Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply ''chaos''.

Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

Applications

Chaos theory is applied in many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, robotics, and meteorology.

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, as well as computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather, the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. There is some controversy over the existence of chaotic dynamics in plate tectonics and in economics.

A successful application of chaos theory is 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, specifically to the prediction of seemingly random seizures by observing initial conditions.

Quantum chaos theory studies how the correspondence between quantum mechanics and classical mechanics works in the context of chaotic systems. Recently, another field, called relativistic chaos, has emerged to describe systems that follow the laws of general relativity.

The motion of ''N'' stars in response to their self-gravity (the gravitational ''N''-body problem) is generically chaotic.

In electrical engineering, chaotic systems are used in communications, random number generators, and encryption systems.

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.

Chaotic dynamics

In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although there is no universally accepted mathematical definition of chaos, a commonly used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:

#it must be sensitive to initial conditions; #it must be topologically mixing; and #its periodic orbits must be dense.

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

Sensitivity to initial conditions

''Sensitivity to initial conditions'' means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour. However, it has been shown that the last two properties in the list above actually imply sensitivity to initial conditions and if attention is restricted to intervals, the second property implies the other two (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list). It is interesting that the most practically significant condition, that of sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) purely topological conditions, which are therefore of greater interest to mathematicians.

Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled ''Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?'' The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

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

The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two trajectories in phase space with initial separation $\backslash delta\; \backslash mathbf\{Z\}\_0$ diverge

:$|\; \backslash delta\backslash mathbf\{Z\}(t)\; |\; \backslash approx\; e^\{\backslash lambda\; t\}\; |\; \backslash delta\; \backslash mathbf\{Z\}\_0\; |\backslash$

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

There are also measure-theoretic mathematical conditions (discussed in ergodic theory) such as mixing or being a K-system which relate to sensitivity of initial conditions and chaos

Topological mixing

''Topological mixing'' (or topological ''transitivity'') means that the system will evolve over time so that any given region or open 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. 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, $\backslash tfrac\{5-\backslash sqrt\{5\}\}\{8\}$ → $\backslash tfrac\{5+\backslash sqrt\{5\}\}\{8\}$ → $\backslash tfrac\{5-\backslash sqrt\{5\}\}\{8\}$ (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).

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

Strange attractors

Some dynamical systems, like the one-dimensional logistic 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

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 found a three dimensional system with just five terms on the right hand side, and with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.

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

History

An early proponent of chaos theory 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. 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. 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, , M.L. Cartwright and J.E. Littlewood, and Stephen Smale. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. 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.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. 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. 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.

The year before, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices. 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. Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional dimension", 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. 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 and in 1958 by R.L. Ives. However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.

In December 1977 the New York Academy of 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. 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".

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. 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'' 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 focused on 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 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 described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by J. Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, 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.

All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point. Thus, given a time series to test for determinism, one can: #pick a test state; #search the time series for a similar or 'nearby' state; and #compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.

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

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

The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in philosophy.

Cultural references

Chaos theory has been mentioned in numerous movies and works of literature. For instance, it was mentioned extensively in Michael Chrichton's novel ''Jurassic Park'' and more briefly in its sequel. Other examples include the film ''Chaos (2006 film)'', ''The Butterfly Effect'', the sitcom ''The Big Bang Theory'', Tom Stoppard's play ''Arcadia'' and the video game Tom Clancy's Splinter Cell: Chaos Theory. The influence of chaos theory in shaping the popular understanding of the world we live in was the subject of the BBC documentary ''High Anxieties - The Mathematics of Chaos'' directed by David Malone. Chaos Theory is also the subject of discussion in the BBC documentary "The Secret Life of Chaos" presented by the physicist Jim Al-Khalili.

See also

;Examples of chaotic systems

Arnold's cat map

Bouncing ball dynamics

Cliodynamics

Coupled map lattice

Chua's circuit

Double pendulum

Dynamical billiards

Economic bubble

Hénon map

Horseshoe map

List of chaotic maps

Logistic map

Rössler attractor

Standard map

Swinging Atwood's machine

Tilt A Whirl

;Other related topics

Butterfly effect

Anosov diffeomorphism

Bifurcation theory

Chaos theory in organizational development

Complexity

Control of chaos

Edge of chaos

Fractal

*Julia set

*Mandelbrot set

Predictability

Quantum chaos

Santa Fe Institute

Synchronization of chaos

Unintended consequence

;People

Ralph Abraham

Michael Berry

Leon O. Chua

Ivar Ekeland

Doyne Farmer

Mitchell Feigenbaum

Martin Gutzwiller

Brosl Hasslacher

Michel Hénon

Edward Lorenz

Aleksandr Lyapunov

Ian Malcolm

Benoît Mandelbrot

Norman Packard

Henri Poincaré

Otto Rössler

David Ruelle

Oleksandr Mikolaiovich Sharkovsky

Robert Shaw

Floris Takens

James A. Yorke

References

Scientific literature

Articles

A.N. Sharkovskii, "Co-existence of cycles of a continuous mapping of the line into itself", Ukrainian Math. J., 16:61–71 (1964)

Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985–92, 1975.

Online version (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online, but which don't provide article views. The online content is identical to the hardcopy text. Citation variations will be related to country of publication).

Kolyada, S. F. "Li-Yorke sensitivity and other concepts of chaos", Ukrainian Math. J. 56 (2004), 1242–1257.

C. Strelioff, A. Hübler (2006). Medium-Term Prediction of Chaos, PRL 96, 044101

A. Hübler, G. Foster, K. Phelps (2007). Managing Chaos: Thinking out of the Box Complexity, vol. 12, pp. 10–13

Textbooks

Semitechnical and popular works

Ralph H. Abraham and Yoshisuke Ueda (Ed.), ''The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory'', World Scientific Publishing Company, 2001, 232 pp.

Michael Barnsley, ''Fractals Everywhere'', Academic Press 1988, 394 pp.

Richard J Bird, ''Chaos and Life: Complexity and Order in Evolution and Thought'', Columbia University Press 2003, 352 pp.

John Briggs and David Peat, ''Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness'', Harper Perennial 1990, 224 pp.

John Briggs and David Peat, ''Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change'', Harper Perennial 2000, 224 pp.

Lawrence A. Cunningham, ''From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis,'' George Washington Law Review, Vol. 62, 1994, 546 pp.

Predrag Cvitanović, ''Universality in Chaos'', Adam Hilger 1989, 648 pp.

Leon Glass and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.

James Gleick, ''Chaos: Making a New Science'', New York: Penguin, 1988. 368 pp.

John Gribbin, ''Deep Simplicity'',

L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.

Arvind Kumar, ''Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature '', National Book Trust, 2003.

Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.

Edward Lorenz, ''The Essence of Chaos'', University of Washington Press, 1996.

Chapter 5 of Alan Marshall (2002) The Unity of nature, Imperial College Press: London

Heinz-Otto Peitgen and Dietmar Saupe (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.

Clifford A. Pickover, ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.

Ilya Prigogine and Isabelle Stengers, ''Order Out of Chaos'', Bantam 1984.

Heinz-Otto Peitgen and P. H. Richter, ''The Beauty of Fractals : Images of Complex Dynamical Systems'', Springer 1986, 211 pp.

David Ruelle, ''Chance and Chaos'', Princeton University Press 1993.

Ivars Peterson, ''Newton's Clock: Chaos in the Solar System'', Freeman, 1993.

David Ruelle, ''Chaotic Evolution and Strange Attractors'', Cambridge University Press, 1989.

Peter Smith, ''Explaining Chaos'', Cambridge University Press, 1998.

Ian Stewart, ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.

Steven Strogatz, ''Sync: The emerging science of spontaneous order'', Hyperion, 2003.

Yoshisuke Ueda, ''The Road To Chaos'', Aerial Pr, 1993.

M. Mitchell Waldrop, ''Complexity : The Emerging Science at the Edge of Order and Chaos'', Simon & Schuster, 1992.

External links

Nonlinear Dynamics Research Group with Animations in Flash

The Chaos group at the University of Maryland

The Chaos Hypertextbook. An introductory primer on chaos and fractals

ChaosBook.org An advanced graduate textbook on chaos (no fractals)

Society for Chaos Theory in Psychology & Life Sciences

Nonlinear Dynamics Research Group at CSDC, Florence Italy

Interactive live chaotic pendulum experiment, allows users to interact and sample data from a real working damped driven chaotic pendulum

Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998.

Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens

Gleick's ''Chaos'' (excerpt)

Systems Analysis, Modelling and Prediction Group at the University of Oxford

A page about the Mackey-Glass equation

High Anxieties - The Mathematics of Chaos (2008) BBC documentary directed by David Malone

* Category:Fundamental physics concepts Category:Non-linear systems Category:Systems Category:Ergodic theory

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Coordinates	36°59′55″N66°41′13″N
name	Tim Sköld
landscape	Yes
background	non_vocal_instrumentalist
birth name	Tim Sköld
alias	Tim Skold, Tim Tim, Skold
born	December 14, 1966Skövde, Sweden
instrument	Bass, guitar, synthesizer, vocals, accordion, programming
occupation	Musician, producer
years active	1987–present
label	Metropolis, Relativity, Wax Trax!, Interscope, Nothing
associated acts	Kingpin, Shotgun Messiah, KMFDM, MDFMK, ohGr, Marilyn Manson, The Newlydeads, Doctor Midnight & The Mercy Cult
website
notable instruments	Gibson Firebird }}

Tim Sköld (born December 14, 1966, in Skövde, Sweden) is a multi-instrumentalist who, in addition to producing solo work, has also collaborated with multiple musical groups including Shotgun Messiah, KMFDM and Marilyn Manson.

Biography

Early life

Tim Sköld was born on December 14, 1966 in Skövde, a small city in Västergötland, Sweden. He grew up in Timmersdala, 21 kilometres north-west of Skövde. He has a sister named Linda. Sköld grew up in a very liberal setting as the son of a 15-year-old mother named Åse and a father trying to get by as a semi-professional drummer.

At age 11, Sköld joined a playback band (where the band mimes over recorded material). Sköld's band would play for people at school meetings with Kiss, Bowie and Sweet songs playing on a cassette player.

Sköld first rented a bass guitar at age 12, and by the time he was 13, he was playing and singing in his first real band. Sköld later met guitarist Harry Cody at a New Year's Eve party, and they formed a creative partnership which would last for many years to come. Their dream was to go to the U.S.A. to be rock stars. At age 17, after 2 years of studying Process Engineering at a boarding school, Sköld moved into his own apartment and took a factory job making military equipment. A half-year later, he was drafted into the army. He brought his bass to the base and sneaked into the shower stalls to practice after everyone went to sleep. Sköld also kept rehearsing with Cody on the weekends, and once out of the army he took on many odd jobs including library assistant, Volvo employee and as a gardener at a bathhouse.

At age 19, Sköld and Cody talked a Swedish record company into allowing them to make a record, which they could use as a demo to reach the American market. Starting off as Shylock, their determination soon paid off in the form of Kingpin.

Kingpin

During the early 80's, bassist Sköld and guitarist Cody formed the glam metal band Kingpin in their native Skövde, Sweden. They were inspired by the flamboyant hard rock bands flowing out of Los Angeles at the time, as well as UK acts such as Sigue Sigue Sputnik and Zodiac Mindwarp.

The band kicked off their career with a 7" single titled "Shout It Out" (which hit #1 in Sweden) in 1987. They then drafted vocalist Zinny J. Zan and drummer Stixx Galore a short time later. Kingpin released their only Swedish album, ''Welcome to Bop City'', the following year, and then moved to Los Angeles.

Shotgun Messiah

Signed by Cliff Cultreri at Relativity Records, Kingpin changed their name to Shotgun Messiah (due to a San Francisco-based band holding the rights to the name Kingpin), remixed and re-packaged ''Welcome To Bop City'' as a self-titled album, and released it as their international debut in 1989.

After having MTV smash hits such as "Don't Care 'Bout Nothin'" and "Shout It Out", whilst the band was in progress of making a follow up to their debut self-titled album, Zan was let go. They hired a new bassist, Bobby Lycon, from New York, and on Cody's suggestion Tim Tim switched to lead vocals. At this time, Tim Tim, whose role was now frontman, went by the name Tim Skold. They released ''Second Coming'' in 1991, which was enthusiastically received by the media, and was their most successful album to date. Two singles were released from that album, "Heartbreak Blvd." and "Living Without You".

Shortly after, the band followed up with an E.P. entitled ''I Want More'', which contained some punk covers along with re-recorded songs from their previous album ''I Want More'', "Babylon" and an acoustic version of "Nobody's Home". Soon after this release, Shotgun Messiah were down to two original members, Sköld and Cody, seeing the departure of Stixx and Lycon.

Still under contract with Relativity Records, Sköld and Cody decided to go back to Sweden for the recording of the third Shotgun Messiah album, ''Violent New Breed'', which leaned towards the industrial end of heavy metal. The album was released in 1993 and is now considered a cult classic due to the "before its time" incorporation of industrial influences. However, at the time of release it received mixed reviews and continued public indifference, which eventually convinced the two to end the band, leading Sköld to embark on his solo project, Skold.

Skold

After the break-up of Shotgun Messiah, Sköld went on to pursue a solo career. Writing all songs and playing all instruments himself, his self-titled debut album, ''Skold'', was released in 1996 by RCA. The album was co-produced with Scott Humphrey.

Promoting his debut album, Sköld and the rest of the live Skold band went on a short tour with Genitorturers. Some of the songs on the ''Skold'' album were used in movies like Disturbing Behavior ("Hail Mary"), Universal Soldier: The Return ("Chaos") and the PlayStation game Twisted Metal 4 ("Chaos"). During his solo career, Sköld also provided remixes for several bands, such as Prong, Nature and Drown. He also met KMFDM front-man Sascha Konietzko during his time in the studio.

Around 2002, Sköld recorded a follow up to his 1996 self-titled album. These songs have never been officially released. Sköld stated in an interview with ''The Sychophant'' that he had made a demo with 10 copies ever made. The demo had 10 songs on it. Six of the 10 were released on file sharing networks without Sköld's permission: "Burn", "Dead God", "I Hate", "Believe", "The Point", and "Don't Pray". The demo is known as the ''Dead God EP'', although Sköld says that it was called ''Disrupting the Orderly Routine of the Institution''. The original artwork was made with an inkjet printer and was the title written in drippy letters. This demo was given to several friends that Sköld considered trustworthy, but his songs appeared on the internet from one of them. Sköld said that "someone suggested to me that I should go back to them and finish it off and release it. And maybe I will one day, who knows?"

On November 3, 2009, "I Will Not Forget", "A Dark Star", and "Bullets Ricochet" were released on iTunes and Amazon as new Skold singles. Also that November, Sköld's official website announced that he was to release his second solo album in early 2010. However, the album remained unreleased in 2010. In January 2011, Metropolis Records announced that they were releasing the new record and single in spring of 2011.

"Anomie" will be made available on May 10 as a 12-track CD or digital download as well as a digital deluxe version with two bonus tracks.

KMFDM & MDFMK

After a short-lived solo career, Sköld joined KMFDM in 1997. His first involvement with KMFDM was on the album ''Symbols''. He wrote and sang the song "Anarchy", which became a hit in clubs, and spawned subsequent remixes of the track done by Sköld himself. His next album with KMFDM, ''Adios'', was released in 1999. Sköld took a more prominent role in the band, not only as co-vocalist, co-writer, and bassist, but also as producer, engineer, and programmer, alongside KMFDM's founder Sascha Konietzko.

Due to turmoil within the band, Konietzko and Sköld ended KMFDM in 1999, and restarted as MDFMK the following year. They released one album, ''MDFMK'', released in 2000 by Universal Records. The band, including Lucia Cifarelli (formerly of Drill), took on a more "futuristic" sound, which contained less of the industrial rock KMFDM was known for, and added a mix of drum & bass, trance and europop, primarily in a production style leaning towards "electronica". MDFMK featured all three members sharing vocal duties. Their song "Missing Time" was used in the animated movie Heavy Metal 2000.

In 2002, the trio reformed KMFDM along with Raymond Watts, and released ''Attak''. Afterwards, Sköld departed the band. Due to a commitment to produce Marilyn Manson's album ''The Golden Age of Grotesque'', Sköld was unable to join KMFDM's 2002 Sturm & Drang tour; he did, however, make two guest appearances at shows in June.

On December 16, 2008, the KMFDM website announced that Sköld and Konietzko would be releasing an album together, titled ''Skold vs. KMFDM''. The album was released on February 24, 2009.

Sköld did production and instrumentation work on KMFDM's album ''Blitz'', released March 24, 2009. Later that same year he contributed a remix to Left Spine Down's 2009 remix album entitled ''Voltage 2.3: Remixed and Revisited'', as well as 16volt's album ''American Porn Songs''.

Newlydeads & Ohgr

Sköld performed a show with Taime Downe's The Newlydeads on December 13, 2000 at the Pretty Ugly Club in Los Angeles. Sköld, on bass, joined Ohgr, a project of Skinny Puppy vocalist Nivek Ogre, for the tour in support of its first album, ''Welt'', in 2001.

Marilyn Manson

Sköld's involvement with Marilyn Manson began as producer for the single "Tainted Love", which is featured in the 2001 teen comedy/parody movie ''Not Another Teen Movie'' and appeared on the soundtrack. Manson and Sköld went on to score the movie ''Resident Evil'', released in 2002. Several tracks are featured on the Resident Evil movie soundtrack.

Sköld officially joined Marilyn Manson in 2002 after the departure of bassist Twiggy Ramirez. At this time, not only was Sköld the bassist for the band, but he was also producing, editing, creating artwork, electronics, programming drums and beats, playing guitar, keyboards, accordion and synthesizer bass for the album ''The Golden Age of Grotesque''.

He is described by Manson as, "the power that attitude brings to an album". On the band's 2004 release, ''Lest We Forget: The Best Of'', Sköld produced, played lead guitar, and sang backup vocals on the cover version of "Personal Jesus", which was also released as a single. Coinciding with the release of ''The Nightmare Before Christmas in 3D'' in October 2006, Manson and Sköld contributed a cover of "This Is Halloween" to ''The Nightmare Before Christmas'' soundtrack, with Sköld taking care of the music while Manson provided the vocals.

This time also saw the start of work on Marilyn Manson’s sixth studio album entitled ''Eat Me, Drink Me''. The album was released worldwide on June 5, 2007.

Sköld played guitar on the band's 2007 world tour, Rape of the World, with Rob Holliday (formerly guitarist/bassist for Curve, Gary Numan, The Mission and The Prodigy) taking over bass duties.

On January 9, 2008 Marilyn Manson posted a bulletin on MySpace which reported that Sköld had left the band and former bassist Twiggy Ramirez had returned to take his place.

Doctor Midnight & The Mercy Cult

In 2009, Sköld formed the Scandinavian supergroup Doctor Midnight & The Mercy Cult alongside Hank von Helvete.

Musical equipment

Throughout his career Sköld has played several different instruments and used equipment from different manufacturers, here is a list of notable equipment.

Guitars and basses

Gibson Les Paul - Black finish

Gibson Firebird V - Cream white finish

Gibson Firebird V - Black finish

Gibson Firebird VII - Red finish

Roland G-707 Guitar Synthesizer

Gretsch Broadkaster G6119B bass - Black finish

Gretsch Broadkaster G6119B bass - Red finish

Gretsch Broadkaster G6119B(Custom) bass - Bamboo finish

King doublebass - White with black flames

King doublebass - Wood

D'Addario XL Strings

Dunlop Tortex Picks

Effects

Dunlop Cry Baby Bass Wah Wah

Dunlop MXR M-103 Blue Box

BOSS GT-6

BOSS GT-6B Multi-Effects Units

Dunlop MXR Smart Gate Pedal

Dunlop MXR 10 Band EQ

Dunlop MXR GT-OD Overdrive

Dunlop MXR Distortion III

Amplification

Ampeg SVT-4PRO head

Ampeg SVT-810E cabinet

Blankenship Variplex (2)

Marshall Amps

Discography

Kingpin

"Shout It Out" (1987)

''Welcome to Bop City'' (1988)

Shotgun Messiah

''Shotgun Messiah'' (September 12, 1989)

"Shout It Out" (1989)

"Don't Care 'Bout Nothin'" (1990)

''Second Coming'' (October 22, 1991)

"Heartbreak Blvd." (1991)

"Red Hot" (1991)

"Living Without You" (1992)

''I Want More'' (November 17, 1992)

''Violent New Breed'' (September 28, 1993)

"Enemy in Me" (1993)

"Violent New Breed" (1993)

Skold

''Skold'' (July 30, 1996)

''Neverland'' (1996) (EP)

''Dead God'' (2002) (Promo EP)

''I Will Not Forget'' (2009) (Single)

''A Dark Star'' (2009) (Single)

''Bullets Ricochet'' (2009) (Single)

''Suck'' (2011) (Single) ''Anomie'' (May 10, 2011)

KMFDM

''Symbols'' (September 23, 1997)

"Megalomaniac" (January 20, 1998)

''Adios'' (April 20, 1999)

"Boots" (February 5, 2002)

''Attak'' (March 19, 2002)

''Blitz'' (March 24, 2009)

MDFMK

''MDFMK'' (March 28, 2000)

Marilyn Manson

"Tainted Love" (2001)

''Resident Evil Soundtrack'' (2002)

''The Golden Age of Grotesque'' (May 13, 2003)

''Lest We Forget: The Best of'' (September 28, 2004)

''Eat Me, Drink Me'' (June 4, 2007)

Skold vs. KMFDM

''Skold vs. KMFDM'' (February 24, 2009)

Doctor Midnight & The Mercy Cult

"(Don't) Waste It" (May 9, 2011)

''I Declare: Treason'' (June 6, 2011)

References

External links

Official website

Official Facebook

Official Myspace

Official Twitter

Category:Industrial musicians Category:KMFDM members Category:Marilyn Manson (band) members Category:Living people Category:Swedish heavy metal bass guitarists Category:Shotgun Messiah Category:1966 births

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