Eco No Physics is an Interdisciplinary Research Field

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Econophysics is an interdisciplinary research field, applying theories and methods originally developed

by physicists in order to solve problems in economics, usually those including uncertainty or stochastic

processes and nonlinear dynamics. Its application to the study of financial markets has also been termed

statistical finance referring to its roots in statistical physics.

Etymology

The correct name from an etymological viewpoint should be economophysics since economics is

a Greek word formed from oikos, meaning home and transformed into eco, and nomos, meaninglaw and transformed into nomy. Thus, economophysics would mean "home-law-physics", while

econophysics is not meaningful from a strictly etymological viewpoint. Nevertheless, the termeconophysics has gained currency, as opposed to economophysics, probably because it is shorter.

[edit] HistoryEconophysics was started in the mid 1990s by several physicists working in the subfield ofstatistical mechanics. They decided to tackle the complex problems posed by economics,

especially by financial markets. Unsatisfied with the traditional explanations of economists, theyapplied tools and methods from physics - first to try to match financial data sets, and then to

explain more general economic phenomena.

One driving force behind econophysics arising at this time was the availability of huge amounts

of financial data, starting in the 1980s. It became apparent that traditional methods of analysiswere insufficient - standard economic methods dealt with homogeneous agents and equilibrium,

while many of the more interesting phenomena in financial markets fundamentally depended onheterogeneous agents and far-from-equilibrium situations.

The term econophysics was coined by H. Eugene Stanley in the mid 1990s, to describe the

large number of papers written by physicists in the problems of (stock) markets, and firstappeared in a conference on statistical physics in Calcutta in 1995 and its following publications.

The inaugural meeting on Econophysics was later organised in Budapest.

Currently, the almost regular meeting series on the topic include: the Nikkei EconophysicsResearch workshop and symposium, APFA, ECONOPHYS-KOLKATA, ESHIA, Econophysics

Colloquium and Bonzenfreies Colloquium.

If "econophysics" is taken to denote the principle of applying statistical mechanics to economic

analysis, as opposed to a particular literature or network, priority of innovation is probably due toFarjoun and Machover (1983). Their bookLaws of Chaos: A Probabilistic Approach to Political

Economy proposes dissolving (their words) the transformation problem in Marx's politicaleconomy by re-conceptualising the relevant quantities as random variables.


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If, on the other side, "econophysics" is taken to denote the application of physics to economics,one can already consider the works ofLon Walras and Vilfredo Pareto as part of it. Indeed, as

shown by Ingrao and Israel, general equilibrium theory in economics is just based on thephysical concept ofmechanical equilibrium.

It should be noted that econophysics has nothing to do with the "physical quantities approach" toeconomics, advocated by Ian Steedman and others associated withNeo-Ricardianism.

[edit] Basic tools

Basic tools of econophysics areprobabilistic and statistical methods often taken from statistical

physics.

Physics models that have been applied in economics includepercolation models, chaotic modelsdeveloped to study cardiac arrest, and models with self-organizing criticality as well as other

models developed forearthquake prediction.[1]

Moreover, there have been attempts to use the

mathematical theory ofcomplexity and information theory, theories developed by manyscientists among whom Murray Gell-Mann and Claude E. Shannon, respectively.

Since economic phenomena are the result of the interaction among many heterogeneous agents,

there is an analogy with statistical mechanics, where many particles interact; but it must be takeninto account that the properties of human beings and particles significantly differ.

There are, however, various other tools from physics that have so far been used with mixed

success, such as fluid dynamics, classical mechanics and quantum mechanics (including so-called classical economy and quantum economy), and thepath integral formulation of statistical

mechanics.

There are also analogies between finance theory and diffusion theory. For instance, the Black-

Scholes equation foroption pricing is a diffusion-advection equation.

[edit] Impact on mainstream economics and finance

Papers on econophysics have been published primarily in journals devoted to physics and

statistical mechanics, rather than in leading economics journals. Mainstream economists havegenerally been unimpressed by this work

[2]. Some Heterodox economists, including Mauro

Gallegati, Steve Keen and Paul Ormerod, have shown more interest, but also criticized trends in

econophysics.

In contrast, econophysics is having some impact on the more applied field ofquantitative

finance, whose scope and aims significantly differ from those ofeconomic theory. Variouseconophysicists have introduced models for price fluctuations in financial markets or original

points of view on established models[3]

[4]


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Thermoeconomics is the name given to a type ofheterodox economic theory that attempts toexplicitly apply theprinciples ofthermodynamics to economics.

[1]The term "thermoeconomics"

was coined in 1962 by American engineerMyron Tribus,[2][3][4]

and developed by the statisticianand economistNicholas Georgescu-Roegen.[5] Thermoeconomics can be thought of as the

statistical physics ofeconomic value.[6]

Thermoeconomics is based on the proposition that the

role ofenergy inbiological evolution should be defined and understood through the second lawof thermodynamics but in terms of such economic criteria asproductivity, efficiency, andespecially the costs and benefits (or profitability) of the various mechanisms for capturing and

utilizing available energy to build biomass and do work.[7][8]

As a result, thermoeconomics areoften discussed in the field ofecological economics, which itself is related to the fields of

sustainability and sustainable development.

In Wealth, VirtualWealth and Debt(George Allen & Unwin 1926), Frederick Soddy turned hisattention to the role of energy in economic systems. He criticized the focus on monetary flows in

economics, arguing that real wealth was derived from the use of energy to transform materialsinto physical goods and services. Soddys economic writings were largely ignored in his time,

but would later be applied to the development ofbioeconomics and ecological economics in thelate 20th century.[9]

Thermoeconomists claim that human economic systems can be modeled as thermodynamic

systems. Then, based on this premise, they attempt to develop theoretical economic analogs ofthe first and second laws of thermodynamics.

[10]In addition, the thermodynamic quantity exergy,

i.e. measure of the useful work energy of a system, is the most important measure ofvalue. Inthermodynamics, thermal systems exchange heat, work, and ormass with their surroundings; in

this direction, relations between the energy associated with theproduction, distribution, andconsumption ofgoods and services can be determined.

[11]

Thermoeconomists argue that economic systems always involve matter, energy, entropy, andinformation.[12]

Moreover, the aim of many economic activities is to achieve a certain structure.

In this manner, thermoeconomics attempts to apply the theories in non-equilibriumthermodynamics, in which structure formations called dissipative structures form, and

information theory, in which information entropy is a central construct, to the modeling ofeconomic activities in which the natural flows of energy and materials function to create scarce

resources.[1]

In thermodynamic terminology, human economic activity may be described as adissipative system, which flourishes by transforming and exchanging resources, goods, and

services.[13]

These processes involve complex networks of flows of energy and materials.

Energy Accounting is a hypothetical system of distribution, proposed by TechnocracyIncorporated in the Technocracy Study Course, which would record the energy used to produce

and distribute goods and services consumed by citizens in a Technate.[14]

Scientists have speculated on different aspects of energy accounting for some time as to how it

might relate to alternatives in social systems.[15]

Many variations of energy accounting are in usenow, as this issue relates to current (price system) economics directly, as well as projected

models in possibleNon-market economics systems.[16]


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Exergy analysis is performed in the field ofindustrial ecology to use energy more efficiently.[17]

The term exergy, was coined by Zoran Rant in 1956, but the concept was developed by J.

Willard Gibbs. In recent decades, utilization of exergy has spread outside of physics andengineering to the fields of industrial ecology, ecological economics, systems ecology, and

energetics.

First law of thermodynamics

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In thermodynamics, the first law of thermodynamics is an expression of the more universalphysical law of the conservation of energy. Succinctly, the first law of thermodynamics states:

"E

nergy can be transformed (changed from one form to another), but it can neither be created nordestroyed."

The increase in the internal energy of a system is equal to the amount of energy added by

heating the system, minus the amount lost as a result of the work done by the system on its

surroundings.

Lawsofthermodynamics

Zeroth Law

First Law

Second Law

Third Law

Fundamental Relation

vde

Contents

[hide]

y 1 Descriptiony 2 Mathematical formulationy 3Types of thermodynamic processes


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y 4 See alsoy 5 Further readingy 6 External links

[edit] Description

The first law of thermodynamics basically states that a thermodynamic system can store or holdenergy and that this internal energy is conserved. Heat is a process by which energy is added to

a system from a high-temperature source, or lost to a low-temperature sink. In addition, energymay be lost by the system when it does mechanical workon its surroundings, or conversely, it

may gain energy as a result of work done on it by its surroundings. The first law states that thisenergy is conserved: The change in the internal energy is equal to the amount added by heating

minus the amount lost by doing work on the environment. The first law can be statedmathematically as:

where dUis a small increase in the internal energy of the system, Q is a small amount of heatadded to the system, and W is a small amount ofworkdone by thesystem.

Notice that a lot of textbooks (e.g., Greiner Neise Stocker) formulate the first law as:

The only difference here is that W is the workdone on thesystem. So, when the system (e.g.

gas) expands the work done on the system is PdVwhereas in the previous formulation of thefirst law, the work done by the gas while expanding isPdV. In any case, both give the same

result when written explicitly as:

The 's before the heat and work terms are used to indicate that they describe an increment ofenergy which is to be interpreted somewhat differently than the dUincrement of internal energy.

Work and heat areprocesses which add or subtract energy, while the internal energy Uis aparticularform of energy associated with the system. Thus the term "heat energy" for Q means

"that amount of energy added as the result of heating" rather than referring to a particular form of

energy. Likewise, the term "work energy" for w means "that amount of energy lost as the resultof work". Internal energy is the property of the system whereas work done or heat supplied isnot. The most significant result of this distinction is the fact that one can clearly state the amount

of internal energy possessed by a thermodynamic system, but one cannot tell how much energyhas flowed into or out of the system as a result of its being heated or cooled, nor as the result of

work being performed on or by the system. The first explicit statement of the first law ofthermodynamics was given by Rudolf Clausius in 1850: "There is a state function E, called


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energy, whose differential equals the work exchanged with the surroundings during anadiabatic process."

[edit] Mathematical formulation

The mathematical statement of the first law is given by:

where dUis the infinitesimal increase in the internal energy of the system, Q is the infinitesimal

amount of heat added to the system, and w is the infinitesimal amount of work done by thesystem. The infinitesimal heat and work are denoted by rather than d because, in mathematical

terms, they are not exact differentials. In other words, they do not describe the state of anysystem. The integral of an inexact differential depends upon the particular "path" taken through

the space of thermodynamic parameters while the integral of an exact differential depends onlyupon the initial and final states. If the initial and final states are the same, then the integral of an

inexact differential may or may not be zero, but the integral of an exact differential will alwaysbe zero. The path taken by a thermodynamic system through state space is known as a

thermodynamic process.

An expression of the first law can be written in terms of exact differentials by realizing that the

work that a system does is, in case of a reversible process, equal to its pressure times theinfinitesimal change in its volume. In other words w =PdVwherePispressure and Vis

volume. Also, for a reversible process, the total amount of heat added to a system can beexpressed as Q = TdSwhere T is temperature and Sis entropy. Therefore, for a reversible

process, :

Since U, S and V are thermodynamic functions of state, the above relation holds also for non-reversible changes. The above equation is known as the fundamental thermodynamic relation.

In the case where the number of particles in the system is not necessarily constant and may be ofdifferent types, the first law is written:

where dNi is the (small) number of type-i particles added to the system, and i is the amount ofenergy added to the system when one type-i particle is added, where the energy of that particle is

such that the volume and entropy of the system remains unchanged. i is known as the chemicalpotential of the type-i particles in the system. The statement of the first law, using exact

differentials is now:


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If the system has more external variables than just the volume that can change, the fundamentalthermodynamic relation generalizes to:

Here the Xi are the generalized forces corresponding to the external variablesxi.

A useful idea from mechanics is that the energy gained by a particle is equal to the force appliedto the particle multiplied by the displacement of the particle while that force is applied. Now

consider the first law without the heating term: dU= PdV. The pressureP can be viewed as aforce (and in fact has units of force per unit area) while dV is the displacement (with units of

distance times area). We may say, with respect to this work term, that a pressure difference

forces a transfer of volume, and that the product of the two (work) is the amount of energytransferred as a result of the process.

It is useful to view the TdS term in the same light: With respect to this heat term, a temperaturedifference forces a transfer of entropy, and the product of the two (heat) is the amount of energy

transferred as a result of the process. Here, the temperature is known as a "generalized" force(rather than an actual mechanical force) and the entropy is a generalized displacement.

Similarly, a difference in chemical potential between groups of particles in the system forces a

trasfer of particles, and the corresponding product is the amount of energy transferred as a resultof the process. For example, consider a system consisting of two phases: liquid water and water

vapor. There is a generalized "force" of evaporation which drives water molecules out of theliquid. There is a generalized "force" of condensation which drives vapor molecules out of thevapor. Only when these two "forces" (or chemical potentials) are equal will there be equilibrium,

and the net transfer will be zero.

The two thermodynamic parameters which form a generalized force-displacement pair aretermed "conjugate variables". The two most familiar pairs are, of course, pressure-volume, and

temperature-entropy.

[edit] Types of thermodynamic processes

Paths through the space of thermodynamic variables are often specified by holding certain

thermodynamic variables constant. It is useful to group these processes into pairs, in which eachvariable held constant is one member of a conjugate pair.

The pressure-volume conjugate pair is concerned with the transfer of mechanical or dynamic

energy as the result of work.


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y An isobaricprocess occurs at constant pressure. An example would be to have a movable pistonin a cylinder, so that the pressure inside the cylinder is always at atmospheric pressure, although

it is isolated from the atmosphere. In other words, the system is dynamicallyconnected, by a

movable boundary, to a constant-pressure reservoir. Like a balloon contracting when the gas

inside cools.

y An isochoric (orisovolumetric)process is one in which the volume is held constant, meaningthat the work done by the system will be zero. It follows that, for the simple system of two

dimensions, any heat energy transferred to the system externally will be absorbed as internal

energy. An isochoric process is also known as an isometric process. An example would be to

place a closed tin can containing only air into a fire. To a first approximation, the can will not

expand, and the only change will be that the gas gains internal energy, as evidenced by its

increase in temperature and pressure. Mathematically, Q = dU. We may say that the system is

dynamicallyinsulated, by a rigid boundary, from the environment

The temperature-entropy conjugate pair is concerned with the transfer of thermal energy as theresult of heating.

y An isothermal process occurs at a constant temperature. An example would be to have asystem immersed in a large constant-temperature bath. Any work energy performed by the

system will be lost to the bath, but its temperature will remain constant. In other words, the

system is thermallyconnected, by a thermally conductive boundary to a constant-temperature

reservoir.

y An isentropicprocess occurs at a constant entropy. For a reversible process this is identical toan adiabatic process (see below). If a system has an entropy which has not yet reached its

maximum equilibrium value, a process of cooling may be required to maintain that value of

entropy.

y An adiabaticprocess is a process in which there is no energy added or subtracted from thesystem by heating or cooling. For a reversible process, this is identical to an isentropic process.

We may say that the system is thermallyinsulated from its environment and that its boundary

is a thermal insulator. If a system has an entropy which has not yet reached its maximum

equilibrium value, the entropy will increase even though the system is thermally insulated.

The above have all implicitly assumed that the boundaries are also impermeable to particles. Wemay assume boundaries that are both rigid and thermally insulating, but are permeable to one or

more types of particle. Similar considerations then hold for the (chemical potential)-(particlenumber) conjugate pairs.

nphysics, the law ofconservation ofenergy states that the total amount ofenergy in an isolatedsystem remains constant. A consequence of this law is that energy cannot be created or

destroyed. The only thing that can happen with energy in an isolated system is that it can changeform, that is to say for instance kinetic energy can become thermal energy. Because energy is

associated with mass in the Einstein's theory of relativity, the conservation of energy also impliesthe conservation of mass in isolated systems (that is, the mass of a system cannot change, so long

as energy is not permitted to enter or leave the system).


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Another consequence of this law is thatperpetual motion machines can only work if they deliverno energy to their surroundings, and also that devices that produce more energy than is put into

them without losing mass (and thus eventually disappearing), are impossible.

Chaos theoryFrom Wikipedia, the free encyclopedia


For other uses, see Chaos Theory (disambiguation).

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

In mathematics, chaos theory describes the behaviour of certain dynamical systems that is,

systems whose states evolve with time that may exhibit dynamics that are highly sensitive toinitial conditions (popularly referred to as thebutterfly effect). As a result of this sensitivity,which manifests itself as an exponential growth of perturbations in the initial conditions, the

behavior of chaotic systems appears to be random. This happens even though these systems aredeterministic, meaning that their future dynamics are fully defined by their initial conditions,

with no random elements involved. This behavior is known as deterministic chaos, or simplychaos.

Chaotic behavior 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 ofphysics that are relevant for the natural system.

Overview

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical

circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices. Observations of chaotic behavior in nature include the dynamics of satellites

in the solar system, the time evolution of the magnetic field of celestial bodies,populationgrowth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations.


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Everyday examples of chaotic systems include weather and climate.[1]

There is some controversyover the existence of chaotic dynamics in theplate tectonics and in economics.

[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 thelaws ofquantum mechanics. Recently, another field, called relativistic chaos,[5]

has emerged todescribe systems that follow the laws ofgeneral relativity.

This article tries to describe limits on the degree of disorder that computers can model with

simple rules that have complex results. For example, the Lorenz system pictured is chaotic, buthas a clearly defined structure.Bounded chaos is a useful term for describing models of disorder.

[edit] History

Fractalfern created using chaos game. Natural forms (ferns, clouds, mountains, etc.) may be

recreated through an Iterated function system (IFS).

The first discoverer of chaos was Henri Poincar. In 1890, while studying the three-body

problem, he found that there can be orbits which are nonperiodic, and yet not forever increasingnor approaching a fixed point.[6] In 1898 Jacques Hadamard published an influential study of the

chaotic motion of a free particle gliding frictionlessly on a surface of constant negativecurvature.

[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.

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 carriedout by G.D. Birkhoff,

[8]A. N. Kolmogorov,

[9][10][11]M.L. Cartwright and J.E. Littlewood,

[12]and

Stephen Smale.[13]

Except for Smale, these studies were all directly inspired by physics: the

three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case ofKolmogorov, and radio engineering in the case of Cartwright and Littlewood.

[citation needed]

Although chaotic planetary motion had not been observed, experimentalists had encounteredturbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a

theory to explain what they were seeing.


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Despite initial insights in the first half of the twentieth century, chaos theory became formalizedas 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 observedbehaviour 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. Oneof the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting

models.

Turbulence in the tip vortex from an aeroplane wing. Studies of the critical point beyond which asystem 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 andFloris Takens later predicted, against Landau, that fluid turbulence could develop through a

strange attractor, a main concept of chaos theory.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about

accidentally through his work on weather prediction in 1961.[14] Lorenz was using a simple

digital computer, a Royal McBeeLGP-30, to run his weather simulation. He wanted to see asequence 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 ofhis 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 computerworked 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 timewould 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).


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The year before, Benot Mandelbrot found recurring patterns at every scale in data on cottonprices.

[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 aconstant thus errors were inevitable and must be planned for by incorporating redundancy.[17]

Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur,

e.g., in a stock's prices after bad news, thus challenging normal distribution theory in statistics,aka Bell Curve) and the "Joseph effect" (in which persistence of a value can occur for a while,yet suddenly change afterwards).

[18][19]In 1967, he published "How long is the coast of Britain?

Statistical self-similarity and fractional dimension," showing that a coastline's length varies withthe scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an

infinitesimally small measuring device.[20]

Arguing that a ball of twine appears to be a pointwhen 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 arerelative 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 Sierpiski gasket). In 1975 Mandelbrot publishedT

heFractal 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 vander 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 computeronNovember 27, 1961. The chaos exhibited by an analog computer is a real phenomenon, in

contrast with those that digital computers calculate, which has a different kind of limit onprecision. Ueda's supervising professor, Hayashi, did not believe in chaos, and thus he prohibited

Ueda from publishing his findings until 1970.[24]

In December 1977 theNew 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 FarmerandNorman 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 meteorologistEdward Lorenz.

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universalityfor a Class of Nonlinear Transformations", where he described logistic maps.

[25]Feigenbaum had

applied fractal geometry to the study of natural forms such as coastlines. Feigenbaum notablydiscovered 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 thebifurcation cascade that leads to chaos and

turbulence in convectiveRayleighBenard systems. He was awarded the Wolf Prize in Physicsin 1986 along with Mitchell J. Feigenbaum "for his brilliant experimental demonstration of the

transition to turbulence and chaos in dynamical systems".[26]


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Then in 1986 the New York Academy of Sciences co-organized with theNational Institute ofMental Health and the Office of Naval Research the first important conference on Chaos in

biology and medicine. Bernardo Huberman thereby presented a mathematical model of the eyetracking disorderamong schizophrenics.[27] Chaos theory thereafter renewedphysiology in the

1980s, for example in the study of pathological cardiac cycles.

In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper inPhysical ReviewLetters

[28]describing for the first time self-organized criticality (SOC), considered to be one of

the mechanisms by which complexity arises in nature. Alongside largely lab-based approachessuch as the BakTangWiesenfeld sandpile, many other investigations have centered around

large-scale natural or social systems that are known (or suspected) to display scale-invariantbehaviour. Although these approaches were not always welcomed (at least initially) by

specialists in the subjects examined, SOC has nevertheless become established as a strongcandidate for explaining a number of natural phenomena, including: earthquakes (which, long

before SOC was discovered, were known as a source ofscale-invariant behaviour such as theGutenbergRichter law describing the statistical distribution of earthquake sizes, and the Omori

law

[29]

describing the frequency of aftershocks); solar flares; fluctuations in economic systemssuch as financial markets (references to SOC are common in econophysics); landscape

formation; forest fires; landslides; epidemics; andbiological evolution (where SOC has beeninvoked, for example, as the dynamical mechanism behind the theory of"punctuated equilibria"

put forward byNiles Eldredge and Stephen Jay Gould). Worryingly, given the implications of ascale-free distribution of event sizes, some researchers have suggested that another phenomenon

that should be considered an example of SOC is the occurrence ofwars. 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 analysisto determine the existence and/or characteristics of natural scaling laws.

The same year, James Gleickpublished Chaos: Makinga New Science, which became a best-

seller and introduced general principles of chaos theory as well as its history to the broad public.At first the domains of work of a few, isolated individuals, chaos theory progressively emerged

as a transdisciplinary and institutional discipline, mainly under the name ofnonlinear systemsanalysis. Alluding to Thomas Kuhn's concept of aparadigm shift exposed in The Structure of

Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimedthat this new theory was an example of such as 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.).

[edit] Chaoticdynamics

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


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Assign z to z minus the conjugate ofz, plus the original value of the pixel for each pixel, thencount how many cycles it took when the absolute value ofzexceeds two; inversion (borders are

inner set), so that you can see that it threatens to fail that third condition, even if it meetscondition two.

1. it must be sensitive to initial conditions,2. it must be topologically mixing, and3. itsperiodic orbits must be dense.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closelyapproximated 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 ofthe title of a paper given by Edward Lorenz in 1972 to the American Association for the

Advancement of Science in Washington, D.C. entitledPredictability: Does the Flap ofaButterflys Wings in Brazil set offaTornado 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.

This article or section may be written in a style that is too abstract to be readilyunderstandable by general audiences.

Please improve it by defining jargon orbuzzwords, and by adding examples.

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 thephase

space of the system. For example, consider the simple dynamical system produced by repeatedlydoubling an initial value (defined by iterating the mapping on the real line that mapsx to 2x).

This system has sensitive dependence on initial conditions everywhere, since any pair of nearbypoints 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 byadding the point at infinity and viewing the result as a circle, the system no longer is sensitive toinitial conditions.[citation needed] For this reason, in defining chaos, attention is normally restricted to

systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.[citation

needed]

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


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between zero and 2. Define a mapping that takes any point (x,y) to (2x,y + a), where a is anynumber such that a/2 is irrational. Because of the doubling in the first coordinate, the mapping

exhibits sensitive dependence on initial conditions. However, because of the irrational rotation inthe second coordinate, there are no periodic orbits, and hence the mapping is not chaotic

according to the definition above.

Topologically mixingmeans that the system will evolve over time so that any given region oropen set of its phase space will eventually overlap with any other given region. Here, "mixing" is

really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is anexample of a chaotic system.

Linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to

be nonlinear. Also, by the PoincarBendixson theorem, a continuous dynamical system on theplane 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 chaoticbehaviour. However, a discrete dynamical system (such as the logistic map) can exhibit chaotic

behaviour in a one-dimensional or two-dimensional phase space.

[edit] Attractors

Some dynamical systems are chaotic everywhere (see e.g. Anosov diffeomorphisms) but in many

cases chaotic behaviour is found only in a subset of phase space. The cases of most interest arisewhen 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 thebasin of attraction of theattractor, 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.

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 theposition of apendulumagainst its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will


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be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called anorbit. Our pendulum has an infinite number of such orbits, forming apencil of nested ellipses

about the origin.

[edit] Strange attractors

While most of the motion types mentioned above give rise to very simple attractors, such aspoints and circle-like curves called limit cycles, chaotic motion gives rise to what are known as

strange attractors, attractors that can have great detail and complexity. For instance, a simplethree-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor.

The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probablybecause 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 isthe Rssler map, which experiences period-two doubling route to chaos, like the logistic map.

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and

in some discrete systems (such as the Hnon map). Other discrete dynamical systems have arepelling 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 Juliasets typically have a fractal structure.

The Poincar-Bendixson theorem shows that a strange attractor can only arise in a continuousdynamical system if it has three or more dimensions. However, no such restriction applies to

discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

The initial conditions of three or more bodies interacting through gravitational attraction (see then-body problem) can be arranged to produce chaotic motion.

[edit] Minimum complexity of a chaoticsystem

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


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Simple systems can also produce chaos without relying on differential equations. An example isthe logistic map, which is a difference equation (recurrence relation) that describes population

growth over time. Another example is the Ricker model of population dynamics.

Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on

initial conditions. Stephen Wolfram has investigated a cellular automaton with this property,termed by him rule 30.

A minimal model for conservative (reversible) chaotic behavior is provided by Arnold's cat map.

[edit] Mathematical theory

Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional

system which exhibits a regular cycle of period three will also display regular cycles of everyother length as well as completely chaotic orbits.

Mathematicians have devised many additional ways to make quantitative statements aboutchaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents,recurrence plots, Poincar maps,bifurcation diagrams, and transfer operator.

[edit] Distinguishing random from chaoticdata

It can be difficult to tell from data whether a physical or other observed process is random orchaotic, 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 timeseries, even if mostly deterministic, will contain some randomness.

[31]

All methods for distinguishing deterministic and stochastic processes rely on the fact that adeterministic 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; and3. compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the timeevolution 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 closeststates 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 onechooses 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 thedimension to obtain a deterministic looking error, then you are done. Though it may sound


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simple it is not really. One complication is that as the dimension increases the search for a nearbystate 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 theembedding 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 inherentnon-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.

[35]In

presence of interactions between nonlinear deterministic components and noise the resultingnonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not

able to capture.[36]

[edit] Applications

Chaos theory is applied in many scientific disciplines: mathematics,biology, computer science,

economics[37][38][39]

,engineering, finance[40][41]

,philosophy,physics,politics,populationdynamics,psychology, and robotics.

[42]

One of the most successful applications of chaos theory has been in ecology, where dynamicalsystems 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 ofepilepsy, specifically to the

prediction of seemingly random seizures by observing initial conditions.[43]

Background

Viewing an atom as a complex system in itself, and magnifying the interactional effects ofsub-atomic particles and waves to reflect the interactions of different elements making up a complex

system, such as an organization, assists us in seeing parallels between quantum physics (namelychaos theory) and organizational relationships. What must be pointed out, however, is that these

"parallels" between organizations and the sub-atomic particles exist largely in terms of analogy(metaphorically) between two very different domains of activity; the interactional effects of sub-

atomic particles, in quantum mechanics, are expressed in terms of math; bringly these theoriesinto the domain of human activity can be seen as problematical. Although these parallels are

easily witnessed in regard to complex organizational systems, it is difficult to see evidence ofirrational quantum-effects in everyday life. If you roll a ball forward, it rolls forward in the

general direction intended. As a whole,Newtonian principles of interaction stand solidly withinthe bounds of macrophysics. But at the sub-atomic level, things do not act as expected. "At the

subatomic level, the objectivity found in classical physics is replaced by quantum subjectivity."(Shelton, 2003) The introduction of chaos theory brings the principles of quantum physics to the


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pragmatic world. These complex systems have a rather random appearance and, until recently,have been labeled and discarded as chaotic and unintelligible. With the advent of computer

systems and powerful processors, it has become easier to map chaotic behaviorand findinteresting underpinnings of order. The newly discovered underlying order to chaos sparked new

interest and inspired more research in the field of chaos theory. The recent focus of most of the

research on chaos theory is primarily rooted in these underlying patterns found in an otherwisechaotic environment, more specifically, concepts such as self-organization, bifurcation, and self-similarity.

[edit] Elements of organization

[edit] Self-organization

Self-organization, as opposed to natural or social selection, is a dynamic change within theorganization where system changes are made by recalculating, re-inventing and modifying its

structure in order to adapt, survive, grow, and develop. Self-organization is the result of re-

invention and creative adaptation due to the introduction of, or being in a constant state of,perturbed equilibrium. One example of an organization which exists in a constant state ofperturbation is that of the learning organization, which is "one that allows self-organization,

rather than attempting to control the bifurcation through planned change." (Dooley, 1995) Being"off-balance" lends itself to regrouping and re-evaluating the systems present state in order to

make needed adjustments and regain control and equilibrium. By understanding and introducingthe element of punctuated equilibrium (chaos) while facilitating networks for growth, an

organization can change gears from "cruise" to "turbo" in regard to speed and intensity oforganizational change. While maintaining an equilibrial state seems to be an intuitively rational

method for enabling an organization to gain a sense of consistency and solidarity, existing on theedge of a chaotic state remains the most beneficial environment for systems to flourish develop

and grow.

For instance, two competing organizations that differ in regard to their levels ofhomeostasis willnot be in competition for long. Generally speaking, the organization with the less-stable structure

will come out ahead while the constant stability of the latter will eventually lead to its owndemise. Although quite similar, small differences in homeostasis levels are enough to make a

tremendous difference in future outcomes for each organization. The notion of similarity inorigin vs. dissimilar results comes to fruition with the emergence of bifurcation.

[edit] Bifurcation

The concept of bifurcation cannot be explained without discussion of the term frequently labeled"sensitivity to initial conditions." Sensitivity to initial conditions refers to the high level ofimportance of primary conditions from which the future path and direction of a system stems.

This sensitivity to initial conditions is commonly referred to as the "ButterflyEffect," in which abutterfly flaps its tiny wings in one end of the world which results in a typhoon or hurricane

somewhere else on the globe. While this is an entertaining notion, sensitivity to initial conditionsremains in reality a very abstract concept without the presence of bifurcation, which is

mathematically labeled as the actual splitting point of two near-identical entities which, due to


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the sensitivity of initial conditions, tend to take two very distinct paths and result in two totallydifferent geographically or even evolutionary places.

Imagine dropping two identical coins from your fingertips off a 25-story balcony at the same

time. Unless they are glued together, they will each take a different path towards the ground.E

ven though the force ofgravity determines their general direction and speed, a host ofuncontrollable variables such as wind and dust particles affect each coin independently. Theinfinitesimal and perhaps unidentifiable difference in starting conditions exponentially amplifies

the effects of all other variables encountered which then feed back and add even more variationto the system resulting in very different paths taken to the ground. The moment the two coins

split paths is known as the bifurcation point. The importance of this point lies in its implicationof change and new direction.

[edit] Self-similarity

[edit] Applications and pitfalls

The primary goal of an organizational development (OD) consultant is to initiate, facilitate, andsupport successful change in an organization. Using chaos theory as the sole model for change

may be far too risky for any stakeholderbuy-in. The concept of uncertainty on which chaostheory relies is not an appealing motive for change compared to many alternative "safer" models

of organizational change which entail less risk. By careful planning and management of disordera successful intervention is possible, but only with a truly dedicated arsenal of talented and

creative resources. By permitting or actively forcing an organization to enter a chaotic state,change becomes inevitable and bifurcation imminent; but the question remains, "Will the new

direction be the one intended?" In order to account for the direction of the new thrust, mostplanning attention should be focused on attractors instead of the initiation of disorder.

Although chaos eventually gives way to self-organization, how can we control the duration,

intensity, and shape of its outcome? It seems that punctuating equilibrium and instilling disorderin an organization is risky business. Throwing an organization off balance could possibly send it

in a downward spiral towards dissemination by ultimately compromising the structural integrity(i.e. identity) of the system to the point of no return. The only way to reap the benefits of chaos

theory in OD while maintaining a sense of security is to adjust the organization towards a state ofexistence which lies on the edge of chaos.

By existing on the edge of chaos, organizations are forced to find new, creative ways to competeand stay ahead. Good examples of such learning organizations are found throughout the field of

technology as well as the airline industry, namely organizations such as Southwest Airlines,which used re-invention not just for survival, but also to prosper in an otherwise dismal market.

In contrast, there are organizations which, due to extended periods of equilibrium, findthemselves struggling for survival. Telephone companies, for instance, were once solid and static

entities that dominated the communication market. While the rest of the world was developingnew communication technology, telephone companies did not creatively grow at the same rate.

The result is an organization that is battling to stay alive unless they embrace the element of


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chaos due to crisis, and allow creative adaptability to function freely so that self-organization andre-invention can occur.

While organizations existing on the edge of chaos are known to be the most creative and

adaptive of organizations, how do their members feel about constant evolution and re-invention?

Is it possible to identify with, and stay loyal to, an organization that constantly changes shape?The short answer is yes. As long as the organization does not change its core essence, itsidentifiable, shared purpose, its members will still experience the organization as a developing

system that changes shape but retains the same familiar face.

Perhaps the safest way to use chaos theory in OD is not in the instigation of organizationalchange, but in the use of its principles in dealing with issues that arise within the organization.

By embracing organizational phenomena previously seen as dysfunctional, such as interpersonalconflict, and using it as a source for transformational change by applying principles found in

chaos theory (Shelton, 2003), an organization can make "lemonade out of lemons" and becomemore responsive to change agents while continuously moving ahead and growing from the inside

out without the fear of complete chaos.

Butterfly effect



For other uses, see Butterfly effect (disambiguation).

Point attractors in 2D phase space.


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The butterfly effect is a phrase that encapsulates the more technical notion ofsensitivedependence on initial conditions in chaos theory. Small variations of the initial condition of a

dynamical system may produce large variations in the long term behavior of the system. This issometimes presented as esoteric behavior, but can be exhibited by very simple systems: for

example, aball placed at the crest of a hill might roll into any of several valleys depending on

slight differences in initial position.

Contents

[hide]

y 1Theoryy 2 Origin of the concept and the termy 3 Illustrationy 4 Mathematical definitiony 5Appearances in popular culturey 6 See alsoy 7 Referencesy 8 Further readingy 9 External links

[edit] Theory

The phrase refers to the idea that abutterfly's wings might create tiny changes in the atmosphere

that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence

of a tornado in a certain location. The flapping wing represents a small change in the initialcondition of the system, which causes a chain of events leading to large-scale alterations ofevents. Had the butterfly not flapped its wings, the trajectory of the system might have been

vastly different. While the butterfly does not cause the tornado, the flap of its wings is anessential part of the initial conditions resulting in a tornado.

Recurrence, the approximate return of a system towards its initial conditions, together withsensitive dependence on initial conditions are the two main ingredients for chaotic motion. They

have the practical consequence of making complex systems, such as the weather, difficult topredict past a certain time range (approximately a week in the case of weather).

[edit] Origin of theconcept and the term

The term "butterfly effect" itself is related to the work ofEdward Lorenz, and is based in ChaosTheory and sensitive dependence on initial conditions, first described in the literature by Jacques

Hadamard in 1890[1] and popularized by Pierre Duhem's 1906 book. The idea that one butterflycould have a far-reaching ripple effect on subsequent events seems first to have appeared in a

1952 short story by Ray Bradbury about time travel (see Literature and print here) althoughLorenz made the term popular. In 1961, Lorenz was using a numerical computer model to rerun


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a weather prediction, when, as a shortcut on a number in the sequence, he entered the decimal.506 instead of entering the full .506127 the computer would hold. The result was a completely

different weather scenario.[2]

Lorenz published his findings in a 1963 paper for theNew YorkAcademy of Sciences noting that "One meteorologist remarked that if the theory were correct,

one flap of a seagull's wings could change the course of weather forever." Later speeches and

papers by Lorenz used the more poeticbutterfly. According to Lorenz, upon failing to provide atitle for a talk he was to present at the 139th meeting of the American Association for theAdvancement of Science in 1972, Philip Merilees concoctedDoes the flap ofa butterflys wings

inBrazilset offa tornado in Texas as a title.

Although a butterfly flapping its wings has remained constant in the expression of this concept,the location of the butterfly, the consequences, and the location of the consequences have varied

widely.[3]

[edit] Illustration

Thebutterflyeffect in the Lorenz attractor

time 0 t 30(larger) z coordinate (larger)

These figures show two segments of the three-dimensional evolution of two trajectories (one in blue,the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that

differ only by 10-5

in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the

small difference between the z coordinate of the blue and yellow trajectories, but for t> 23 the

difference is as large as the value of the trajectory. The final position of the cones indicates that the two

trajectories are no longer coincident at t=30.


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AJava animation of the Lorenz attractor shows the continuous evolution.

[edit] Mathematical definition

A dynamical system with evolution mapftdisplays sensitive dependence on initial conditions ifpoints arbitrarily close become separate with increasing t. IfMis the state space for the mapf

t,

thenftdisplays sensitive dependence to initial conditions if there is a >0 such that for every

pointxM and any neighborhoodNcontainingx there exist a pointy from that neighborhoodN

and a time such that the distance

The definition does not require that all points from a neighborhood separate from the base pointx.

[edit] Appearancesin popular culture

Main article: Butterfly effect in popularculture

The term is sometimes used in popular media dealing with the idea oftime travel, usuallyinaccurately. Most time travel depictions simply fail to address butterfly effects. According to

the actual theory, if history could be "changed" at all (so that one is not invoking something liketheNovikov self-consistency principle which would ensure a fixed self-consistent timeline), the

mere presence of the time travelers in the past would be enough to change short-term events(such as the weather) and would also have an unpredictable impact on the distant future.

Therefore, no one who travels into the past could ever return to the same version of reality he orshe had come from and could have therefore not been able to travel back in time in the first

place, which would create a phenomenon known as a time paradox.

[edit] See also

Domino effect



This article is aboutchain reactions. Forthepolitical theory, see Domino Theory.


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Dominoes waiting to fall

Domino effect events

(Top) Dominoes are standing. (Bottom) Dominoes are in motion.

The domino effect is a chain reaction that occurs when a small change causes a similar changenearby, which then will cause another similar change, and so on in linearsequence. The term is

best known as a mechanical effect, and is used as an analogy to a falling row ofdominoes. Ittypically refers to a linked sequence of events where the time between successive events is

relatively small. It can be used literally (an observed series of actual collisions) or metaphorically(complex systems such as global finance, or in politics, where linkage is only a hypothesis).


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[edit] Demonstrations of theeffect

The classic demonstration involves setting up a chain of dominoes stood on end, and toppling thefirst domino. That domino topples the one next to it, and so on. In theory, however long the chain

the dominoes will still fall. This is because the energy required to topple each domino is less than

the energy transferred by each impact, so the chain is self-sustaining. Energy is stored by settingeach domino in the metastable upright position, and that energy is what keeps the chain toppling.

There are many demonstrations of the effect involving more complex systems. Currently popular

is the Diet Coke and Mentos film on YouTube, where a chain ofDiet Coke and Mentos eruptionsis demonstrated. Although apparently complex and lacking the purity of a simple chain, this

involves a simple physical linkage whereby each eruption triggers the next.

chain reaction is a sequence ofreactions where a reactive product or by-product causesadditional reactions to take place. In a chain reaction,positive feedbackleads to a self-

amplifying chain of events. Examples of chain reactions include:

y The neutron-fission chain reaction: a neutron plus a fissionableatom causes a fissionresulting in a larger number of neutrons than was consumed in the initial reaction.

y Every step ofH2 + Cl2 chain reaction consumes one molecule of H2 or Cl2, one freeradical H or Cl producing one HCl molecule and another free radical.

y Electron avalanche process: Collisions offree electrons in a strong electric field releasing"new" electrons to undergo the same process in successive cycles.

y A cascading failure, a failure in a system of interconnected parts, for example apowertransmission grid, where the service provided depends on the operation of a precedingpart, and the failure of a preceding part can trigger the failure of successive parts.

y Polymerase chain reaction, a technique used in molecular biology to amplify (make manycopies of) a piece ofDNA by in vitroenzymaticreplication using a DNA polymerase.

Complexity and modeling

A way of modelling Complex Adaptive System


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One ofHayek's main contributions to early complexity theory is his distinction between thehuman capacity to predict the behaviour of simple systems and its capacity to predict the

behaviour of complex systems through modeling. He believed that economics and the sciences ofcomplex phenomena in general, which in his view includedbiology,psychology, and so on,

could not be modeled after the sciences that deal with essentially simple phenomena like

physics.

[5]

Hayek would notably explain that complex phenomena, through modeling, can onlyallow pattern predictions, compared with the precise predictions that can be made out of non-complex phenomena.

[6]

[edit] Complexity and chaos theory

Complexity theory is rooted in Chaos theory, which in turn has its origins more than a century

ago in the work of the French mathematician Henri Poincar. Chaos is sometimes viewed asextremely complicated information, rather than as an absence of order.

[7]The point is that chaos

remains deterministic. With perfect knowledge of the initial conditions and of the context of anaction, the course of this action can be predicted in chaos theory. As argued by Prigogine,[8]

Complexity is non-deterministic, and gives no way whatsoever to predict the future. Theemergence of complexity theory shows a domain between deterministic order and randomness

which is complex.[9] This is referred as the 'edge of chaos'.[10]

A plot of the Lorenz attractor

When one analyses complex systems, sensitivity to initial conditions, for example, is not an issueas important as within the chaos theory in which it prevails. As stated by Colander,

[11]the study

of complexity is the opposite of the study of chaos. Complexity is about how a huge number of

extremely complicated and dynamic set of relationships can generate some simple behaviouralpatterns, whereas chaotic behaviour, in the sense of deterministic chaos, is the result of a

relatively small number of non-linear interactions.[9]

Therefore, the main difference between Chaotic systems and complex systems is their history.[12]

Chaotic systems dont rely on their history as complex ones do. Chaotic behaviour pushes a

system in equilibrium into chaotic order, which means, in other words, out of what wetraditionally define as 'order'. On the other hand, complex systems evolve far from equilibrium at


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the edge of chaos. They evolve at a critical state built up by a history of irreversible andunexpected events. In a sense chaotic systems can be regarded as a subset of complex systems

distinguished precisely by this absence of historical dependence. Many real complex systems are,in practice and over long but finite time periods, robust. However, they do possess the potential

for radical qualitative change of kind whilst retaining systemic integrity. Metamorphosis serves

as perhaps more than a metaphor for such transformations.

Complexity economics is the application ofcomplexity science to the problems ofeconomics. It

is one of the four C's of a new paradigm surfacing in the field of economics. The four C's arecomplexity, chaos, catastrophe and cybernetics. This new mode of economic thought rejects

traditional assumptions that imply that the economy is a closed system that eventually reaches anequilibrium. Instead, it views economies as open complex adaptive systems with endogenous

evolution. Complex systems do not necessarily settle to equilibrium even ideal deterministicmodels may exhibit chaos, which is distinct from both random (nondeterministic) and analytic

behavior.[1]

Contents

[hide]

y 1 Introductiony 2 Behavior of complex systemsy 3 Comparison with traditional economicsy 4 Examplesy 5 See alsoy 6 Referencesy 7 External links

[edit] Introduction

Complexity economics rejects many aspects of traditional economic theory. The mathematical

models used by traditional economics were formulated in an analogy with early models ofthermodynamics. These mathematical models of economics were substantially based on the first

law of thermodynamics, equilibrium.[1]

Later, the second law of thermodynamics, concerning thegrowing amount ofentropy in any spontaneous physical process, was formulated by Rudolf

Clausius. Proponents of complexity economics claim that traditional economic models never

adapted to the latter discovery and thus remain incomplete models of reality, and thatmainstream economists are yet to introduce information entropy to their models. Informationentropy was developed in 1949 by C. Shannon and W. Weaver, based on Boltzmann'sstatistical

thermodynamics, as "information uncertainty", associated with any probability distribution.Entropy has been used at least since 1988 to formulate the important concepts of organization

and disorder, viewed as basic state parameters, in describing/simulating the evolution of complexsystems (including economic systems).


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In the light of the new concepts introduced, economic systems shall no more be considered as"naturally" inclined to achieve equilibrium states. On the contrary, economic systems - like most

complex and self-organized systems - are intrinsically evolutionary systems, which tend todevelop, prevailingly toward levels of higher internal organization; though the possibility of

involution processes - or even of catastrophic events - remains immanent. Traditional economic

models have been constructed by allowing only a very small amount ofdegrees of freedom, inorder to simplify models. For example, the relation ofunemployment and inflation istraditionally considered to be a simple function with one degree of freedom, allowing for very

little entropy.

As to the practicability of theoretical instruments, there is also a crucial difference to allow for:traditional economics was conceived before computers had been invented. Computational

simulations have made it possible to demonstrate macro-level rules using only micro-levelbehaviors without assuming idealized market actors. For example, Pareto's law can be

demonstrated to arise spontaneously.

Complexity economics is built on foundations which draw inspiration from areas such asbehavioral economics, institutional economics, Austrian economics, and evolutionary

economics. Complexity incorporates components from each of these areas of economic thought,though the complex perspective includes many more characteristics to describe a dynamic

system such as emergence, sensitive dependence on initial conditions, red queen behavior, andcomplex systems usually incorporate a selection mechanism as described by most general

evolutionary models. There is no widely accepted specific definition for complexity as it pertainsto economic systems. This is largely due to the fact that the field as a whole is still under

construction.

[edit] Behavior ofcomplexsystems

Brian Arthur, Steven N. Durlauf, and David A. Lane, of the Santa Fe Institute define six featuresof complex systems that have presented significant trouble for traditional mathematics.

[2]

1. Dispersed Interaction What happens in the economy is determined by the interactionof many dispersed, possibly heterogeneous, agents acting in parallel. The action of anygiven agent depends upon the anticipated actions of a limited number of other agents and

on the aggregate state these agents co-create.2. No Global Controller No global entity controls interactions. Instead, controls are

provided by mechanisms of competition and coordination between agents. Economicactions are mediated by legal institutions, assigned roles, and shifting associations. Nor is

there a universal competitor a single agent that can exploit all opportunities in theeconomy.

3. Cross-cutting Hierarchical Organization The economy has many levels oforganization and interaction. Units at any given level behaviors, actions, strategies,

products typically serve as "building blocks" for constructing units at the next higherlevel. The overall organization is more than hierarchical, with many sorts of tangling

interactions (associations, channels of communication) across levels.


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4. Continual Adaptation Behaviors, actions, strategies, and products are revisedcontinually as the individual agents accumulate experience the system constantly

adapts.5. Perpetual Novelty Niches These are continually created by new markets, new

technologies, new behaviors, new institutions. The very act of filling a niche may provide

new niches. The result is ongoing, perpetual novelty.6. Out-of-Equilibrium Dynamics Because new niches, new potentials, newpossibilities, are continually created, the economy operates far from any optimum or

global equilibrium. Improvements are always possible and indeed occur regularly.

[edit] Comparison with traditional economics

The table below illustrates the differences between the complexity perspective and classical

economics. Eric Beinhockerproposes five concepts that distinguish complexity economics fromtraditional economics. The first five categories are Beinhocker's synthesis, the last four are from

W. Brian Arthur as reprinted in David Colander's The Complexity Vision.[3]

Complexity Economics Traditional Economics

DynamicOpen, dynamic, non-linear systems,

far from equilibriumClosed, static, linear systems in equilibrium

Agents

Modelled individually; useinductive rules of thumb to make

decisions; have incompleteinformation; are subject to errors

and biases; learn to adapt over time;heterogeneous agents

Modelled collectively; use complexdeductive calculations to make decisions;

have complete information; make no errorsand have no biases; have no need for

learning or adaptation (are already perfect),mostly homogeneous agents

Networks

Explicitly model bi-lateralinteractions between individualagents; networks of relationshipschange over time

Assume agents only interact indirectlythrough market mechanisms (e.g. auctions)

Emergence

No distinction between micro/macro

economics; macro patterns areemergent result of micro level

behaviours and interactions.

Micro-and macroeconomics remain separatedisciplines

Evolution

The evolutionary process ofdifferentiation, selection and

amplification provides the system

with novelty and is responsible forits growth in order and complexity

No mechanism for endogenously creating

novelty, or growth in order and complexity

TechnologyTechnology fluid, endogenous tothe system

Technology as given or selected oneconomic basis

Preferences

Formulation of preferences becomes

central; individuals not necessarilyselfish

Preferences given; Individuals selfish


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Origins from

PhysicalSciences

Based on Biology (structure,pattern, self-organized, life cycle)

Based on 19th-century physics (equilibrium,stability, deterministic dynamics)

Elements Patterns and Possibilities Price and Quantity

CHAOSAND THE STOCKMARKET

According to respected authorities, stock markets are non-linear, dynamic systems. Chaos theory

is the mathematics of studying such non-linear, dynamic systems. Chaos analysis has determined

that market prices are highly random, but with a trend. The amount of the trend varies frommarket to market and from time frame to time frame. A concept involved in chaotic systems is

fractals. Fractals are objects which are "self-similar" in the sense that the individual parts arerelated to the whole. A popular example of this is a tree. While the branches get smaller and

smaller, each is similar in structure to the larger branches and the tree as a whole. Similarly, inmarket price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure

has a similar appearance. Just as with natural objects, as you move in closer and closer, you seemore and more detail. Another characteristic of chaotic markets is called "sensitive dependence

on initial conditions." This is what makes dynamic market systems so difficult to predict.Because we cannot accurately describe the current situation band because errors in the

description are hard to find due to the system's overall complexity, accurate predictions become

impossible. Even if we could predict tomorrow's stock market change exactly (which we can't),we would still have zero accuracy trying to predict only twenty days ahead.

A number of thoughtful traders and experts have suggested that those trading with intra day data

such as five-minute bar charts are trading random noise and thus wasting their time. Over time,

they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they

follow trends. The question naturally arises how can short-term data be random and longer-termdata not be in the same market? If short-term (random) data accumulates to form long-term data,

wouldn't that also have to be random? As it turns out, such a paradox can exist. A system can berandom in the short-term and deterministic in the long term.

CHAOSAND THE STOCKMARKET

The following material is from an independent study by John Matthews and is used here with his

permission.


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Background

It started off with an assignment to do extrapolations when I was still under-graduate. I used

some of the Johannesburg StockExchange (JSE) data as basis for input, and the regressionmethodologies. It soon turns out that these methodologies have some inherent shortcomings

regarding extrapolations. This triggered off the development of my own extrapolation algorithm.After 6 years of development and refinement I had a workable methodology. It worked fine,

except every so often my predictions were wrong - I did not know at the time that it was chaosthat hit me, I called it investor madness! This was something that I could not wish away, so some

8 more years of development and refinement follows in which the accuracy went up from 30% to90%, with the occasional 100%.

With this accuracy I decided to invest some of my own money on the JSE, using my algorithm.That was January 1996. During 1996 the JSE had a disman performance, ending the year on -4%

for the year. By using my algorithm I was able to make more that 37% net profit during the sameperiod. We all know that in a declining market there are always individual shares that will rise,

the algorithm was therefore quite successful in picking out the correct shares to buy and sell atthe right time. From the beginning of this year until the end of June, net profit is more that 43%.

Development andApplication

It is common knowledge that share prices run in cycles, with each share having its own

individual cycle or set of cycles - almost like the fingerprints of a human being. This is the baseof the algorithm - given the closing prices of any share oer a time period, the algorithm finds all

the possible cycles that exist in the given data by using my own developed fractals. Once thecycles have been found, the strongest cycle identified (which I call the chaos cycle, as all shares

and indices follow this cycle), extrapolation becomes a trivial Algebra exercise. This this, the

behavior of each individual share is predicted a day, a week, a month, or even a year in advance.

As once-offf exercise, I used the terminal of a local stockbroker and feed in prices for 50

randomly selected shares on an hourly basis. After each input the extrapolation algorithm wasrun to predict what will happen in the next hour. Success rate was a staggering 95% correct. As

rules and regulations on the JSE prohibits this kind of activity (for a member of the public to usethe terminal of a stockbroker) I could not repeat this - I had to deal with a lot of angry people

when they "caught" me, but my joy with the success I had overshadowed my problems withseveral orders of magnitude!

As I could not repeat the above exercise, I turned to some other (I thought, more accessible!)

data. I changed the input and output format, keep the algorithm the same, and feed in data aboutaircraft accidents. After accumulating some input, the algorithm was able to predict that kind ofaccident (i.e. a flat tire), can be expected, what kind of aircraft (i.e. twin engine, prop) will be

involved, geographically where this accident will occur, and in what time window (i.e. between10h00 - 11h00) it will occur, with more than 90% accuracy. Unfortunately I had to stop that too

as all these data are considered sensitive and not for public consumption. So the Air TrafficController (ATC) that supplied me with the data and gave me feedback and myself were persona

non grata for months.


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The Future

As many investors use the so-called conventional indicators (moving average,

overbought/oversold and a host of others) for decision making in buying and selling, stockmarkets all over the world react to this kind of behavior. In an effort to push up the success rate

of the algorithm, several of these indicators are currently used in conjunction with theextrapolations. Initial success rate increased from one-day-in-a-week 100% correct to two days-

in-a-week 100% correct. However it is early days, as the new set of indicators are only installedfor 2 months, and we only had one bad month (May) thus far on the JSE.

As proof of the algorithm and of the work I'm doing, the following graphs are of the Dow Joneson Wallstreet, as well as MACMED (some other share on the JSE in the parmaceutical sector).

The entire system is PC-based and written in PASCAL by myself. The graphs are screendumpsfrom my program.

The first graph is the Dow-Jones

extrapolated to the end of July 1997.The second graph is the last 20

trading days, extrapolated to thenext 20 trading days of the Dow

Jones. The two graphs that followare of MACMED extrapolated to the

end of July 1997, and a zoom-in onthe last 20 trading days of

MACMED. These graphs are VERYBUSY graphs, but follow the thin

red sinus-curve-like line on the

zoomed-in graphs as this is thechaos line. All calculations are doneevery time new input is received,

and it usually changes theextrapolation curves significantly.

So these graphs are only valid forone day, in this case for June 30,

1997.

Graph 1

Graph 2

Graph 3

Graph 4

In the graphs of the Dow Jones, the chaos cycles are not very well developed (of low

prominence) which indicate instability as it is approaching an upper-trend line (not shown for

clarity). The graph of MACMED is included as an example to illustrate well developed chaoscycles.

SOLAR SYSTEM CHAOS


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Chaos theory isn't new to astronomers. Most have long known that the solar system does not "runwith the precision of a Swiss watch." Astronomers have uncovered certain kinds of instabilities

that occur throughout the solar system -- in the motions of Saturn's moon Hyperion, in gaps inthe asteroid belt between Mars and Jupiter, and in the orbits of the system's planets themselves.

As used by astronomers, the word chaos denotes an abrupt change in some property o

Eco No Physics is an Interdisciplinary Research Field

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