common chronic Neurological disorder that is characterized by recurrent unprovoked seizures. ^[36]

Chaos theory in the media

Movies

• Jurassic Park (1993)
• π (1998)
• The Butterfly Effect (2004)
• The Science of Sleep (2006)
• Chaos (2006)
• Chaos Theory (2007)

Books

• Michael Crichton's Jurassic Park and The Lost World
• Ray Bradbury's A Sound of Thunder
• James Gleick Chaos: Making a New Science

See also

References

Literature

Articles

• Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos. Tien-Yien Li (李天岩 (born 1945 is a University Distinguished Professor of Mathematics at Michigan State University and a Guggenheim Fellow. This article is about the 20th century mathematician There is also an 18th century clergyman James Yorke James A " American Mathematical Monthly 82, 985-992, 1975. The American Mathematical Monthly ( is a mathematical journal founded by Benjamin Finkel in 1894.
• Kolyada, S. F. "Li-Yorke sensitivity and other concepts of chaos", Ukrainian Math. J. 56 (2004), 1242--1257.
• C.E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal, vol. Claude Elwood Shannon (April 30 1916 – February 24 2001 an American Electronic engineer and Mathematician, is "the father of Information "A Mathematical Theory of Communication" is an influential 1948 article by Mathematician Claude E Bell Labs Technical Journal is the in-house journal for scientists of Bell Labs / Alcatel-Lucent. 27, pp. 379-423, 623-656, July, October, 1948

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. Michael Fielding Barnsley is a mathematician researcher and an Entrepreneur who has worked on Fractal compression; he holds several Patents on the technology
• 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 may refer to Johnny Briggs (cricketer (1862-1902 English cricketer Johnny Briggs (actor (born 1935 English actor who
• John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper Perennial 2000, 224 pp. John Briggs may refer to Johnny Briggs (cricketer (1862-1902 English cricketer Johnny Briggs (actor (born 1935 English actor who
• 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. James Gleick (born August 1, 1954) is an author journalist and biographer whose books explore the cultural ramifications of science and technology Chaos 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 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. Edward Norton Lorenz ( May 23, 1917) was an American Mathematician and meteorologist, and a pioneer of Chaos theory.
• 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. Clifford A Pickover is an American author editor and columnist in the fields of Science, Mathematics, and Science fiction, and is employed at the
• Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984. Ilya Viscount Prigogine (Илья́ Рома́нович Приго́жин ( January 25, 1917 &ndash May 28, 2003) was a Russian
• 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 Pierre Ruelle (b August 20, 1935 Ghent, Belgium) is a Belgian-French Mathematical physicist.
• David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989. David Pierre Ruelle (b August 20, 1935 Ghent, Belgium) is a Belgian-French Mathematical physicist.
• Peter Smith, Explaining Chaos, Cambridge University Press, 1998.
• Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990. Ian Nicholas Stewart (born 1945) is a professor of Mathematics at University of Warwick, England and a widely known popular-science and science-fiction
• Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003. Steven H Strogatz (born August 13 1959 is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University.
• 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.
• Society for Chaos Theory in Psychology & Life Sciences
• 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.