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Chaos theory

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and 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. The theory was summarized by Edward Lorenz as:

"Chaos: When the present determines the future but the approximate present does not approximately determine the future."

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. It also occurs spontaneously in some systems with artificial components, such as road traffic. This behavior can be studied through the analysis of a chaotic mathematical model or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory and self-assembly processes.

Introduction
Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.

Chaos theory is a method of qualitative and quantitative analysis to investigate the behavior of dynamic systems that cannot be explained and predicted by single data relationships, but must be explained and predicted by whole, continuous data relationships.

Chaotic dynamics
In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:

 * 1) it must be sensitive to initial conditions,
 * 2) it must be topologically transitive,
 * 3) it must have dense periodic orbits.

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In the discrete-time case, this is true for all continuous maps on metric spaces. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.

If attention is restricted to intervals, the second property implies the other two. An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.

Sensitivity to initial conditions
Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.

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 that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.

As suggested in Lorenz's book entitled The Essence of Chaos, published in 1993, "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions. A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).

A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead. This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach 100 C or fall below -130 C on earth (during the current geologic era), but we cannot predict exactly which day will have the hottest temperature of the year.

In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions. More specifically, given two starting trajectories in the phase space that are infinitesimally close, with initial separation $$\delta \mathbf{Z}_0$$, the two trajectories end up diverging at a rate given by

 * $$ | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,$$

where $$t$$ is the time and $$\lambda$$ is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.

Non-periodicity
A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior.

Topological mixing

Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps 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 only 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 eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

Topological transitivity
A map $$f:X \to X$$ is said to be topologically transitive if for any pair of non-empty open sets $$U, V \subset X$$, there exists $$k > 0$$ such that $$f^{k}(U) \cap V \neq \emptyset$$. Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point x and a region V, there exists a point y near x whose orbit passes through V. This implies that it is impossible to decompose the system into two open sets.

An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.

Density of periodic orbits
For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. 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, $$\tfrac{5-\sqrt{5}}{8}$$ → $$\tfrac{5+\sqrt{5}}{8}$$ → $$\tfrac{5-\sqrt{5}}{8}$$ (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 continuous one-dimensional system that 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 behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions leads 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 is 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 that, with a little imagination, looks like the wings of a butterfly.

Unlike fixed-point attractors and limit cycles, the attractors that 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 the fractal dimension can be calculated for them.

Coexisting attractors
In contrast to single type chaotic solutions, recent studies using Lorenz models have emphasized the importance of considering various types of solutions. For example, coexisting chaotic and non-chaotic may appear within the same model (e.g., the double pendulum system) using the same modeling configurations but different initial conditions. The findings of attractor coexistence, obtained from classical and generalized Lorenz models, suggested a revised view that "the entirety of weather possesses a dual nature of chaos and order with distinct predictability", in contrast to the conventional view of "weather is chaotic".

Minimum complexity of a chaotic system

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must 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 below is generated by a system of three differential equations such as:
 * $$ \begin{align}

\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. \end{align} $$ where $$x$$, $$y$$, and $$z$$ make up the system state, $$t$$ is time, and $$\sigma$$, $$\rho$$, $$\beta$$ are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott found a three-dimensional system with just five terms, that had only one nonlinear term, 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 shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can still exhibit some chaotic properties. Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional. A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.

The above set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model. Since 1963, higher-dimensional Lorenz models have been developed in numerous studies  for examining the impact of an increased degree of nonlinearity, as well as its collective effect with heating and dissipations, on solution stability.

Infinite dimensional maps
The straightforward generalization of coupled discrete maps is based upon convolution integral which mediates interaction between spatially distributed maps: $$\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}$$,

where kernel $$K(\vec r - \vec r^{,},t)$$ is propagator derived as Green function of a relevant physical system, $$ f [\psi_{n}(\vec r,t) ] $$ might be logistic map alike $$ \psi \rightarrow G \psi [1 - \tanh (\psi)]$$ or complex map. For examples of complex maps the Julia set $$ f[\psi] = \psi^2$$ or Ikeda map $$ \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} $$ may serve. When wave propagation problems at distance $$L=ct$$ with wavelength $$\lambda=2\pi/k$$ are considered the kernel $$K$$ may have a form of Green function for Schrödinger equation:.

$$ K(\vec r - \vec r^{,},L) = \frac {ik\exp[ikL]}{2\pi L}\exp[\frac {ik|\vec r-\vec r^{,}|^2}{2 L} ]$$.

Jerk systems
In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form
 * $$J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0$$

are sometimes called jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behavior. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.

One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another example of a jerk equation with nonlinearity in the magnitude of $$x$$ is:
 * $$\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.$$

Here, A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes:

In the above circuit, all resistors are of equal value, except $$R_A=R/A=5R/3$$, and all capacitors are of equal size. The dominant frequency is $$1/2\pi R C$$. The output of op amp 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative.

Similar circuits only require one diode or no diodes at all.

See also the well-known Chua's circuit, one basis for chaotic true random number generators. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system.

Spontaneous order
Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system. Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.

Moreover, from the theoretical physics standpoint, dynamical chaos itself, in its most general manifestation, is a spontaneous order. The essence here is that most orders in nature arise from the spontaneous breakdown of various symmetries. This large family of phenomena includes elasticity, superconductivity, ferromagnetism, and many others. According to the supersymmetric theory of stochastic dynamics, chaos, or more precisely, its stochastic generalization, is also part of this family. The corresponding symmetry being broken is the topological supersymmetry which is hidden in all stochastic (partial) differential equations, and the corresponding order parameter is a field-theoretic embodiment of the butterfly effect.

History

James Clerk Maxwell first emphasized the "butterfly effect", and is seen as being one of the earliest to discuss chaos theory, with work in the 1860s and 1870s. 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 that 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, called "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.

Chaos theory began in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by George David Birkhoff, Andrey Nikolaevich Kolmogorov,  Mary Lucy Cartwright and John Edensor Littlewood, and Stephen Smale. 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 to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "noise" was considered by chaos theorists as a full component of the studied systems. In 1959 Boris Valerianovich Chirikov proposed a criterion for the emergence of classical chaos in Hamiltonian systems (Chirikov criterion). He applied this criterion to explain some experimental results on plasma confinement in open mirror traps. This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos.

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. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 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.

Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz and his collaborator Ellen Fetter and Margaret Hamilton were using a simple digital computer, a Royal McBee LGP-30, to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They 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 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions.

In 1963, Benoit Mandelbrot, studying information theory, discovered that noise in many phenomena (including stock prices and telephone circuits) was patterned like a Cantor set, a set of points with infinite roughness and detail 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). 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 as 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 (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or snowflake, which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619). In 1982, Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory.

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, and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections. Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the 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 their inspiring achievements.

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 dysfunction among people with schizophrenia. 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 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 behavior. 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 behavior 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). 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 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. Also in 1987 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. Initially the domain 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 Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, involving many different disciplines such as mathematics, topology, physics, social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, pandemic crisis management, etc.

Lorenz's pioneering contributions to chaotic modeling
Throughout his career, Professor Lorenz authored a total of 61 research papers, out of which 58 were solely authored by him. Commencing with the 1960 conference in Japan, Lorenz embarked on a journey of developing diverse models aimed at uncovering the SDIC and chaotic features. A recent review of Lorenz's model progression spanning from 1960 to 2008 revealed his adeptness at employing varied physical systems to illustrate chaotic phenomena. These systems encompassed Quasi-geostrophic systems, the Conservative Vorticity Equation, the Rayleigh-Bénard Convection Equations, and the Shallow Water Equations. Moreover, Lorenz can be credited with the early application of the logistic map to explore chaotic solutions, a milestone he achieved ahead of his colleagues (e.g. Lorenz 1964 ).

In 1972, Lorenz coined the term "butterfly effect" as a metaphor to discuss whether a small perturbation could eventually create a tornado with a three-dimensional, organized, and coherent structure. While connected to the original butterfly effect based on sensitive dependence on initial conditions, its metaphorical variant carries distinct nuances. To commemorate this milestone, a reprint book containing invited papers that deepen our understanding of both butterfly effects was officially published to celebrate the 50th anniversary of the metaphorical butterfly effect.

A popular but inaccurate analogy for chaos
The sensitive dependence on initial conditions (i.e., butterfly effect) has been illustrated using the following folklore:

"For want of a nail, the shoe was lost.

For want of a shoe, the horse was lost.

For want of a horse, the rider was lost.

For want of a rider, the battle was lost.

For want of a battle, the kingdom was lost.

And all for the want of a horseshoe nail."

Based on the above, many people mistakenly believe that the impact of a tiny initial perturbation monotonically increases with time and that any tiny perturbation can eventually produce a large impact on numerical integrations. However, in 2008, Lorenz stated that he did not feel that this verse described true chaos but that it better illustrated the simpler phenomenon of instability and that the verse implicitly suggests that subsequent small events will not reverse the outcome. Based on the analysis, the verse only indicates divergence, not boundedness. Boundedness is important for the finite size of a butterfly pattern. In a recent study, the characteristic of the aforementioned verse was recently denoted as "finite-time sensitive dependence".

Applications

Although chaos theory was born from observing weather patterns, it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are geology, mathematics, biology, computer science, economics,  engineering,  finance,     meteorology, philosophy, anthropology, physics,   politics,  population dynamics, and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing.

Cryptography
Chaos theory has been used for many years in cryptography. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of cryptographic primitives. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers, watermarking, and steganography. The majority of these algorithms are based on uni-modal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys. From a wider perspective, without loss of generality, the similarities between the chaotic maps and the cryptographic systems is the main motivation for the design of chaos based cryptographic algorithms. One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory. Another type of computing, DNA computing, when paired with chaos theory, offers a way to encrypt images and other information. Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.

Robotics
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model. Chaotic dynamics have been exhibited by passive walking biped robots.

Biology
For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous, but recently scientists have been able to implement chaotic models in certain populations. For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth. Chaos can also be found in ecological systems, such as hydrology. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory. Another biological application is found in cardiotocography. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of fetal hypoxia can be obtained through chaotic modeling.

As Perry points out, modeling of chaotic time series in ecology is helped by constraint. There is always potential difficulty in distinguishing real chaos from chaos that is only in the model. Hence both constraint in the model and or duplicate time series data for comparison will be helpful in constraining the model to something close to the reality, for example Perry & Wall 1984. Gene-for-gene co-evolution sometimes shows chaotic dynamics in allele frequencies. Adding variables exaggerates this: Chaos is more common in models incorporating additional variables to reflect additional facets of real populations. Robert M. May himself did some of these foundational crop co-evolution studies, and this in turn helped shape the entire field. Even for a steady environment, merely combining one crop and one pathogen may result in quasi-periodic- or chaotic- oscillations in pathogen population.

Economics
It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task. Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.

Chaos could be found in economics by the means of recurrence quantification analysis. In fact, Orlando et al. by the means of the so-called recurrence quantification correlation index were able detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (regular) to turbulent (chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics. Finally, chaos theory could help in modeling how an economy operates as well as in embedding shocks due to external events such as COVID-19.

AI-Extended Modeling Framework
In AI-driven large language models, responses can exhibit sensitivities to factors like alterations in formatting and variations in prompts. These sensitivities are akin to butterfly effects. Although classifying AI-powered large language models as classical deterministic chaotic systems poses challenges, chaos-inspired approaches and techniques (such as ensemble modeling) may be employed to extract reliable information from these expansive language models (see also "Butterfly Effect in Popular Culture").

Other areas
In chemistry, predicting gas solubility is essential to manufacturing polymers, but models using particle swarm optimization (PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck. In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets. Four of the five moons of Pluto rotate chaotically. In quantum physics and electrical engineering, the study of large arrays of Josephson junctions benefitted greatly from chaos theory. Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately.

Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been attempted). Glass and Mandell and Selz have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior.

Researchers have continued to apply chaos theory to psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion's theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member.

Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.

In their 1995 paper, Metcalf and Allen maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r.

Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model.

By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions. Modern organizations are increasingly seen as open complex adaptive systems with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance, team building and group development is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.

Some say the chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.

Traffic forecasting may benefit from applications of chaos theory. Better predictions of when a congestion will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the BML traffic model at right).

Chaos theory has been applied to environmental water cycle data (also hydrological data), such as rainfall and streamflow. These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.

Articles

 * 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 that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
 * 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 that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
 * 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 that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
 * 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 that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).

Semitechnical and popular works

 * Christophe Letellier, Chaos in Nature, World Scientific Publishing Company, 2012, ISBN 978-981-4374-42-2.
 * 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.
 * 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.
 * 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.
 * David Peak and Michael Frame, Chaos Under Control: The Art and Science of Complexity, Freeman, 1994.
 * Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp.
 * Nuria Perpinya, Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político, Páginas de Espuma, 2021.
 * Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991.
 * Clifford A. Pickover, Chaos in Wonderland: Visual Adventures in a Fractal World, St Martins Pr 1994.
 * Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984.
 * David Ruelle, Chance and Chaos, Princeton University Press 1993.
 * Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.
 * Manfred Schroeder, Fractals, Chaos, and Power Laws, Freeman, 1991.
 * 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.
 * Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012.
 * Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993.
 * Manfred Schroeder, Fractals, Chaos, and Power Laws, Freeman, 1991.
 * 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.
 * Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012.
 * 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.
 * Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012.

title: Chaos theory


text:
Talk:Chaos theory/Archive 5

Confusing
Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth. I think that this article does not explain well how were all these systems determined to be chaotic.Lakinekaki 00:14, 14 February 2006 (UTC)

 * The atmosphere is chaotic because its dynamical behavior is governed by the Navier-Stokes equations, shown to have chaotic regimes (references in Ruelle's book); the dynamics of the solar system is governed by an n-body system of equations, known to have homoclinic tangles and to have a positive Lyapunov exponent under careful numerical simulations (references in Gutzwiller's book); for the Earth's core dynamics, an example is given by the dynamo equation; and for population growth the predator-prey equations are chaotic (most books discuss it). For the economy there is no generally accepted set of equations, but different time-series show predictability and chaotic features. References for the other items can easily be found. – XaosBits 13:50, 14 February 2006 (UTC)

 * Thanks a lot for your answers. However, I fail to see how are economic time series tested for chaotic features when definitions in the article apply to systems whose parameters are known, while we do not know parameters for economic systems. This question also applies to the Earth's core dynamics. Lakinekaki

 * Economics uses many different types of equations; some are classical differential equations, many are stochastic differential equations. Stochasic diff-eq are ordinary diff-eq with additional noise terms. They ave chaotic behaviours as well. These have been discussed in the Federal Reserve Economic Review (the subscription used to be free, once upon a time). Earth's core dynamics are beleived to be describably by means of fluid dynamics. Certainly, Earth's magnetization reversal time series exhibits chaotic behaviour. 22:11, 14 February 2006 (UTC)

 * But stochastic differential equations in economics seems characterises the behavior of the continuous time stochastic process Xt. I thought that stochastic process was different from chaotic one.Lakinekaki

 * Economic time-series helped motivate the development of phase space reconstruction techniques. These time-series were assumed to come from low-dimensional chaotic systems.  The effectiveness (or not) of the approach is today hidden behind proprietary methods employed by quants in Wall Street (and beyond).  There is another aspect in which chaotic dynamical systems are used in economics:  iterated games have chaotic dynamics.  (Look at Saari's work or at cond-mat/9806359.)

 * The dynamics of the Earth's interior is dominated by convective processes, which geophysicists model in chaotic regimes. – XaosBits 12:19, 15 February 2006 (UTC)

 * I interviewed for the job of a quant once. I had to bone up on Ito's integral before the interview. One paper, which at first seemed terribly boring, was about drill-bit breakage at a printed-circuit manufacturing plant. On the last page was a graph showing money saved per month by applying the solution to the stoch diff eq for drill breakage. It was labelled in units of $10K/month and I slapped my forehead, "holy shit, this kind of money will pay for a few PhD's in no time." linas 23:13, 15 February 2006 (UTC)

 * BTW, if you want to make money working on bios theory, get a job as a quant. These companies are willing to try any kind of math at all, any idea, as long as it can "predict" the stock market. linas 23:17, 15 February 2006 (UTC)

 * Thanks for answers and suggestion! Actually, I have a friend who is already trying to use ideas from bios to 'predict' the stock market. I don't know how succesful is he in his explorations. Near future will tell! There certainly are few features in bios that are absent in chaos and that economists could find useful. Since you mentioned money, I will not tell here which are those features! Interested parties can contact me for more details. :O) Lakinekaki

 * P.S. my scepticism about the claim that economic series are chaotic seems to be justified! From your answers, it seems more to be assumption than the fact, and more stochastic than chaotic. But that's OK. I'm sure chaos analysis is also useful in analyses of these series. Lakinekaki

 * Well really, the entire chaos theory is founded upon the idea, and presumably factual notion that different variables can and will yield different results, even if they are seemingly insignificant to the human eye. This concept applies to every scale... microscopic, atomic, sub-atomic, etc., and so, in the purely physical sense, it can be applied to any discipline.  As far as the economy goes, I suppose it depends on your point of view.  Economics is, in a sense, a purely abstract concept, whereas chaos theory is meant to be applied to -physical and tangible- instances, so if you are thinking of the economy in an abstract way, then no, chaos theory does not apply; however, in reality chaos theory does apply to the actual economy as it does everything else.  As I'm sure you can imagine, even the weather effects the economy.

Kachizzle

Verbiage on Attractors
Why is so much space spent in this article on the concept of an attractor? That's a general dynamical systems concept that doesn't have an inherent connection to chaos. Could it be refactored into a separate article, which could then be referenced when talking about strange attractors?

 * There are separate articles onm attractor and strange attractors. Why are they so detailed here, and not referenced?Lakinekaki 04:36, 7 April 2006 (UTC)

Sensitive dependence
Recent update added:

 * Consider the dynamical system defined on the real line by mapping x to 2x. This system has sensitive dependence on initial conditions everywhere ...

Hmmm ... I don't think this is true. The map x->2x has two fixed points - a repelling fixed point at 0 and an attractive fixed point at infinity. All initial points apart from 0 head towards infinity. With an appropriate metric (e.g. by mapping projective real line onto a circle) we can even say that any trajectory that does not start at 0 converges to infinity. That doesn't look like sensitive dependence to me. Maybe another example would be a better illustration of the point being made ? Gandalf61 17:08, 3 March 2006 (UTC)

 * I assumed we were using the usual metric on the real line, and that it did not include the point at infinity. Doesn't this go without saying? Maybe not... if that really seems unclear, I can state it explicitly. --Experiment123 19:07, 3 March 2006 (UTC)

 * You're both right. The x->2x is a terrible example; although one may argue it has a positve lyapunov exponent, the system is not a bounded system, duhh, so the positive exponent does not imply topological mixing or even ergodicity, so it sure won't be chaotic. linas 23:12, 3 March 2006 (UTC)

 * The point of the x->2x example is exactly what you're saying: it's clearly not chaotic, so sensitive dependence on initial conditions does not imply chaos. That's why it's a good example, not a terrible one! --Experiment123 02:21, 4 March 2006 (UTC)

 * I have added the definition of sensitive dependence on initial conditions to the page butterfly effect. It's from Devaney's book. — XaosBits 03:13, 4 March 2006 (UTC)

I'm just a newbie and not a very highly schooled mathematician, but I have two suggestions regarding this quote:

 * The systems behave identically only if their initial configurations are exactly the same.

First, I think it should read "system behaves." It refers to the trajectory after two different initial points, yet within the same chaotic system.

Second, I think that the words "behave(s) identically" is slightly misleading. The word "behavior" seems to invoke a concept of what the system does after it has been seeded with an initial condition. I think we should take care to avoid the implication that what follows from two different initial conditions cannot be identical.

Thoughts? --afuturehead 18:13, 26 June 2006 (UTC)

What concerns me about the example x -> 2x is that it is being used to justify the questionable claim that sensitive dependence "in itself is not particularly interesting". This is probably not a fair statement, and seems contradicted by this very discussion. One interesting thing about sensitive dependence is that it depends on the metric being used, which is implicit but unremarked in the definition in butterfly effect. Since the mapping x -> 2x fails to exhibit sensitive dependence when viewed in the compactified line, and indeed is not a chaotic map, this would seem to suggest that definition of "sensitive dependence" should be applied only to bounded metrics. Orbits converging to infinity are not in the spirit of the concept, since these are convergent in the compactification.

Part of the difficulty here is that the article is trying to achieve a non-technical discussion of a topic really requiring precise technicalities to get right. A proper discussion should probably take place within the setting of a compact invariant set -- but these words are not really lay terms. Maybe I'll have a go at it and you can see what you think. --Widenet 02:04, 28 June 2006 (UTC)

Dissipative
I just googled to find some more info on "chaos theory" in conjunction with dissipative structures. I found an old cached version of this article there, but that info has been deleted from this section, and the old cached version did not say much. Perhaps one of you experts on the area could direct me to a suitable website or book? I have some familiarity with both, but I am looking for some more examples here.DanielDemaret 11:06, 5 March 2006 (UTC)

Chaos theory and complex systems
I just take a brief look to this topic, and I found something to review about categorization. I only add an external link to Santa Fe Institute, but I thing that the links between "chaos theory" and "complex systems theory" are really deeper and that their topics and sub-topics deserve much more cross-linking (perhaps in a common category) just to make browsing easier (I can't do it by myself, since I am far from competent about and I'm working about "very complex system" of horsemanship...)--Alex brollo 15:39, 9 March 2006 (UTC)

 * I beleive that maybe 20 or 30 years ago, "chaos theory" and "complex systems" were almost synonyms. I don't beleive that's true anymore; chaos has a much narrower definition: these days, it tends to focus on simple systems, not complex systems. This is in part needed to make headway: one wants to study the simplest possible system that is still chaotic. By contrast, in the field of "complex systems", one is instead looking for ways of discerning the simple parts of a complex system. These simple parts may, or may not be chaotic. So they'r two differnt fields of study. linas 02:01, 10 March 2006 (UTC)

 * This is part of why mathematicians increasingly prefer the term nonlinear dynamics rather than chaos theory to describe what they study: it's less likely to be confused with something else, in addition to sounding less metaphysical. --Delirium 08:44, 15 March 2006 (UTC)

Poll: Should 'See also' section include the link Bios theory or not?
Hello! As a result of prior and ongoing discussions, this poll is to identify a course of action regarding the link on Bios theory in the 'See also' section of Chaos theory article.

At least 2 options are below. Wikipedians can only choose one option and should indicate their preference by signing in the appropriate section with four tildes (~) followed by an optional, one-sentence reasoning. Please only assert positive votes, do not indicate negative ones.

Voting will continue to 1 April 2006 23:59 UTC, but may be extended beyond that if any option does not garner a clear plurality of support.

Thanks for your co-operation! Lakinekaki 18:59, 14 March 2006 (UTC)

Option 1: 'See also' section should include Bios theory link.

 * David Kernow 19:53, 14 March 2006 (UTC). I'd say there needs to be some sort of link, even if to indicate why "bios theory" shouldn't be freely associated with chaos theory. From perusing Bios theory, it seems to me that "bios theory" is a subset of chaos theory, namely all those (models of) instances occuring in biologically-related areas. (Re the article's current tag, I'll offer to rephrase parts of it if that approach is acceptable.)
 * — Arthur Rubin | (talk) 18:45, 15 March 2006 (UTC). Bios theory seems to be a fork of a subset of chaos theory.  If it's legitimate (which I cannot determine without reading articles in journals I don't have access to), and is to be retained as a separate article, there should be a link.
 * That's what categories are for. Septentrionalis 20:45, 15 March 2006 (UTC)
 * Salix alba (talk) 10:09, 21 March 2006 (UTC). A "See also" link seems to be the optimal solution. Bios theory is distinct enough to warrant its own page as it deals with a specific class of chaotic behaviour. More extensive coverage here would give it undue prominence.

Option 2: 'See also' section should not include Bios theory link.

 * Experiment123 19:09, 14 March 2006 (UTC) "See also" links should relate to topics of sufficient importance to be covered in standard textbooks on the subject.
 * WP:DNFT Septentrionalis 20:23, 14 March 2006 (UTC)
 * XaosBits 22:32, 15 March 2006 (UTC)  I have explained in some detail the issues with Bios theory: the problem in the definition of novelty, one of the main distinguishing feature between chaos and bios; the lack of literature in journals with impact; the lack of mention of Bios theory in Mathematical Reviews in the seven years since the first article; and the problems with the Bios theory entry.  These topics are discussed in the Talk:Chaos theory and the Talk:Bios theory pages
 * Especially in major "subject overview" articles like this one, IMO "see also" should link to related articles as defined by what major textbooks and review/survey papers on the subject consider related. I'm not aware of "Bios theory" having such a status. --Delirium 04:04, 16 March 2006 (UTC)
 * I believe the Chaos Theory is a very complex and evasive theory..a theory also known as the butterfly effect, plays a big role in our society..there is much chaos in the world today, and much of this is none to be random..and in the chaos theory, chaos is considered to be random..so i feel there is a connection between these two.. Josh
 * Gandalf61 08:25, 18 March 2006 (UTC) Was undecided on this, but seeing the merge proposal appear again has made my mind up. Please don't merge, don't even link, keep Bios theory entirely separate so that Lakinekaki can keep his own private playground and the rest of us don't have to waste our time on what is, at best, an incoherent minority POV, and at worst verges on monomania and trolling.

Option 3: anyone who likes Bios theory should go and improve it instead of having polls

 * Wikipedia is not a democracy. We shouldn't even be voting on this issue. The whole thing smells bad, we've hashed this into little itty-bitty bits on the various talk pages, and absolutely zero effort has been made to fix the dozens of problems that the bios theory page has. Until that page is fixed, we shouldn't even be going here. linas 00:23, 16 March 2006 (UTC)
 * Agree William M. Connolley 16:17, 16 March 2006 (UTC)
 * Hi guys, I just invested quite a time in improving it. Please tell me what you think. Lakinekaki
 * I'm sorry, but the bios article makes no sense. After staring hard at it I can't even tell whether the definitions are supposed to apply to time series or to dynamical systems. For example, the first criterion for biotic motion is sensitivity to initial conditions--ok, so the author is talking about dynamical systems. But the later we see, "the distribution of biotic series is assymetrical [sic]". The more I thought about this, the less sense it made. There may perhaps be something to bios theory, but if so, this article does not do it justice. --Experiment123 22:08, 18 March 2006 (UTC)
 * The series generated by these systems have asymetric distribution. What is so nonsensical about that?!? Lakinekaki
 * That requires a sensible definition, also. — Arthur Rubin | (talk) 16:23, 19 March 2006 (UTC)

 * Please don't expect me to find answers to all your questions. I wrote Bios theory article as best as I could. I am sure if I spend more time reading the sources that I could make that article even better. However, what is important for me is that it is clear enough and understandable (and I think it is). Reader can get an idea about the subject, and if interested, will go and look for the sources him/herself. Probably some definitions that I wrote are not perfect, if you know how to improve them, please do so. Maybe even bios theory definitions are not perfect, and will evolve further. So what? Even the fundamental theorem of algebra went thru so many revisions by the best mathematicians. That's what science is all about. Wikipedia is about presenting current state of science and its history.

 * This quote is from the book Chaos and time-series analysis, Julien Clinton Sprott, Oxford University Press. 2003. (page 104, first paragraph):
 * Although there is no universally accepted definition of chaos, most experts would concur that chaos is the aperiodic, long-term behaviour of a bounded, deterministic system that exhibits sensitive dependence on initial conditions.

 * Although introduction to this article gives few definitions, it is not clear how well are all of them accepted. Lakinekaki 21:04, 20 March 2006 (UTC)

 * So I think we see the problem: its not just that you can't spell asymmetric, its that you've got no idea what an asymmetric distribution is. Thats fine, in general, you don't have to know these things. But you *do* need to know them if you're writing articles about them. William M. Connolley 21:53, 20 March 2006 (UTC)

 * You are twisting what I said. Lakinekaki

 * My point is that chaos is not so well defined, and yet, you expect bios to be. In the Mathematical bios paper by Kauffman and Sabelli, it sais: In contrast to symmetric random processes, periodic processes and chaos, bios generates asymmetric statistical distribution. They did not define asimmetry there, and I am not going to do it here either. That would be Original Research. Few of you are accusing me of being a troll, and yet, you are behaving like that, not me. Stop bothering me! It's not that you don't understand things, you don't want to understand them. You are trying to dissaprove something, by doing OR, that does not fit your view of the world. I don't care! Things have been published like that, and I'm not gonna fix them for you. Lakinekaki

I think you'll find that Chaos is well defined, but you neither know nor understand the definition. Don't you get bored typing all those colons and long for a fresh start sometimes? Asymmetry: well done, you spelt it right 1 out of 2, thats progress. But you still don't know what it means. Defining what it means in this context would *not* be original research - unless you do what you appear to propose, ie, make the definition up out of your own head. What you need to do is understand what your sources mean by it. And you should do that *before* you write about it. I become more and more convinc3ed that you haven't got a clue what any of this is about, but you just like the words for some reason. Wiki is vast: why not write articles about things you *do* understand? William M. Connolley 19:53, 21 March 2006 (UTC)

 * My answer is here, for those who have patience and time to read. Lakinekaki 23:19, 21 March 2006 (UTC)

 * I would have no problem linking to Bios theory, if that article were even remotely coherent. --- GWO

Serious problems concerning Bios theory and User:Lakinekaki
Look out! In Talk:Bios theory, I have just listed troubling evidence suggesting serious problems with the claims made in Bios theory and an apparent conflict of interest on the part of User:Lakinekaki. See also WP:NPOV [[WP:VAIN WP:VERIFY and WP:RS. ---CH 06:36, 12 May 2006 (UTC)

Merging
I just wanted to voice my opposition to merging bios theory with this article. Bios theory is such a fringe topic, if anything at all, that it does not deserve that much attention in an article on such a broad field as chaos theory. -- Jitse Niesen (talk) 11:08, 18 March 2006 (UTC)

I agree with Jitse Niesen. One link to it should be enough.Lakinekaki

Wow - what a discussion
Guys, although Chaos and Bios theories may be treated as related they are slightly different in scope, how they are used and their targets. They themselves and articles about them will likely evolve further, possibly away from one another in actual subject area. As such these two articles should not be merged.

Should they be listed in one another's "see also" section? Well, tell me, in general, what needs to be listed in any article's "See also" section and then we'll be able to give an answer. I don't have a clear definition myself, but can say that "See also" entries are as frequently useful as they are not and that is quite OK. It is supposed to be that way because of different interests. "See also" sections frequently include sub-classifications, uses or similarly related articles not listed anywhere else in the article we started from. Chaos theory is used for many things and it does not hurt to mention it. It gives the reader more ways to comprehend why it is there and what purposes it serves. In this regard, I think "Bios" theory should be included in the See also section of this article.

"See also" section should also be short. Having too many entries defies its main purpose - giving the reader a finite set of paths to follow up. In that regard, if we have articles A, B and C such that article B appears in A's "see also" list and C appears in B's "see also" list, then we might not need to also list C in A if the list is already long. Example? See, for example, article Bird. One can probably find better examples, but this is the one I just thought of.

Since "See also" of this article is not that long yet, I still don't see any problem in including "Bios" in its "See also" list. When and if it starts growing, this may need to be reconsidered and, perhaps, more infrastructure articles added for this alone - e.g. "List of theories based on Chaos Theory" vs. "List of direct uses Chaos theory", etc. Until then just cool down.

--Aleksandar Šušnjar 18:25, 21 March 2006 (UTC)

 * Did you actually read the page on bios theory? Maybe there is such a thing, but the current WP page on it is utter bunkum. Until someone can describe bios theory so that it doesn't sound like pseudoscience, it certainly should not be referenced here. linas 15:49, 22 March 2006 (UTC)

What shell we do now?
Shell we have the link on Tuesdays, Thursdays and Saturdays, and not have it on Mondays, Wednesdays, Fridays and Sundays? :O) Lakinekaki
 * Let's see. You called a poll; it came out 6-3 against you, and your efforts to suggest a merge for your faorite theory are consistently overridden. You now want equal time. No, we shall not violate WP:NPOV. Verbum sap. sufficit. Septentrionalis 05:51, 5 April 2006 (UTC)
 * It's more like 6-5 (if you count Aleksandar's and my vote), or 8-5 if option 3 (improve) votes are considered against. From WP:NPOV: ...Articles that compare views need not give minority views as much or as detailed a description as more popular views... Noone is comparing views here, or trying to put description of bios in chaos article. We are talking about a link in a See also section.Lakinekaki 16:01, 5 April 2006 (UTC)

 * No matter which way you cut it, the majority result of the poll was a decision not to include a link to the bios theory article. Why did you propose a poll if you are not prepared to accept its result ? Are you simply trying to stir up controversy ? Gandalf61 09:05, 6 April 2006 (UTC)

 * I am accepting result of the poll. I was just hoping more people would vote and give some option significant majority. This way it's more like no concensus.Lakinekaki 14:48, 6 April 2006 (UTC)

3 body and attractors
Did I misplaced the sentence about 3 body problem? Are they behaving as chaotic attractor, or just chaotic? I know that Solar system is attractor. Lakinekaki 06:04, 7 April 2006 (UTC)

 * The word chaotic often substitutes for another technical term. Different dynamical systems are chaotic in different ways: as  Axiom A systems, as  mixing systems, as  uniform hyperbolic, etc.  So when one reads in non-technical discussions that the three body problem is a chaotic dynamical system, what is meant is that the phase space of the three body problem is a mixed phase space with regions of periodic behavior and regions that have embedded horseshoes.  That means that there is a neighborhood of initial conditions of the three body problem that behave just like a uniform hyperbolic system.  So rather than say all this, it is common to say that the system is chaotic.  It is similar to usage of smooth when referring to differentiable functions. — XaosBits 22:35, 7 April 2006 (UTC)

Commentary on definition
The text that was moved:

Gleick (1989, p. 306) wrote that No one [of the chaos scientists he interviewed] could quite agree on [a definition of] the word [chaos] itself,. That's why he gave descriptions from a number of practitioners in the field some of which are: the complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems (refering to "chaotic" behaviour); a kind of order without periodicity (refering to chaos), etc.

Textbooks devoted to chaos do not really define the term. Some descriptions include:

 * A dynamical system displaying sensitive dependence on initial conditions on a closed invariant set (which consists of more than one orbit) will be called chaotic.

 * ''By a chaotic solution to a deterministic equation we mean a solution whose outcome is very sensitive to initial conditions (i.e., small changes in initial conditions lead to great differences in outcome) and whose evolution through phase space appears to be quite random."

 * The very use of the word 'chaos' implies some observation of a system, perhaps through measurement, and that these observations or measurements vary unpredictably. We often say observations are chaotic when there is no discernible regularity or order.

 * Chaotic behavior can be predicted only if the initial conditions are known to an infinite degree of accuracy, which is impossible.(Jerry Gollub and Thomas Solomon, Academic American Encyclopedia.)

Rather than to give a precise definition of chaos it is much easier to list properties that a system described as "chaotic" has.

(Moved by XaosBits 22:49, 7 April 2006 (UTC))

 * The text that XaosBits has removed is very close to a straight lift from mathworld, and so could be copyvio. If we are going to include a discussion of the definition of chaos, at least let's find our own quotes. Gandalf61 11:05, 8 April 2006 (UTC)

I was asked by Lakinekaki to explain why I moved the text.

The issue of the definition of chaos has been worked out in the literature. As I mentioned above, the apparent difficulty is that the term chaos is used, when referring to dynamical systems, the same way the term smooth is used when referring to the differentiability of functions. It serves as a shorthand for other technical terms that mean almost the same thing. One author may use the term chaotic for a uniform hyperbolic system, whereas another author may use the term chaotic for a Hamiltonian system that has developed Cantori.

Gleick's book, while a bestseller and helpful in providing an overview of the excitement in the field in the late eighties, is not a mathematical textbook. When trying to ascertain the definition of chaos from textbook, one needs to take into considerations the compromises authors are faced when balancing simplicity of explanation with precision.

Current authors seem to have settled on Devaney's definition, which is the one in the Wikipedia article. In the mid-nineties that definition was simplified in terms of periodic orbits. Devaney's definition focuses on the points of the phase space. If instead the focus is on the measures supported by the dynamics, then two possible definitions for a chaotic dynamical system are common: (1) it has a mixing measure; or the more restrictive (2) it has an SRB measure.

One difficulty does arise when studying systems numerically. It is common to take the concepts that are well defined for an Axiom A system, say the concept of shadowing, and use the concept in a system that appears to be Axiom A. Usually the numerical experiments will show that the concept could be extended to this new situation in some probabilistic sense. The difficult task of making clear the probabilistic sense is left for future work. It is a hard task to define the class of systems used in these numerical experiments. — XaosBits 13:08, 8 April 2006 (UTC)

Newton
Now here I have to disagree. The roots of theories don't originate in solutions to problems, but in the formulation of problems themselves. Henry was solving the problem that dated back to Newton and that others could not solve. Question was asked by Newton, answer was given by Henry. Lakinekaki 15:09, 8 April 2006 (UTC)

 * I have not read Principia Mathematica, only little bits, but it does not seem to me that Newton would be discussing predictability. Maybe Lakinekaki could quote. — XaosBits 02:43, 9 April 2006 (UTC)

 * Other readers should be aware that many of the historical statements inserted by user Lakinekaki are incorrect.  He is prone to reverse wars, which makes it difficult to fix the main text. XaosBits 04:49, 9 April 2006 (UTC)

 * This is an accusation. Please be specific which historical statements that I wrote are incorect. Lakinekaki 06:27, 9 April 2006 (UTC)

 * I can find no evidence that Newton studied the three-body problem. Wikipedia and scienceworld both agree that the first published studies were by Euler in 1760 and Lagrange in 1772, by which time Newton was dead. I will look through my translation of the Principia to see I can find anything, but I think Newton's character makes it unlikely he would openly discuss a problem that he could not solve. Lakinekaki - do you have a reference ? If we have no evidence, I think it is going too far to say that Newton tried to solve the three-body problem - the most we can say is that he may have studied the three-body problem, since it is a natural next step after the two-body problem. Gandalf61 09:43, 9 April 2006 (UTC)

 * Here are refs:

 * +math magazine

 * Astronomy & Geophysics, Volume 41 Page 6.21 - December 2000

 * Success and failure in Newton's lunar theory, Prof. Sir Alan Cook FRAS

 * 'Newton's theory of the Moon's orbit is often seen as a failure, but that is misleading. As the first statement of the three-body problem and the first attempt at a solution, it was for the most part remarkably successful. The outstanding defect was the theory of the rotation of the apse line. Halley (1731), towards the end of his life, asserted that Newton's theory was correct to 2 arcmin, and materials of his in the archives of the Royal Observatory, recently analysed, show that estimate to be justified for the period 1725 to 1730 (Cook 1996, 1998). What did Newton achieve in his lunar theory, why did he fail conspicuously in one problem, and why should we still be interested in his achievements?'

 * Lakinekaki 15:45, 9 April 2006 (UTC)

 * know what guys, do whatever you want. i thought i was improving the article, but you don't seem to like my edits. you can also erase that animated image if you like, and sarkovskii's connection, and Yoshisuke Ueda paragraph, and every other thing i added. i dont' care, you've been quite hostile to me, and i'm not gonna bother any more. Lakinekaki 15:53, 9 April 2006 (UTC)

 * Another ref for newton having though about the three body problem is http://www-groups.dcs.st-and.ac.uk/~history/PrintHT/Orbits.html from the farly respecatable st-andrews histoy of maths site
 * In the Principia Newton also deduced Kepler's third law. He looked briefly (in Propositions 65 and 66) at the problem of three bodies. However Newton later said that an exact solution for three bodies "exceeds, if I am not mistaken, the force of any human mind. "
 * --Salix alba (talk) 15:58, 9 April 2006 (UTC)

 * Thank you - that was exactly what I was looking for. Going back to the Principia, these parts actually make interesting reading. In Proposition 64 Newton solves the three-body problem for the case where mutual attraction is proportional to separation - an academic exercise, since this is not a physically realistic model. Then in Propositions 65 and 66 he talks about the three-body problem for an inverse square law force. In contrast to the very precise geometric proofs in the rest of Book 1, these Propositions are qualitative discussions of special cases - what we would today call "hand waving". Gandalf61 09:31, 10 April 2006 (UTC)

I've reverted to XB. It seems clear enough that whether or not Newton studied the 3-body problem, he didn't study chaos. Identifiying a problem as difficult is quite a different matter. William M. Connolley 16:38, 9 April 2006 (UTC)

Shall we have a Cultural Refrences Section?
There is a book that has a refrence to Chaos theory. I think it was Jurrasic Park (god I hope I spelt that right). 68.228.33.74 08:00, 30 May 2006 (UTC)

I have created a page about the Chaos Theory of Literature which makes reference to JP, in the context of Harriett Hawkins' book Strange Attractors. -- Scartol 17:03, 29 June 2006 (UTC)

Irrelevant links
I removed the following external links, which I considered to be irrelevant:
 * Nonlinear Dynamics at Drexel
 * Bristol Centre for Applied Nonlinear Mathematics
 * Linear and Nonlinear Dynamics and Vibrations Laboratory at the University of Illinois

If anyone disagrees, let me know. If these links are to be used at all, I would recommend putting them in some kind of "list of Nonlinear Dynamics research groups" or something. I don't think they would be immediately useful to someone looking up information about chaos theory. --BryanD 20:04, 6 June 2006 (UTC)

Chaos for the future
If you could calculate the chaos of one person, could you determine ones future?

 * I guess in theory you could, but then again no one knows if chaos theory even exists. That's why it's called chaos theory, not the Law of Chaos or something. It would also be impossible to predict the "chaos" of a person. Chaos is usually figured around things that have set paths to choose. For instance, the water mill example. The water mill, even if it's chaotically rotating from left to right when the water flow is increased to a high amount, the mill only has two directions to turn no matter what. It can only turn left and right, so the chaos of which direction is contained within those two directions. Humans, however, having freedom of choice, always have many options to choose from. For instance, let's say you were in a room and there were two doors leading separate ways. You could go through one of the doors, you could go into a door then come back out, you could use the other door, you could open a door and then shut it without going through it, you could do nothing at all, you could jog in place, etc. As you can see, the possibilities for us are endless, and thus the chaos is not contained by any specific perameters. Thus, even if chaos theory were true, it would be impossible to actually read one's future by determining their "chaos". 24.15.53.225 02:38, 3 June 2007 (UTC)

Chaos theory and Theory of biological evolution
I have tried to find articles about how Chaos Theory can be applied on Evolution. I wish to use an argument that Evolution by Natural Selection from a Common decent is possible because certain systems in physics bring Chaos from Order as the Double Pendulum. Is this a valid argument?

--Msitua 17:33, 6 August 2006 (UTC)

Three body GIF
I find that the GIF animation illustrating the three body problem adds very little to the text, so I have removed it. The three body problem played an important role in the development of dynamical systems and it could provide a great illustration. A direct simulation may be interesting, if run for a longer time. This may be impractical with a GIF, but there may be other solutions that do not overload the page. – XaosBits 02:56, 7 August 2006 (UTC)

The Chaos movie about edward lorenz
Is not about edward lorenz its an action movie...

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Chaos theory/Archive 5


text:
Cupola

In architecture, a cupola is a relatively small, most often dome-like, tall structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome.

The word derives, via Italian, from lower Latin cupula (classical Latin cupella), (Latin cupa), indicating a vault resembling an upside-down cup.

Background
The cupola evolved during the Renaissance from the older oculus. Being weatherproof, the cupola was better suited to the wetter climates of northern Europe. The chhatri, seen in Indian architecture, fits the definition of a cupola when it is used atop a larger structure.

Cupolas often serve as a belfry, belvedere, or roof lantern above a main roof. In other cases they may crown a spire, tower, or turret. Barns often have cupolas for ventilation.

Cupolas can also appear as small buildings in their own right.

The square, dome-like segment of a North American railroad train caboose that contains the second-level or "angel" seats is also called a cupola.

On armoured vehicles
The term cupola can also refer to the protrusions atop an armoured fighting vehicle due to their distinctive dome-like appearance. They allow crew or personnel to observe, offering very good all round vision, or even field weaponry, without being exposed to incoming fire. Later designs, however, became progressively flatter and less prominent as technology evolved to allow designers to reduce the profile of their vehicles.

title: Cupola