Chaos theory is a field of study in mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, economics Economics is the social science that is concerned with the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)". Current and philosophy Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect The butterfly effect is a metaphor that encapsulates the concept of sensitive dependence on initial conditions in chaos theory; namely that small differences in the initial condition of a dynamical system may produce large variations in the long term behavior of the system. Although this may appear to be an esoteric and unusual behavior, it is. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.[1] This happens even though these systems are deterministic In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. Deterministic models thus produce the same output for a given starting condition, meaning that their future behaviour is fully determined by their initial conditions, with no random Randomness is a concept of non-order or non-coherence in a sequence of symbols or steps, such that there is no intelligible pattern or combination. Randomness has somewhat disparate meanings as used in several different fields. It also has common meanings which may have loose connections with some of those more definite meanings. The Oxford elements involved.[2] In other words, the deterministic nature of these systems does not make them predictable.[3] This behavior is known as deterministic chaos, or simply chaos Chaos typically means a state lacking order or predictability. In ancient Greece, it first meant the initial state of the universe, and, by extension, space, darkness, or an abyss (the antithetical, or possibly complementary, concept was cosmos), but later uses of the term by philosophers varied over time. In modern English, the word is used in.

Chaotic behavior can be observed in many natural systems, such as the weather.[4] Explanation of such behavior may be sought through analysis of a chaotic mathematical model A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed mathematical modelling . Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics,, or through analytical techniques such as recurrence plots and Poincaré maps.

Contents

Applications

Chaos theory is applied in many scientific disciplines: mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, programming Computer programming is the process of designing, writing, testing, debugging / troubleshooting, and maintaining the source code of computer programs. This source code is written in a programming language. The code may be a modification of an existing source or something completely new. The purpose of programming is to create a program that, microbiology Microbiology is the study of microorganisms, which are unicellular or cell-cluster microscopic organisms. This includes eukaryotes such as fungi and protists, and prokaryotes. Viruses, though not strictly classed as living organisms, are also studied. In short; microbiology refers to the study of life and organisms that are too small to be seen, biology Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy, computer science Computer science or computing science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe, and transform information. Computer science, economics Economics is the social science that is concerned with the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek οἰκονομία from οἶκος (oikos, "house") + νόμος (nomos, "custom" or "law"), hence "rules of the house(hold)". Current,[5][6][7] engineering Engineering is the discipline, art and profession of acquiring and applying technical, scientific, and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or invention,[8] finance Finance is the science of funds management. The general areas of finance are business finance, personal finance, and public finance. Finance includes saving money and often includes lending money. The field of finance deals with the concepts of time, money, and risk and how they are interrelated. It also deals with how money is spent and budgeted,[9][10] philosophy Philosophy is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language. It is distinguished from other ways of addressing fundamental questions by its critical, generally systematic approach and its reliance on rational argument. The word "philosophy" comes from the, physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, politics Politics , is a process by which groups of people make collective decisions. The term is generally applied to behavior within civil governments, but politics has been observed in other group interactions, including corporate, academic, and religious institutions. It consists of "social relations involving authority or power" and refers, population dynamics Population dynamics is the branch of life sciences that studies short- and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes. Population dynamics deals with the way populations are affected by birth and death rates, and by immigration and emigration, and, psychology Psychology is the scientific study of human or other animal mental functions and behaviors. In this field, a professional practitioner or researcher is called a psychologist. Psychologists are classified as social or behavioral scientists. Psychological research can be considered either basic or applied. Psychologists attempt to understand the, and robotics The word "beam" in BEAM robotics is an acronym for Biology, Electronics, Aesthetics, and Mechanics. This is a term that refers to a style of robotics that primarily uses simple analogue circuits, such as comparators, instead of a microprocessor in order to produce an unusually simple design that trades flexibility for robustness and.[11]

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources, and switches, lasers Light Amplification by Stimulated Emission of Radiation is a mechanism for emitting electromagnetic radiation, typically light or visible light, via the process of stimulated emission. The emitted laser light is (usually) a spatially coherent, narrow low-divergence beam, that can be manipulated with lenses. In laser technology, "coherent, oscillating Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes is used to be synonymous with "oscillation.& chemical reactions A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, often coming about only after the input of some type of energy, viz. heat, light or electricity. Classically, chemical reactions encompass, fluid dynamics In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications,, and mechanical and magneto-mechanical devices, as well as computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather,[4] the dynamics of satellites in the solar system The Solar System[a] consists of the Sun and those celestial objects bound to it by gravity, all of which were formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. Of the many objects that orbit the Sun, most of the mass is contained within eight relatively solitary planets[e] whose orbits are almost circular and, the time evolution of the magnetic field of celestial bodies, population growth in ecology Ecology is the scientific study of the distributions, abundance and relations of organisms and their interactions with the environment. Ecology includes the study of plant and animal populations, plant and animal communities and ecosystems. Ecosystems describe the web or network of relations among organisms at different scales of organization, the dynamics of the action potentials An action potential is a short-lasting event in which the electrical membrane potential of a cell rapidly rises and falls, following a stereotyped trajectory. Action potentials occur in several types of excitable cells, including neurons, muscle cells, and endocrine cells. In neurons, they play a central role in cell-to-cell communication. In in neurons A neuron (pronounced /ˈnjʊərɒn/ NOOR-on, also known as a neurone or nerve cell) is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core, and molecular vibrations A molecular vibration is excited when the molecule absorbs a quantum of energy, E, corresponding to the vibration's frequency, ν, according to the relation E=hν, where h is Planck's constant. A fundamental vibration is excited when one such quantum of energy is absorbed by the molecule in its ground state. When two quanta are absorbed the first. There is some controversy over the existence of chaotic dynamics in plate tectonics Plate tectonics is a scientific theory which describes the large scale motions of Earth's lithosphere. The theory builds on the older concepts of continental drift, developed during the first decades of the 20th century by Alfred Wegener, and seafloor spreading, developed in the 1960s and in economics.[12][13][14]

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.

Chaos theory is also currently being applied to medical studies of epilepsy Epilepsy (from the Ancient Greek ἐπιληψία — "to seize") is a common chronic neurological disorder characterized by recurrent unprovoked seizures. These seizures are transient signs and/or symptoms of abnormal, excessive or synchronous neuronal activity in the brain. About 50 million people worldwide have epilepsy, with almost 9, specifically to the prediction of seemingly random seizures by observing initial conditions.[15]

A related field of physics called quantum chaos Quantum chaos is a branch of physics which studies how chaotic classical systems can be shown to be limits of quantum-mechanical systems. The phenomena covered by quantum chaos so far are mainly related to wave theory. An alternative name, proposed by Sir Michael Berry, is quantum chaology. Michael Berry argues that there is no quantum chaos, in theory investigates the relationship between chaos and quantum mechanics Quantum mechanics , also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales. In advanced topics of QM, some of these. The correspondence principle In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers states that classical mechanics is a special case of quantum mechanics, the classical limit. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos.[16] Recently, another field, called relativistic chaos,[17] 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 1915. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a geometric property of space and time, or spacetime.

The motion of N stars in response to their self-gravity (the gravitational N-body problem The general n-body problem of celestial mechanics is an initial-value problem for ordinary differential equations. Given initial values for the positions and velocities of n particles with for all mutually distinct j and k , find the solution of the second order system) is generically chaotic.[18]

Chaotic dynamics

The map defined by x → 4 x (1 – x) and y → x + y if x + y < 1 (x + y – 1 otherwise) displays sensitivity to initial conditions. Here two series of x and y values diverge markedly over time from a tiny initial difference.

In common usage, "chaos Chaos typically means a state lacking order or predictability. In ancient Greece, it first meant the initial state of the universe, and, by extension, space, darkness, or an abyss (the antithetical, or possibly complementary, concept was cosmos), but later uses of the term by philosophers varied over time. In modern English, the word is used in" means "a state of disorder",[19] but the adjective "chaotic" is defined more precisely in chaos theory. Although there is no universally accepted mathematical definition of chaos, a commonly used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:[20]

  1. it must be sensitive to initial conditions;
  2. it must be topologically mixing In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc; and
  3. its periodic orbits In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the phase space. Understanding the properties of orbits must be dense In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point in X can be "well-approximated" by points in A in the sense that any point in X is either an element or a limit point of A.

The requirement for sensitive dependence on initial conditions implies that there is a set of initial conditions of positive measure which do not converge to a cycle of any length.

Sensitivity to initial conditions

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour. However, it has been shown that the last two properties in the list above actually imply sensitivity to initial conditions[21][22] and if attention is restricted to intervals In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers , the, the second property implies the other two[23] (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list[24]). It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) purely topological Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation conditions, which are therefore of greater interest to mathematicians.

Sensitivity to initial conditions is popularly known as the "butterfly effect The butterfly effect is a metaphor that encapsulates the concept of sensitive dependence on initial conditions in chaos theory; namely that small differences in the initial condition of a dynamical system may produce large variations in the long term behavior of the system. Although this may appear to be an esoteric and unusual behavior, it is", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science The American Association for the Advancement of Science is an international non-profit organization with the stated goals of promoting cooperation between scientists, defending scientific freedom, encouraging scientific responsibility, and supporting scientific education and science outreach for the betterment of all humanity. It is the world's 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 (even the evolution of simple discrete systems, such as cellular automata A cellular automaton is a discrete model studied in computability theory, mathematics, physics, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off". The grid can be in any finite number of dimensions. For each cell, a set of, can heavily depend on initial conditions, and Stephen Wolfram Stephen Wolfram is a British physicist, software developer, mathematician, author and businessman, known for his work in theoretical particle physics, cosmology, cellular automata, computational complexity theory, computer algebra and the Wolfram Alpha computational knowledge engine has investigated a cellular automaton with this property, termed by him rule 30).

A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This is most familiar in the case of weather, which is generally predictable only about a week ahead.[25]

The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two trajectories in phase space with initial separation diverge

where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

Topological mixing

The map defined by x → 4 x (1 – x) and y → x + y if x + y < 1 (x + y – 1 otherwise) also displays topological mixing. Here the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of points scattered across the space.

Topological mixing (or topological transitivity) 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. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behaviour: all points except 0 tend to infinity.

Density of periodic orbits

Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may not be chaotic. For example, an irrational rotation of the circle is topologically transitive, but does not have dense periodic orbits, and hence does not have sensitive dependence on initial conditions.[26] The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).[27]

Sharkovskii's theorem is the basis of the Li and Yorke[28] (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.

Strange attractors

The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.

Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behaviour is found only in a subset of phase space. The cases of most interest arise when the chaotic behaviour takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.

An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly.

Unlike fixed-point attractors and limit cycles, the attractors which arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both 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, and a fractal dimension can be calculated for them.

Minimum complexity of a chaotic system

Bifurcation diagram of the logistic map x → r x (1 – x). Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. However, the Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions. Finite dimensional linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be either nonlinear, or infinite-dimensional.

The Poincaré–Bendixson theorem states that a two dimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations with a total of seven terms on the right hand side, five of which are linear terms and two of which are quadratic (and therefore nonlinear). Another well-known chaotic attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is (quadratic) nonlinear. Sprott[29] found a three dimensional system with just five terms on the right hand side, and with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel[30][31] showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.

While the Poincaré–Bendixson theorem means that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behaviour.[citation needed] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional.[32] A theory of linear chaos is being developed in the functional analysis, a branch of mathematical analysis.

History

Barnsley fern created using the 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 the 1880s, 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.[33][34] 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.[35] 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.

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,[36] A. N. Kolmogorov,[37][38][39] M.L. Cartwright and J.E. Littlewood,[40] and Stephen Smale.[41] 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.[citation needed] 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. What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated 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.

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.[42] 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.

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.[43] Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot in general make long-term weather predictions. Weather is usually predictable only about a week ahead.[25]

The year before, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices.[44] 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.[45] 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).[46][47] 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.[48] 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.

Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol[49] and in 1958 by R.L. Ives.[50][51] However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.[52][53]

In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James A. 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, California), and the meteorologist Edward Lorenz.

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps.[54] 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.

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".[55]

Then in 1986 the New York Academy of Sciences co-organized with the National Institute of Mental Health and the Office of Naval Research the first important conference on Chaos in biology and medicine. There, Bernardo Huberman presented a mathematical model of the eye tracking disorder among schizophrenics.[56] This led to a renewal of physiology in the 1980s through the application of chaos theory, for example in the study of pathological cardiac cycles.

In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[57] 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 centered 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[58] 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.

The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public. At first the domain of work of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to Thomas 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 a shift, a thesis upheld by J. Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, topology, physics, population biology, biology, meteorology, astrophysics, information theory, etc.).

Distinguishing random from chaotic data

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.[59][60]

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.[59][61] 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.[62]

Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' state (e.g., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on phase space embedding methods.[63] Typically one chooses an embedding dimension, and investigates the propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small (less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the dimension too large – the method will work.

When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and permanent distortions. Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback system.[64] In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.[65]

Chaos in searching for roots of equations

In numerical analysis, the Newton-Raphson method of approximating the roots of a function can lead to chaotic iterations if the function has no real roots.[66]

Cultural references

Chaos theory has been mentioned in numerous movies and works of literature. Examples include the film Jurassic Park and Tom Stoppard's play Arcadia.

See also

Examples of chaotic systems
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References

  1. ^ Stephen H. Kellert, In the Wake of Chaos: Unpredictable Order in Dynamical Systems, University of Chicago Press, 1993, p 32, ISBN 0-226-42976-8.
  2. ^ Kellert, p. 56.
  3. ^ Kellert, p. 62.
  4. ^ a b Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background", Environmetrics, vol. 8, no. 5, pages 517–532.
  5. ^ Kyrtsou, C. and W. Labys, (2006). Evidence for chaotic dependence between US inflation and commodity prices, Journal of Macroeconomics, 28(1), pp. 256–266.
  6. ^ Kyrtsou, C. and W. Labys, (2007). Detecting positive feedback in multivariate time series: the case of metal prices and US inflation, Physica A, 377(1), pp. 227–229.
  7. ^ Kyrtsou, C., and Vorlow, C., (2005). Complex dynamics in macroeconomics: A novel approach, in New Trends in Macroeconomics, Diebolt, C., and Kyrtsou, C., (eds.), Springer Verlag.
  8. ^ Applying Chaos Theory to Embedded Applications
  9. ^ Hristu-Varsakelis, D., and Kyrtsou, C., (2008): Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns, Discrete Dynamics in Nature and Society, Volume 2008, Article ID 138547, 7 pages, doi:10.1155/2008/138547.
  10. ^ Kyrtsou, C. and M. Terraza, (2003). Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series, Computational Economics, 21, 257–276.
  11. ^ Metaculture.net, metalinks: Applied Chaos, 2007.
  12. ^ Apostolos Serletis and Periklis Gogas,Purchasing Power Parity Nonlinearity and Chaos, in: Applied Financial Economics, 10, 615–622, 2000.
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  66. ^ Strang, Gilbert, "A chaotic search for i," The College Mathematics Journal 22(1), January 1991, 3-12.

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How does By the waters of babylon by Stephen Vincent Benet relate to the Chaos theory and the butterfly effect?
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A. "By the Waters of Babylon" is a post-apocalyptic short story by Stephen Vincent Benet first published July 31, 1937, in The Saturday Evening Post as "The Place of the Gods". It was republished in 1943 in The Pocket Book of Science Fiction, John tells the tale of his exploration of the forbidden "Place of the Gods" in a world that has nearly forgotten the existence of 20th century civilization. John is the son of a priest of a tribe of hunters, heirs to a global catastrophe, whose curiosity takes him on a journey of discovery and search for truth about his civilized ancestors.
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