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Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic
laws Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. ...
of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s that are highly sensitive to initial conditions, and 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 Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled c ...
, 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 that 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 Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, 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 In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
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: 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 the
road traffic Traffic comprises pedestrians, vehicles, ridden or herded animals, trains, and other conveyances that use public ways (roads) for travel and transportation. Traffic laws govern and regulate traffic, while rules of the road include 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 Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
,
anthropology Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, societies, and linguistics, in both the present and past, including past human species. Social anthropology studies patterns of behavi ...
, sociology,
environmental science Environmental science is an interdisciplinary academic field that integrates physics, biology, and geography (including ecology, chemistry, plant science, zoology, mineralogy, oceanography, limnology, soil science, geology and physical geograp ...
, computer science, engineering, economics, ecology, and pandemic
crisis management Crisis management is the process by which an organization deals with a disruptive and unexpected event that threatens to harm the organization or its stakeholders. The study of crisis management originated with large-scale industrial and envir ...
. 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 In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. Use ...
. 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 Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
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 Robert Luke Devaney (born 1948) is an American mathematician, the Feld Family Professor of Teaching Excellence at Boston University. His research involves dynamical systems and fractals.. Education and career Devaney was born on April 9, 1948, a ...
, says that to classify a dynamical system as chaotic, it must have these properties: # it must be sensitive to initial conditions, # it must be topologically transitive, # it must have
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
maps on
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s. 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 The American Association for the Advancement of Science (AAAS) is an American international non-profit organization with the stated goals of promoting cooperation among scientists, defending scientific freedom, encouraging scientific responsi ...
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 or fall below 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 In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
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 A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traje ...
in the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
that are infinitesimally close, with initial separation \delta \mathbf_0, the two trajectories end up diverging at a rate given by : , \delta\mathbf(t) , \approx e^ , \delta \mathbf_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 The K-system is an audio level measuring technique proposed by mastering engineer Bob Katz in the paper "An integrated approach to Metering, Monitoring and Levelling". It proposes a studio monitor calibration system and a set of meter ballistic ...
.


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 In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
initial conditions, the variable evolves chaotically with non-periodic behavior.


Topological mixing

Topological mixing In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appear ...
(or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
of its
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored
dye A dye is a colored substance that chemically bonds to the substrate to which it is being applied. This distinguishes dyes from pigments which do not chemically bind to the material they color. Dye is generally applied in an aqueous solution and ...
s 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 set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
s U, V \subset X, there exists k > 0 such that f^(U) \cap V \neq \emptyset. Topological transitivity is a weaker version of
topological mixing In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appear ...
. 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 In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \ma ...
, 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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
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 → \tfrac → \tfrac (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 In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the ...
). 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 In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
, 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 In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
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 Lorenz is an originally German name derived from the Roman surname Laurentius, which means "from Laurentum". Given name People with the given name Lorenz include: * Prince Lorenz of Belgium (born 1955), member of the Belgian royal family by ...
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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
structure, and the
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is mea ...
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. Universality of one-dimensional maps with parabolic maxima and
Feigenbaum constants In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. Histo ...
\delta=4.669201...,\alpha=2.502907... is well visible with map proposed as a toy model for discrete laser dynamics: x \rightarrow G x (1 - \mathrm (x)), where x stands for electric field amplitude, G is laser gain as bifurcation parameter. The gradual increase of G at interval
_with_qualitatively_the_same_
_with_qualitatively_the_same_bifurcation_diagram">,_\infty)_changes_dynamics_from_regular_to_chaotic_one_with_qualitatively_the_same_bifurcation_diagram_as_those_for__logistic_map. In_contrast,_for_continuous_function_(topology).html" "title="bifurcation_diagram.html" ;"title=", \infty) changes dynamics from regular to chaotic one with qualitatively the same bifurcation diagram">, \infty) changes dynamics from regular to chaotic one with qualitatively the same bifurcation diagram as those for logistic map. In contrast, for continuous function (topology)">continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
dynamical systems, the
Poincaré–Bendixson theorem In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an o ...
shows that a strange attractor can only arise in three or more dimensions.
Finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
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 In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. Theorem Given a differentiable real dynamical system defined on an o ...
states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s such as: : \begin \frac &= \sigma y - \sigma x, \\ \frac &= \rho x - x z - y, \\ \frac &= x y - \beta z. \end 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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 Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior. 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 elegant 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_(\vec r,t) = \int K(\vec r - \vec r^,t) f psi_(\vec r^,t) ^, where kernel K(\vec r - \vec r^,t) is propagator derived as Green function of a relevant physical system, f psi_(\vec r,t) might be logistic map alike \psi \rightarrow G \psi - \tanh (\psi)/math> or complex map. For examples of complex maps the Julia set f psi= \psi^2 or
Ikeda map In physics and mathematics, the Ikeda map is a discrete-time dynamical system given by the complex map : z_ = A + B z_n e^ The original map was proposed first by Kensuke Ikeda as a model of light going around across a nonlinear optical res ...
\psi_ = A + B \psi_n e^ 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 \exp frac /math>.


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,\ddot,\dot ,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+A\frac+\frac-, 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 An operational amplifier (often op amp or opamp) is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op amp produces an output potential (relative to c ...
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 Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic ...
, 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 Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of
Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists ...
' pendulums, fireflies, neurons, the
London Millennium Bridge The Millennium Bridge, officially known as the London Millennium Footbridge, is a steel suspension bridge for pedestrians crossing the River Thames in London, England, linking Bankside with the City of London. It is owned and maintained by Brid ...
resonance, and large arrays of Josephson junctions.


History

An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits 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 In physics and mathematics, the Hadamard dynamical system (also called Hadamard's billiard or the Hadamard–Gutzwiller model) is a chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamard in 1898, and studied by Ma ...
". 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 In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
. 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 Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, Mary Lucy Cartwright and John Edensor Littlewood, and Stephen Smale. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident 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 Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
" was considered by chaos theorists 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. 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 Ellen Cole Fetter Gille is an American computer scientist. She worked with Edward Norton Lorenz on chaos theory. Early life and education Fetter was born to Frank Whitson Fetter and Elizabeth Garrett Pollard. Her mother created an endowment ...
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 attractor The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the L ...
s, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions. In 1963,
Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
found recurring patterns at every scale in data on cotton prices. Beforehand he had studied information theory and concluded noise was patterned like a
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). This challenged the idea that changes in price were normally distributed. In 1967, he published "
How long is the coast of Britain? Statistical self-similarity and fractional dimension "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper by mathematician Benoit Mandelbrot, first published in ''Science'' on 5 May 1967. In this paper, Mandelbrot discusses self-similar curves that ...
", 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
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...
ly 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 In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...
(examples include the
Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional S ...
, the Sierpiński gasket, and the Koch curve or ''snowflake'', which is infinitely long yet encloses a finite space and has a
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is mea ...
of circa 1.2619). In 1982, Mandelbrot published ''
The Fractal Geometry of Nature ''The Fractal Geometry of Nature'' is a 1982 book by the Franco-American mathematician Benoît Mandelbrot. Overview ''The Fractal Geometry of Nature'' is a revised and enlarged version of his 1977 book entitled ''Fractals: Form, Chance and Dimen ...
'', 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 Pierre C. Hohenberg (3 October 1934 – 15 December 2017) was a French-American theoretical physicist, who worked primarily on statistical mechanics. Hohenberg studied at Harvard, where he earned his bachelor's degree in 1956 and a master's degree ...
, presented his experimental observation of the
bifurcation Bifurcation or bifurcated may refer to: Science and technology * Bifurcation theory, the study of sudden changes in dynamical systems ** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow * River bifurcation, the for ...
cascade that leads to chaos and turbulence in
Rayleigh–Bénard convection In fluid thermodynamics, Rayleigh–Bénard convection is a type of natural convection, occurring in a planar horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. ...
systems. He was awarded the
Wolf Prize in Physics The Wolf Prize in Physics is awarded once a year by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Mathematics, Medicine and Arts ...
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 Bernardo Huberman is a Fellow and vice president of the Next-Gen Systems Team aCableLabs He is also a Consulting Professor in the Department of Applied Physics and the Symbolic System Program at Stanford University. He received his Ph.D. in Physi ...
presented a mathematical model of the
eye tracking Eye tracking is the process of measuring either the point of gaze (where one is looking) or the motion of an eye relative to the head. An eye tracker is a device for measuring eye positions and eye movement. Eye trackers are used in research ...
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 cycle The cardiac cycle is the performance of the human heart from the beginning of one heartbeat to the beginning of the next. It consists of two periods: one during which the heart muscle relaxes and refills with blood, called diastole, following ...
s. In 1987, Per Bak,
Chao Tang Tang Chao (; born 1958) is a Chair Professor of Physics and Systems Biology at Peking University. Education He had his undergraduate training at the University of Science and Technology of China, then went to the United States through the CU ...
and Kurt Wiesenfeld published a paper in ''
Physical Review Letters ''Physical Review Letters'' (''PRL''), established in 1958, is a peer-reviewed, scientific journal that is published 52 times per year by the American Physical Society. As also confirmed by various measurement standards, which include the '' Jour ...
'' describing for the first time self-organized criticality (SOC), considered one of the mechanisms by which
complexity Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generally used to ch ...
arises in nature. Alongside largely lab-based approaches such as the
Bak–Tang–Wiesenfeld sandpile The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, ...
, 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 In seismology, the Gutenberg–Richter law (GR law) expresses the relationship between the magnitude and total number of earthquakes in any given region and time period of ''at least'' that magnitude. : \!\,\log_ N = a - b M or : \!\,N = 10^ wh ...
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,
epidemic An epidemic (from Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of patients among a given population within an area in a short period of time. Epidemics of infectious d ...
s, and
biological evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
(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. In 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. 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 ''The Structure of Scientific Revolutions'' (1962; second edition 1970; third edition 1996; fourth edition 2012) is a book about the history of science by philosopher Thomas S. Kuhn. Its publication was a landmark event in the history, philosophy ...
'' (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 model A population model is a type of mathematical model that is applied to the study of population dynamics. Rationale Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can ...
ing,
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
,
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
,
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
, information theory,
computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to u ...
, pandemic
crisis management Crisis management is the process by which an organization deals with a disruptive and unexpected event that threatens to harm the organization or its stakeholders. The study of crisis management originated with large-scale industrial and envir ...
, etc.


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 Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, computer science, economics, engineering, finance,
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
, philosophy,
anthropology Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, societies, and linguistics, in both the present and past, including past human species. Social anthropology studies patterns of behavi ...
, 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 Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of
cryptographic primitive Cryptographic primitives are well-established, low-level cryptographic algorithms that are frequently used to build cryptographic protocols for computer security systems. These routines include, but are not limited to, one-way hash functions an ...
s. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers,
watermarking A watermark is a recognizable image or pattern in paper used to determine authenticity. Watermark or watermarking may also refer to: Technology * Digital watermarking, a technique to embed data in digital audio, images or video ** Audio water ...
, 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 Symmetric-key algorithms are algorithms for cryptography that use the same cryptographic keys for both the encryption of plaintext and the decryption of ciphertext. The keys may be identical, or there may be a simple transformation to go between th ...
, 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 model A population model is a type of mathematical model that is applied to the study of population dynamics. Rationale Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can ...
s. Most models are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
, but recently scientists have been able to implement chaotic models in certain populations. For example, a study on models of
Canadian lynx The Canada lynx (''Lynx canadensis''), or Canadian lynx, is a medium-sized North American lynx that ranges across Alaska, Canada, and northern areas of the contiguous United States. It is characterized by its long, dense fur, triangular ears w ...
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 Cardiotocography (CTG) is a technique used to monitor the fetal heartbeat and the uterine contractions during pregnancy and labour. The machine used to perform the monitoring is called a cardiotocograph. Fetal heart sounds was described as earl ...
. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of
fetal hypoxia Intrauterine hypoxia (also known as fetal hypoxia) occurs when the fetus is deprived of an adequate supply of oxygen. It may be due to a variety of reasons such as prolapse or occlusion of the umbilical cord, placental infarction, maternal dia ...
can be obtained through chaotic modeling.


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 could help in modeling how economy operate as well as in embedding shocks due to external events such as COVID-19.


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 Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, 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 The dwarf planet Pluto has five natural satellites. In order of distance from Pluto, they are Charon, Styx, Nix, Kerberos, and Hydra. Charon, the largest, is mutually tidally locked with Pluto, and is massive enough that Pluto–Charon is some ...
rotate chaotically. In
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
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 Wilfred Ruprecht Bion DSO (; 8 September 1897 – 8 November 1979) was an influential English psychoanalyst, who became president of the British Psychoanalytical Society from 1962 to 1965. Early life and military service Bion was born in M ...
'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 system A complex adaptive system is a system that is ''complex'' in that it is a dynamic network of interactions, but the behavior of the ensemble may not be predictable according to the behavior of the components. It is ''adaptive'' in that the individ ...
s with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance,
team building Team building is a collective term for various types of activities used to enhance social relations and define roles within teams, often involving collaborative tasks. It is distinct from team training, which is designed by a combine of business ...
and
group development The goal of most research on group development is to learn why and how small groups change over time. To quality of the output produced by a group, the type and frequency of its activities, its cohesiveness, the existence of group conflict. A numb ...
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 Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
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.


See also

Examples of chaotic systems * Advected contours *
Arnold's cat map In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. Thinking of the torus \mathbb^2 as the quotient space ...
*
Bifurcation theory Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
* Bouncing ball dynamics *
Chua's circuit Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic ...
*
Cliodynamics Cliodynamics () is a transdisciplinary area of research that integrates cultural evolution, economic history/cliometrics, macrosociology, the mathematical modeling of historical processes during the '' longue durée'', and the construction and anal ...
* Coupled map lattice * Double pendulum * Duffing equation *
Dynamical billiards A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ...
* Economic bubble * Gaspard-Rice system * Hénon map * Horseshoe map *
List of chaotic maps In mathematics, a chaotic map is a map (namely, an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated fun ...
* Rössler attractor * Standard map *
Swinging Atwood's machine The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters an ...
*
Tilt A Whirl Tilt-A-Whirl is a flat ride similar to the Waltzer in Europe, designed for commercial use at amusement parks, fairs, and carnivals, in which it is commonly found. The rides are manufactured by Larson International of Plainview, Texas. Descrip ...
Other related topics * Amplitude death *
Anosov diffeomorphism In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "cont ...
* Catastrophe theory * Causality * Chaos machine * Chaotic mixing * Chaotic scattering * Control of chaos * Determinism * Edge of chaos * Emergence * Mandelbrot set * Kolmogorov–Arnold–Moser theorem * Ill-conditioning * Ill-posedness * Nonlinear system * Patterns in nature * Predictability * Quantum chaos * Santa Fe Institute * Synchronization of chaos * Unintended consequence * Supersymmetric theory of stochastic dynamics, Chaos as topological supersymmetry breaking People * Ralph Abraham (mathematician), Ralph Abraham * Michael Berry (physicist), Michael Berry * Leon O. Chua * Ivar Ekeland * Doyne Farmer * Martin Gutzwiller * Brosl Hasslacher * Michel Hénon * Aleksandr Lyapunov * Norman Packard * Otto Rössler * David Ruelle * Oleksandr Mikolaiovich Sharkovsky * Robert Shaw * Floris Takens * James A. Yorke * George M. Zaslavsky


References


Further reading


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). * * * * *


Textbooks

* * * * * * * * * * * * * * * * * * * * * * * * *


Semitechnical and popular works

* Christophe Letellier, ''Chaos in Nature'', World Scientific Publishing Company, 2012, . * * * * John Briggs (author), 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 (mathematician), 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.


External links

*
Nonlinear Dynamics Research Group
with Animations in Flash
The Chaos group at the University of Maryland

The Chaos Hypertextbook
An introductory primer on chaos and fractals
ChaosBook.org
An advanced graduate textbook on chaos (no fractals)
Society for Chaos Theory in Psychology & Life Sciences


Florence Italy
Nonlinear dynamics: how science comprehends chaos
talk presented by Sunny Auyang, 1998.
Nonlinear Dynamics
Models of bifurcation and chaos by Elmer G. Wiens


Systems Analysis, Modelling and Prediction Group
at the University of Oxford
A page about the Mackey-Glass equation

High Anxieties — The Mathematics of Chaos
(2008) BBC documentary directed by David Malone (independent filmmaker), David Malone
The chaos theory of evolution
– article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos. * Jos Leys, Étienne Ghys et Aurélien Alvarez
''Chaos, A Mathematical Adventure''
Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
"Chaos Theory"
BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (''In Our Time'', May 16, 2002)
Chaos: The Science of the Butterfly Effect
(2019) an explanation presented by Derek Muller {{DEFAULTSORT:Chaos Theory Chaos theory, Complex systems theory Computational fields of study