AND after these things I saw another angel come down from heaven, having great power; and the earth was lightened with his glory…

.

My baloney has a first name, it's m-i-g-u-e- - -

Meh, I'd rather believe in the Chaos Theory

Chaos theory

From Wikipedia, the free encyclopedia

For other uses, see Chaos Theory (disambiguation).

A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, 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.[1] 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.[2] In other words, the deterministic nature of these systems does not make them predictable.[3][4] This behavior is known as deterministic chaos, or simply chaos.

Chaotic behavior can be observed in many natural systems, such as the weather.[5] 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.

Contents
[hide] 1 Applications
2 Chaotic dynamics 2.1 Sensitivity to initial conditions
2.2 Topological mixing
2.3 Density of periodic orbits
2.4 Strange attractors
2.5 Minimum complexity of a chaotic system

3 History
4 Distinguishing random from chaotic data
5 Cultural references
6 See also
7 References
8 Scientific literature 8.1 Articles
8.2 Textbooks
8.3 Semitechnical and popular works

9 External links

[edit] Applications

A conus textile shell, similar in appearance to Rule 30, a cellular automaton with chaotic behaviour.
Chaos theory is applied in many scientific disciplines: geology, mathematics, programming, microbiology, biology, computer science, economics,[6][7][8] engineering,[9] finance,[10][11] meteorology, philosophy, physics, politics, population dynamics, psychology, and robotics.

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,[5] 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.[12][13][14]

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[citation needed].

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.[15]

Quantum chaos theory studies how the correspondence between quantum mechanics and classical mechanics works in the context of chaotic systems.[16] Recently, another field, called relativistic chaos,[17] 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.[18]

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.[19]

[edit] Chaotic dynamics

The map defined by x → 4 x (1 – x) and y → x + y if x + y < 1 (x + y – 1 otherwise) displays sensitivity to initial conditions. Here two series of x and y values diverge markedly over time from a tiny initial difference.
In common usage, "chaos" means "a state of disorder".[20] 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:[21]
1.it must be sensitive to initial conditions;
2.it must be topologically mixing; and
3.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.

[edit] 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[22][23] and if attention is restricted to intervals, the second property implies the other two[24] (an alternative, and in general weaker, definition of chaos uses only the first two properties in the above list[25]). 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.[26]

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

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

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.[4]

[edit] Topological mixing

The map defined by x → 4 x (1 – x) and y → x + y if x + y < 1 (x + y – 1 otherwise) also displays topological mixing. Here the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of points scattered across the space.
Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

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

[edit] 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.[27] The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, → → (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).[28]

Sharkovskii's theorem is the basis of the Li and Yorke[29] (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.

[edit] Strange attractors

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

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

Unlike fixed-point attractors and limit cycles, the attractors which arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points – Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and a fractal dimension can be calculated for them.

[edit] Minimum complexity of a chaotic system

Bifurcation diagram of the logistic map x → r x (1 – x). Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.
Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. However, the Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions. Finite dimensional linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be either nonlinear, or infinite-dimensional.

The Poincaré–Bendixson theorem states that a two dimensional differential equation has very regular behavior. The Lorenz attractor discussed above is generated by a system of three differential equations with a total of seven terms on the right hand side, five of which are linear terms and two of which are quadratic (and therefore nonlinear). Another well-known chaotic attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is (quadratic) nonlinear. Sprott[30] 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[31][32] showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved.

While the Poincaré–Bendixson theorem means that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behaviour.[citation needed] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional.[33] A theory of linear chaos is being developed in the functional analysis, a branch of mathematical analysis.

[edit] History

Barnsley fern created using the chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an Iterated function system (IFS).
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.[34][35] 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.[36] 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,[37] A. N. Kolmogorov,[38][39][40] M.L. Cartwright and J.E. Littlewood,[41] and Stephen Smale.[42] Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.[citation needed] Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied systems.

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems.

Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence were important for Chaos theory, analyzed for example by the Soviet physicist Lev Landau who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory.
An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961.[43] 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.[44] 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.[26]

The year before, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices.[45] 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.[46] 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).[47][48] 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.[49] 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[50] and in 1958 by R.L. Ives.[51][52] 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.[53][54]

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.[55] 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 Rayleigh–Bénard convection 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".[56]

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.[57] 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[58] 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[59] 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). 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, (though his history under-emphasized important Soviet contributions). 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.).

[edit] 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.[60][61]

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.[60][62] Thus, given a time series to test for determinism, one can:
1.pick a test state;
2.search the time series for a similar or 'nearby' state; and
3.compare their respective time evolutions.

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby state. A deterministic system will have an error that either remains small (stable, regular solution) or increases exponentially with time (chaos). A stochastic system will have a randomly distributed error.[63]

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.[64] 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.[65] 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.[66]

The question of how to distinguish deterministic chaotic systems from stochastic systems has also been discussed in philosophy.[67]

[edit] Cultural references

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (July 2011)

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, The Butterfly Effect, the sitcom The Big Bang Theory, Tom Stoppard's play Arcadia and the video games Tom Clancy's Splinter Cell: Chaos Theory and Assassin's Creed (video game). 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.

Courtesy of Wikipedia
:D

I think that just about covers it.

نظرية الفوضى
نظرية الفوضى حقل الدراسة في الرياضيات، مع تطبيقات في تخصصات عدة، بما في ذلك الفلسفة، الاقتصاد، البيولوجيا، الفيزياء والهندسة. نظرية الفوضى دراسات السلوك للكويكبات النظم التي شديدة الحساسية للشروط الأولية، أثرا شعبيا المشار إليها كتأثير الفراشة. صغيرة من الاختلافات في الظروف الأولية (مثل تلك التي تعزى إلى أخطاء في الحساب العددي التقريب) غلة متباينة على نطاق واسع نتائج لنظم الفوضى، يجعل من المستحيل التنبؤ بالمدى الطويل بشكل عام.[1] وهذا يحدث على الرغم من أن هذه النظم القطعية، بمعنى أن سلوكهم المستقبل يحدده تماما عن الظروف الأولية، مع لا عشوائية العناصر المتورطة.[2] بعبارة أخرى، طبيعة هذه النظم القطعية لا تجعلها يمكن التنبؤ بها.[3][4] هذا السلوك المعروف كالفوضى القطعية، أو مجرد الفوضى.

ويمكن ملاحظة السلوك الفوضوي في كثير من النظم الطبيعية، مثل الطقس.[5] يمكن التماس، تعليلا لمثل هذا السلوك من خلال تحليل نموذج رياضي الفوضى، أو من خلال التقنيات التحليلية مثل تكرار المؤامرات وخرائط بوانكاريه.

محتويات
[إخفاء] 1 التطبيقات
2 ديناميات الفوضى 2.1 الحساسية للشروط الأولية
2.2 خلط الطبولوجيا
2.3 كثافة من مدارات الدوري
2.4 ضخت غريبة
2.5 الحد الأدنى من تعقيد نظام الفوضى

التاريخ 3
4 التمييز بين عشوائية من البيانات الفوضى
5 المراجع الثقافية
6 انظر أيضا
7 مراجع
8 مقالات المؤلفات العلمية 8.1
8.2 الكتب المدرسية
8.3 الأشغال سيميتيتشنيكال وشعبية

9 وصلات خارجية

[تحرير] تطبيقات

كونوس المنسوجات قذيفة، مماثلة في شكلها للمادة 30، اﻹنسان خلوية مع السلوك الفوضوي.
يتم تطبيق نظرية الفوضى في العديد من التخصصات العلمية: "الجيولوجيا" و "الرياضيات"، "البرمجة" و "علم الأحياء المجهرية،" و "بيولوجيا" و "علم الحاسوب" أو "اقتصاديات الهندسة والمالية [9]، [10] [11] [6] [7] [8] الأرصاد الجوية، والفلسفة الفيزياء،، الديناميات السكانية، وعلم النفس، والسياسة الروبوتيات.

وقد لوحظ السلوك الفوضوي في المختبر في مجموعة متنوعة من النظم بما في ذلك الدوائر الكهربائية، أشعة الليزر، تتأرجح التفاعلات الكيميائية، ديناميات السوائل، والأجهزة الميكانيكية والأجهزة الميكانيكية-مغناطيسي، فضلا عن نماذج حاسوبية للعمليات الفوضوية. تشمل الملاحظات للسلوك الفوضوي في طبيعة التغييرات في الطقس، [5] ديناميات السواتل في النظام الشمسي، وتطور الزمن المجال المغناطيسي السماوية، والنمو السكاني في الإيكولوجيا، القوى المحركة لإمكانات العمل في الخلايا العصبية، والذبذبات الجزيئية. هناك بعض الجدل حول وجود ديناميات الفوضى في التكتونية وفي الاقتصاد.[12][13][14]

تطبيق ناجح لنظرية الفوضى في علم البيئة التي استخدمت فيها نظم للكويكبات مثل نموذج Ricker لإظهار كيف يمكن أن يؤدي النمو السكاني تحت التبعية الكثافة إلى ديناميات الفوضى [اقتباس حاجة].

ويجري حاليا أيضا تطبيق نظرية الفوضى للدراسات الطبية من الصرع، على وجه التحديد إلى التنبؤ بالمضبوطات التي تبدو عشوائية بمراعاة الشروط الأولية.[15]

نظرية الفوضى الكم دراسات كيفية عمل المراسلات بين ميكانيكا الكم والميكانيكا الكلاسيكية في سياق نظم الفوضى.[16] وحقل آخر، دعا الفوضى النسبية، [17] برز مؤخرا لوصف النظم التي تتبع قوانين النسبية العامة.

اقتراح n النجوم في استجابة لخطورتها المتمتعة بالحكم الذاتي (مشكلة الهيئة ن الجاذبية) الفوضى عموما.[18]

في الهندسة الكهربائية، تستخدم نظم الفوضى في الاتصالات ومولدات الأرقام العشوائية، ونظم التشفير.

في التحليل العددي، يمكن أن يؤدي طريقة نيوتن-رافسون تقارب جذور دالة على التكرارات الفوضى إذا الدالة لها لا جذور حقيقية.[19]

[تحرير] ديناميات الفوضى

الخريطة التي يحددها x → 4 x (س 1 –) و y → x + y إذا x + y < 1 (x + y-1 إلا) يعرض الحساسية للشروط الأولية. سلسلة هنا اثنين من س وقيم ص تتباعد ملحوظ مع مرور الوقت من فرق صغيرة أولية.
مشتركة الاستخدام، "الفوضى" ويعني "حالة من الفوضى".[20] ولكن في نظرية الفوضى، يعرف المصطلح أكثر دقة. على الرغم من أن هناك أي تعريف الحسابية المقبولة عالمياً للفوضى، يقول تعريف الشائع استخدامها، لنظام حركي تصنف باعتبارها فوضوية، يجب أن تكون له الخصائص التالية: [21]
1. يجب أن تراعي الظروف الأولية؛
2. أنه يجب خلط توبولوجيكالي؛ و
3. به المدارات الدوري يجب أن تكون كثيفة.

الحاجة إلى اعتماد الحساسة على الشروط الأولية تعني أن هناك مجموعة من الشروط الأولية لتدبير إيجابي لم تتقارب بدوره من أي طول.

[تحرير] الحساسية للشروط الأولية

الحساسية للشروط الأولية يعني أن كل نقطة في هذا نظام هو تقريبها تعسفاً عن كثب بنقاط أخرى مع مسارات المستقبل تختلف اختلافاً كبيرا. وهكذا، اضطرابات صغيرة تعسفاً من مساره الحالي قد يؤدي إلى

Courtesy of Bing & Wikipedia

Miguel, (G'zuz) b- nutz! If you da light, thank the lard for darkness! If u b da way, we is better off LOST! Hey I have an idea, you can prove it to us doubter by getting yourself curcified, and coming back in three days to say "see? I told you so!" Worth a shot?

DID YOU MISSS ME?

The Generation of Vipers shall not escape the damnation of HELL

Zion, January, 10^th 2012

As is the mother, so is her daughter.

Dear MSNBC and/or the Newsvine code of honor guardians, even to Turner and Sally and to Sally and Turner and all those great and small who represent these houses of press for that matter.

The reason I am writing to you today, is to ask you in a solemn and satisfactory manner to please reinstate my account at your news vine site on account of the spiritual importance of my work regardless attention or belief. YOU DO NOT HAVE TO DO IT BUT I AM KINDLY ASKING FOR IT. The choice is always yours. And this is the reason why I ask for it, whether you are initiated or not in the heavenly art of prophecy and revelation.

You have chosen to ban me or have suspended me from using my account in your MSNBC Newsvine site as a result of truth. In other words, you have chosen a plague of scarab scoundrels and mocking birds over the true rain to the vine. They are the ones you should have suspended and not me. If you just knew how they expressed themselves bout you after you suspended me. That alone should tell you the plague of terror you are dealing with. In any case, it was not the vine that chose me, but it was I who chose the vine to conduct the marvelous and strange work of the LORD as a special witness unto the truth.

My work may not be important or attractive to the masses, but it serves a higher purpose behind the scenes. It confesses boldly the sins of the people and the consequences thereof, both temporal and eternal, so that they are not suddenly taken by the devil and go into captivity in ignorance by an enemy that they never met. It is a social service of great import in view of the ignorance of many. And by shutting me down, you are yielding to terror, and asking for torment, for those scoundrels that I have mentioned who you have preferred to listen over me, who are indeed a plague to the vine do wish the heavens to be shut down so that there is no spiritual rain. Even though most do not care, there must be one soul here or there that is searching for what I have to offer. This in consequence, endangers the spiritual and temporal well being of the elderly, the youth and even the raising generation. And that will bring torment to a great many souls as a result of spiritual darkness. And they have made you and your house, that is MSNBC AND the NEWSVINE, an accessory to their wicked purpose whether they or you know it or not. For I know that the devil is whispering in their ears lies and deceptions, which they freely publish and in so doing they carry this people into captivity. For if your support their work you support them.

Now, if you knew who writes upon these walls or virtual portals, you would move out of the way indeed give complete clearance. My work is genuine, unedited, meaningful and full of purpose. I use my true and legal name and no pseudo names, for I am responsible before man and GOD for every word that proceeds from my hand. And everything I do, I ascribe it with proper authority to Jesus Christ, which is greater than mine. I am serious and I am not here to entertain or to play with fire.

You as representatives of journalism, are committing a greater crimes than what I may have done breaking your so called code of honor. Since when do serpents have honor? Are you going to shut me down because I call you serpents or the more subtle creatures that exists in nature? If you do, it is because you believe that you are or maybe you do not believe it. As far as I know, you are human beings not serpents. It is a true figurative speech; and as such you may or may not understand it. Nonetheless is the way of expression I utilize for myself in order to transact or rain ideas and knowledge from heaven in the world. This is the refreshment or the latter rain and yet you will not hear me. But if you shut me down, you have and are censoring the truth, infringing upon freedom of speech and obstructing divine justice. In other words, you are committing crimes against journalism; or you are eating your very own tail in a vicious cycle as a counterproductive snake. You are infringing upon the civil freedom and liberty civil of an American citizen who is expressing his beliefs and views with tested words on a public site that you use for gain or commercial purposes. Despite your so called codes of honor, what you have done is a violation of local and federal law. And if I can keep digging, you may also be committing discrimination on the basis of gender, race, and religious belief.

In any case, if you are bored, I think you should consider or pass my words to someone with more authority in your organization and who can have the intellect to considerer the gravity of your offenses. For I have no need to break your laws or codes, unless they are unrighteous, for I keep the laws of God, which are more strict and higher than the laws of men and even the laws of the land. That means that I am a law abiding citizen, who is minding his own business whose right is guaranteed by the laws of the land.

Now Tuner, I hope you have heard the proverb that says like father like son. I must turn my attention to you particularly and give you a portion of torment by making you and Sally responsible for what will shortly come to pass. And I will ascribe unto you the sins of all these scarabs, scoundrels and mocking birds that you have listened to. For of this it is written:

Behold, the days come, saith the Lord GOD, that I will send a famine in the land, not a famine of bread, nor a thirst for water, but of hearing the words of the LORD: And they shall wander from sea to sea, and from the north even to the east, they shall run to and fro to seek the word of the LORD, and shall not find it. In that day shall the fair virgins and young men faint for thirst. They that swear by the sin of Samaria, and say, Thy god, O Dan, liveth; and, The manner of Beer-sheba liveth; even they shall fall, and never rise up again.

And it shall come to pass in that day, saith the Lord GOD, that I will cause the sun to go down at noon, and I will darken the earth in the clear day: And I will turn your feasts into mourning, and all your songs into lamentation; and I will bring up sackcloth upon all loins, and baldness upon every head; and I will make it as the mourning of an only son, and the end thereof as a bitter day.

(Old Testament | Amos 8:9 - 14)

I am a witness of Jesus Christ and I am given liberty to ask and receive whatever I want from heaven, even to smite the earth with famine and plagues as often as I want it. And it will be granted. But, I wish no evil, for I am not totally exempt from suffering peril also, for the LORD is not a respecter of persons and not everyone in my family and my friends are obedient. This is why I preach repentance day and night, so that I am not compelled to ask for mischief when warranted. But the judgments usually come by the voice of the people and the desire in their hearts. And those scarab scoundrels want a famine. They represent the wickedness of the world you live in; and that is why I was concentrated in addressing them, so that they and not the people do suffer what the judgments of God approach us. Are you going to allow them to speak for you and drag you with them to woe and misery? Now of this it is also written:

And there were exceedingly many prophets among us. And the people were a stiffnecked people, hard to understand. And there was nothing save it was exceeding harshness, preaching and prophesying of wars, and contentions, and destructions, and continually reminding them of death, and the duration of eternity, and the judgments and the power of God, and all these things—stirring them up continually to keep them in the fear of the Lord.

I say there was nothing short of these things, and exceedingly great plainness of speech, would keep them from going down speedily to destruction. And after this manner do I write concerning them.

(Book of Mormon | Enos 1:22 - 23)

Now it is your turn, Sally, and as I said to turner, I hope you have heard the old proverb that says: As is the mother, so is her daughter, for I will judge you as women that break wedlock and shed blood are judged.

In other words I will compare you to Babylon. We in truth would have healed the world or Babylon from mischief with balm, from Gilead; yea, I am speaking about that great harlot that has spread corruption on the earth. And we are commanded to forsake her for her iniquity has reached up heaven. This is the fate that awaits her, for of her it is also written:

Babylon hath been a golden cup in the LORD's hand, that made all the earth drunken: the nations have drunken of her wine; therefore the nations are mad. Babylon is suddenly fallen and destroyed: howl for her; take balm for her pain, if so be she may be healed.

We would have healed Babylon, but she is not healed: forsake her, and let us go every one into his own country: for her judgment reacheth unto heaven, and is lifted up even to the skies.

The LORD hath brought forth our righteousness: come, and let us declare in Zion the work of the LORD our God.

Make bright the arrows; gather the shields: the LORD hath raised up the spirit of the kings of the Medes: for his device is against Babylon, to destroy it; because it is the vengeance of the LORD, the vengeance of his temple.

Set up the standard upon the walls of Babylon, make the watch strong, set up the watchmen, prepare the ambushes: for the LORD hath both devised and done that which he spake against the inhabitants of Babylon.

O thou that dwellest upon many waters, abundant in treasures, thine end is come, and the measure of thy covetousness. The LORD of hosts hath sworn by himself, saying, Surely I will fill thee with men, as with caterpillers; and they shall lift up a shout against thee. He hath made the earth by his power, he hath established the world by his wisdom, and hath stretched out the heaven by his understanding. When he uttereth his voice, there is a multitude of waters in the heavens; and he causeth the vapours to ascend from the ends of the earth: he maketh lightnings with rain, and bringeth forth the wind out of his treasures. Every man is brutish by his knowledge; every founder is confounded by the graven image: for his molten image is falsehood, and there is no breath in them. They are vanity, the work of errors: in the time of their visitation they shall perish.

(Old Testament | Jeremiah 51:7 - 18)

Wherefore, O harlot, hear the word of the LORD: Thus saith the Lord GOD; Because thy filthiness was poured out, and thy nakedness discovered through thy whoredoms with thy lovers, and with all the idols of thy abominations, and by the blood of thy children, which thou didst give unto them; Behold, therefore I will gather all thy lovers, with whom thou hast taken pleasure, and all them that thou hast loved, with all them that thou hast hated; I will even gather them round about against thee, and will discover thy nakedness unto them, that they may see all thy nakedness. And I will judge thee, as women that break wedlock and shed blood are judged; and I will give thee blood in fury and jealousy. And I will also give thee into their hand, and they shall throw down thine eminent place, and shall break down thy high places: they shall strip thee also of thy clothes, and shall take thy fair jewels, and leave thee naked and bare.

They shall also bring up a company against thee, and they shall stone thee with stones, and thrust thee through with their swords. And they shall burn thine houses with fire, and execute judgments upon thee in the sight of many women: and I will cause thee to cease from playing the harlot, and thou also shalt give no hire any more. So will I make my fury toward thee to rest, and my jealousy shall depart from thee, and I will be quiet, and will be no more angry. Because thou hast not remembered the days of thy youth, but hast fretted me in all these things; behold, therefore I also will recompense thy way upon thine head, saith the Lord GOD: and thou shalt not commit this lewdness above all thine abominations. Behold, every one that useth proverbs shall use this proverb against thee, saying, As is the mother, so is her daughter.

(Old Testament | Ezekiel 16:35 - 44)

And another angel shall sound his trump, saying: That great church, the mother of abominations, that made all nations drink of the wine of the wrath of her fornication, that persecuteth the saints of God, that shed their blood—she who sitteth upon many waters, and upon the islands of the sea—behold, she is the tares of the earth; she is bound in bundles; her bands are made strong, no man can loose them; therefore, she is ready to be burned. And he shall sound his trump both long and loud, and all nations shall hear it. And there shall be silence in heaven for the space of half an hour; and immediately after shall the curtain of heaven be unfolded, as a scroll is unfolded after it is rolled up, and the face of the Lord shall be unveiled;

(Doctrine and Covenants | Section 88:94 - 95)

Therefore, oh inhabitants of the earth, fear God and give Glory to him, for the hour of his judgment is come. These words have come through inspiration so that you know what awaits you as individuals, as a nation and as a world. I was not expecting to begin my work so soon, but it has come as a result of your own actions. In other words, you have unleashed a sequence of events that in the due time of the LORD will bring about the next apocalypse. For the law and the testimony has already been sealed even with blood.

Now therefore be ye not mockers, lest your bands be made strong: for I have heard from the Lord GOD of hosts a consumption, even determined upon the whole earth. Give ye ear, and hear my voice; hearken, and hear my speech. Doth the plowman plow all day to sow? doth he open and break the clods of his ground? When he hath made plain the face thereof, doth he not cast abroad the fitches, and scatter the cummin, and cast in the principal wheat and the appointed barley and the rie in their place? For his God doth instruct him to discretion, and doth teach him. For the fitches are not threshed with a threshing instrument, neither is a cart wheel turned about upon the cummin; but the fitches are beaten out with a staff, and the cummin with a rod. Bread corn is bruised; because he will not ever be threshing it, nor break it with the wheel of his cart, nor bruise it with his horsemen.

This also cometh forth from the LORD of hosts, which is wonderful in counsel, and excellent in working.

(Old Testament | Isaiah 28:22 - 29)

And these things I do not say of myself, but as the LORD has spoken, he shall also fulfill, and I do what I do, with a single eye for the honor and glory of God, with proper authority and in the Holy Name of Jesus Christ, amen.

Attentively, yours, an unworthy servant in the hands of Christ

Miguel Angel Tinoco Rodriguez

And I saw another angel fly in the midst of heaven, having the everlasting gospel to preach unto them that dwell on the earth, and to every nation, and kindred, and tongue, and people, Saying with a loud voice, Fear God, and give glory to him; for the hour of his judgment is come: and worship him that made heaven, and earth, and the sea, and the fountains of waters. And there followed another angel, saying, Babylon is fallen, is fallen, that great city, because she made all nations drink of the wine of the wrath of her fornication.

(New Testament | Revelation 14:6 - 8)