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Chaos theory is an
interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of multiple academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields like sociology, anthropology, psychology, ec ...
area of
scientific study Scientific study is a kind of study that involves scientific theory, scientific models, experiments and physical situations. It may refer to: *Scientific method, a body of techniques for investigating phenomena, based on empirical or measurabl ...
and branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s that are highly sensitive to
initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
, 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 ...
, repetition,
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
,
fractals 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 illus ...
, and
self-organization Self-organization, also called spontaneous order in the social sciences, is a process where some form of overall order arises from local interactions between parts of an initially disordered system. The process can be spontaneous when suff ...
. 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 Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
can cause a
tornado A tornado is a violently rotating column of air that is in contact with both the surface of the Earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. It is often referred to as a twister, whirlwind or cyclone, altho ...
in
Texas Texas (, ; Spanish language, Spanish: ''Texas'', ''Tejas'') is a state in the South Central United States, South Central region of the United States. At 268,596 square miles (695,662 km2), and with more than 29.1 million residents in 2 ...
. 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 :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
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 Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and ...
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 l ...
. This behavior can be studied through the analysis of a chaotic
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
, or through analytical techniques such as
recurrence plot In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits roug ...
s and
Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
s. 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 Sociology is a social science that focuses on society, human social behavior, patterns of Interpersonal ties, social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of Empirical ...
,
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 Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
,
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
,
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...
, and
pandemic A pandemic () is an epidemic of an infectious disease that has spread across a large region, for instance multiple continents or worldwide, affecting a substantial number of individuals. A widespread endemic (epidemiology), endemic disease wi ...
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 The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order ...
theory, and
self-assembly Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the ...
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 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, 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 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 appea ...
, # 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. Mathematically ...
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given a ...
s. 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 ...
maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
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 settin ...
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 Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
, 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 Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and ...
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 respons ...
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 The geologic time scale, or geological time scale, (GTS) is a representation of time based on the rock record of Earth. It is a system of chronological dating that uses chronostratigraphy (the process of relating strata to time) and geochr ...
), 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 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 traj ...
in the phase space 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 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 appea ...
(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 suf ...
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 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 suf ...
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 appea ...
. 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 \mat ...
,
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, then topological transitivity implies the existence of a
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
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. Mathematically ...
periodic orbits In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynami ...
means that every point in the space is approached arbitrarily closely by periodic orbits. The one-dimensional
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
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 r ...
). 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 The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
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 h ...
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 cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
s, 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 ...
dynamical systems (such as the Lorenz system) and in some
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
systems (such as the
Hénon map The Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (''xn'', ''yn'') in ...
). Other discrete dynamical systems have a repelling structure called a
Julia set In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
, 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 illu ...
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 me ...
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 The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
, can exhibit strange attractors whatever their
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
ality. 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_ The_logistic_map_is_a_polynomial__mapping_(equivalently,_recurrence_relation)_of__degree_2,_often_referred_to_as_an_archetypal_example_of_how_complex,_chaotic_behaviour_can_arise_from_very_simple__non-linear_dynamical_equations._The_map_was_popular_...
. 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 The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
. 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 ...
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 op ...
shows that a strange attractor can only arise in three or more dimensions. Dimension (vector space), Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
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 op ...
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
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. 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 In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. 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 In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
f
psi Psi, PSI or Ψ may refer to: Alphabetic letters * Psi (Greek) (Ψ, ψ), the 23rd letter of the Greek alphabet * Psi (Cyrillic) (Ѱ, ѱ), letter of the early Cyrillic alphabet, adopted from Greek Arts and entertainment * "Psi" as an abbreviatio ...
= \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 reso ...
\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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
:. K(\vec r - \vec r^,L) = \frac \exp
frac Frac or FRAC may refer to: * Frac or fraccing, short name for Hydraulic fracturing, a method for extracting oil and natural gas * FRAC Act, United States legislation proposed in 2009 to regulate hydraulic fracturing * Frac module, a format for ...
/math>.


Jerk systems

In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, jerk is the third derivative of
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
, 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 ...
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 The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used to describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated b ...
, 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 of ...
' pendulums, fireflies,
neuron A neuron, neurone, or nerve cell is an electrically excitable cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous tissue in all animals except sponges and placozoa. N ...
s, 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 In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mech ...
.


History

An early proponent of chaos theory was
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
. In the 1880s, while studying the
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898,
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
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 chaos theory, chaotic dynamical system, a type of dynamical billiards. Introduced by Jacques Hadamard in 1898, and ...
". 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 Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
. Later studies, also on the topic of nonlinear
differential equations 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 ...
, were carried out by
George David Birkhoff George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and durin ...
,
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 Dame Mary Lucy Cartwright, (17 December 1900 – 3 April 1998) was a British mathematician. She was one of the pioneers of what would later become known as chaos theory. Along with J. E. Littlewood, Cartwright saw many solutions to a proble ...
and
John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
, and
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
. 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 In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or ...
, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the
logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
. 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 arise ...
" 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 Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next 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 Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and ...
was an early pioneer of the theory. His interest in chaos came about accidentally through his work on
weather prediction Weather is the state of the atmosphere, describing for example the degree to which it is hot or cold, wet or dry, calm or stormy, clear or cloudy. On Earth, most weather phenomena occur in the lowest layer of the planet's atmosphere, the tr ...
in 1961. Lorenz and his collaborator Ellen Fetter were using a simple digital computer, a
Royal McBee The Royal Typewriter Company is a manufacturer of typewriters founded in January 1904. It was headquartered in New York City with its factory in Hartford, Connecticut. History The Royal Typewriter Company was founded by Edward B. Hess and Lewis ...
LGP-30 The LGP-30, standing for Librascope General Purpose and then Librascope General Precision, was an early off-the-shelf computer. It was manufactured by the Librascope company of Glendale, California (a division of General Precision Inc.), and so ...
, 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 Lo ...
s, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions. In 1963, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand he had studied
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
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. Thr ...
: 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 referr ...
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 illu ...
(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 Si ...
, the Sierpiński gasket, and the
Koch curve The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
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 me ...
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 The New York Academy of Sciences (originally the Lyceum of Natural History) was founded in January 1817 as the Lyceum of Natural History. It is the fourth oldest scientific society in the United States. An independent, nonprofit organization wit ...
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 Mitchell Jay Feigenbaum (December 19, 1944 – June 30, 2019) was an American mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants. Early life Feigenbaum was born in Philadelphia, Pe ...
'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 Albert Joseph Libchaber (born 23 October 1934, Paris) is a Detlev W. Bronk Professor at The Rockefeller University. He won the Wolf Prize in Physics in 1986. In 1999 he received the Prix des Trois Physiciens from the Fondation de France. Educatio ...
, 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 degr ...
, 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 ...
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 Mitchell Jay Feigenbaum (December 19, 1944 – June 30, 2019) was an American mathematical physics, mathematical physicist whose pioneering studies in chaos theory led to the discovery of the Feigenbaum constants. Early life Feigenbaum was bor ...
for their inspiring achievements. In 1986, the New York Academy of Sciences co-organized with the
National Institute of Mental Health The National Institute of Mental Health (NIMH) is one of 27 institutes and centers that make up the National Institutes of Health (NIH). The NIH, in turn, is an agency of the United States Department of Health and Human Services and is the prima ...
and the
Office of Naval Research The Office of Naval Research (ONR) is an organization within the United States Department of the Navy responsible for the science and technology programs of the U.S. Navy and Marine Corps. Established by Congress in 1946, its mission is to plan ...
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 Schizophrenia is a mental disorder characterized by continuous or relapsing episodes of psychosis. Major symptoms include hallucinations (typically hearing voices), delusions, and disorganized thinking. Other symptoms include social withdra ...
. This led to a renewal of
physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
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 Per Bak (8 December 1948 – 16 October 2002) was a Danish theoretical physicist who coauthored the 1987 academic paper that coined the term "self-organized criticality." Life and work After receiving his Ph.D. from the Technical University of ...
,
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 C ...
and
Kurt Wiesenfeld Kurt Wiesenfeld is an American physicist working primarily on non-linear dynamics. His works primarily concern stochastic resonance, spontaneous synchronization of coupled oscillators, and non-linear laser dynamics. Since 1987, he has been profe ...
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 ''Journa ...
'' describing for the first time
self-organized criticality Self-organized criticality (SOC) is a property of dynamical systems that have a critical point as an attractor. Their macroscopic behavior thus displays the spatial or temporal scale-invariance characteristic of the critical point of a phase ...
(SOC), considered one of the mechanisms by which
complexity Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence. The term is generall ...
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, C ...
, 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
earthquake An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
s, (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^ wher ...
describing the statistical distribution of earthquake sizes, and the Omori law describing the frequency of aftershocks),
solar flare A solar flare is an intense localized eruption of electromagnetic radiation in the Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar particle events, and other solar phe ...
s, fluctuations in economic systems such as
financial market A financial market is a market in which people trade financial securities and derivatives at low transaction costs. Some of the securities include stocks and bonds, raw materials and precious metals, which are known in the financial markets ...
s (references to SOC are common in
econophysics Econophysics is a Heterodox economics, heterodox interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes ...
), landscape formation,
forest fire A wildfire, forest fire, bushfire, wildland fire or rural fire is an unplanned, uncontrolled and unpredictable fire in an area of Combustibility and flammability, combustible vegetation. Depending on the type of vegetation present, a wildfire ...
s,
landslide Landslides, also known as landslips, are several forms of mass wasting that may include a wide range of ground movements, such as rockfalls, deep-seated grade (slope), slope failures, mudflows, and debris flows. Landslides occur in a variety of ...
s,
epidemic An epidemic (from Ancient Greek, 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 ...
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 t ...
(where SOC has been invoked, for example, as the dynamical mechanism behind the theory of " punctuated equilibria" put forward by
Niles Eldredge Niles Eldredge (; born August 25, 1943) is an American biologist and paleontologist, who, along with Stephen Jay Gould, proposed the theory of punctuated equilibrium in 1972. Education Eldredge began his undergraduate studies in Latin at Columb ...
and
Stephen Jay Gould Stephen Jay Gould (; September 10, 1941 – May 20, 2002) was an American paleontologist, evolutionary biologist, and historian of science. He was one of the most influential and widely read authors of popular science of his generation. Gould sp ...
). 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
war War is an intense armed conflict between states, governments, societies, or paramilitary groups such as mercenaries, insurgents, and militias. It is generally characterized by extreme violence, destruction, and mortality, using regular o ...
s. 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 James Gleick (; born August 1, 1954) is an American author and historian of science whose work has chronicled the cultural impact of modern technology. Recognized for his writing about complex subjects through the techniques of narrative nonficti ...
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 Thomas Samuel Kuhn (; July 18, 1922 – June 17, 1996) was an American philosopher of science whose 1962 book ''The Structure of Scientific Revolutions'' was influential in both academic and popular circles, introducing the term '' paradigm ...
's concept of a
paradigm shift A paradigm shift, a concept brought into the common lexicon by the American physicist and philosopher Thomas Kuhn, is a fundamental change in the basic concepts and experimental practices of a scientific discipline. Even though Kuhn restricted t ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
social systems In sociology, a social system is the patterned network of relationships constituting a coherent whole that exist between individuals, groups, and institutions. It is the formal Social structure, structure of role and status that can form in a smal ...
,
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 i ...
,
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 h ...
,
information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
,
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 A pandemic () is an epidemic of an infectious disease that has spread across a large region, for instance multiple continents or worldwide, affecting a substantial number of individuals. A widespread endemic (epidemiology), endemic disease wi ...
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 Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 i ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
,
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
,
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
,
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 Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
,
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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
politics Politics (from , ) is the set of activities that are associated with making decisions in groups, or other forms of power relations among individuals, such as the distribution of resources or status. The branch of social science that studies ...
,
population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
, and
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
. 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 adver ...
. 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 A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually ...
, 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 Steganography ( ) is the practice of representing information within another message or physical object, in such a manner that the presence of the information is not evident to human inspection. In computing/electronic contexts, a computer file, ...
. 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 t ...
, relies on
diffusion and confusion In cryptography, confusion and diffusion are two properties of the operation of a secure cipher identified by Claude Elwood Shannon, Claude Shannon in his 1945 classified report ''A Mathematical Theory of Cryptography'.'' These properties, when ...
, which is modeled well by chaos theory. Another type of computing,
DNA computing DNA computing is an emerging branch of unconventional computing which uses DNA, biochemistry, and molecular biology hardware, instead of the traditional electronic computing. Research and development in this area concerns theory, experiments, a ...
, 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 Predictive modelling uses statistics to predict outcomes. Most often the event one wants to predict is in the future, but predictive modelling can be applied to any type of unknown event, regardless of when it occurred. For example, predictive mod ...
. 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 ...
, 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 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 ...
. 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 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 Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space tr ...
. 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 A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic an ...
, 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, qua ...
and
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the study of large arrays of
Josephson junctions In physics, the Josephson effect is a phenomenon that occurs when two superconductors are placed in proximity, with some barrier or restriction between them. It is an example of a macroscopic quantum phenomenon, where the effects of quantum mech ...
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 num ...
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 BML may refer to: Businesses and organizations * Bank of Maldives Limited * Big Mouth Loud, a member of the Global Professional Wrestling Alliance * Bill Me Later, now PayPal Credit * BML Munjal University, a university in Sidhrawali, Haryana, Ind ...
at right). Chaos theory has been applied to environmental
water cycle The water cycle, also known as the hydrologic cycle or the hydrological cycle, is a biogeochemical cycle that describes the continuous movement of water on, above and below the surface of the Earth. The mass of water on Earth remains fairly cons ...
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 *
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 The physics of a bouncing ball concerns the physical behaviour of bouncing balls, particularly its motion before, during, and after impact against the surface of another body. Several aspects of a bouncing ball's behaviour serve as an introd ...
* Chua's circuit *
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 analy ...
*
Coupled map lattice A coupled map (mathematics), map lattice (group), lattice (CML) is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the Chaos theory, c ...
*
Double pendulum In physics and mathematics, in the area of dynamical systems, a double pendulum also known as a chaos pendulum is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamic behavior with a ...
*
Duffing equation The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by :\ddot + \delta \dot + \ ...
*
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 An economic bubble (also called a speculative bubble or a financial bubble) is a period when current asset prices greatly exceed their intrinsic valuation, being the valuation that the underlying long-term fundamentals justify. Bubbles can be c ...
* Gaspard-Rice system *
Hénon map The Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (''xn'', ''yn'') in ...
*
Horseshoe map In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavi ...
*
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 functi ...
*
Rössler attractor The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s... These differential equations define a continuous-time dynamical ...
* Standard map * Swinging Atwood's machine *
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. Descripti ...
Other related topics *
Amplitude death In the theory of dynamical systems, amplitude death is complete cessation of 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 ...
*
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 Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
*
Chaos machine In mathematics, a chaos machine is a class of algorithms constructed on the base of chaos theory (mainly deterministic chaos) to produce pseudo-random oracle. It represents the idea of creating a universal scheme with modular design and customiza ...
*
Chaotic mixing In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, s ...
*
Chaotic scattering Chaotic scattering is a branch of chaos theory dealing with scattering systems displaying a strong sensitivity to initial conditions. In a classical scattering system there will be one or more ''impact parameters'', ''b'', in which a particle is ...
*
Control of chaos In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of ...
*
Determinism Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
*
Edge of chaos The edge of chaos is a transition space between order and disorder that is hypothesized to exist within a wide variety of systems. This transition zone is a region of bounded instability that engenders a constant dynamic interplay between order ...
*
Emergence In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
*
Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This ...
*
Kolmogorov–Arnold–Moser theorem The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory ...
*
Ill-conditioning In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...
*
Ill-posedness The mathematics, mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that: # a Solution (mat ...
*
Nonlinear system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
*
Patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foa ...
*
Predictability Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relationship with predictability. Perf ...
*
Quantum chaos Quantum chaos is a branch of physics which studies how chaos theory, chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship betwee ...
*
Santa Fe Institute The Santa Fe Institute (SFI) is an independent, nonprofit theoretical research institute located in Santa Fe, New Mexico, United States and dedicated to the multidisciplinary study of the fundamental principles of complex adaptive systems, includ ...
* Synchronization of chaos *
Unintended consequence In the social sciences, unintended consequences (sometimes unanticipated consequences or unforeseen consequences) are outcomes of a purposeful action that are not intended or foreseen. The term was popularised in the twentieth century by Ameri ...
* Chaos as topological supersymmetry breaking People * Ralph Abraham * Michael Berry * Leon O. Chua *
Ivar Ekeland Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis, the calculus of variations, and mathematical economics, as well a ...
*
Doyne Farmer J. Doyne Farmer (born 22 June 1952) is an American complex systems scientist and entrepreneur with interests in chaos theory, complexity and econophysics. He is Baillie Gifford Professor of Mathematics at Oxford University, where he is also Direct ...
*
Martin Gutzwiller Martin may refer to: Places * Martin City (disambiguation) * Martin County (disambiguation) * Martin Township (disambiguation) Antarctica * Martin Peninsula, Marie Byrd Land * Port Martin, Adelie Land * Point Martin, South Orkney Islands Austral ...
*
Brosl Hasslacher Brosl Hasslacher (May 13, 1941 – November 11, 2005) was a theoretical physicist. Brosl Hasslacher obtained a bachelor's in physics from Harvard University in 1962. He did his Ph.D. with D.Z. Freeman and Yang Chen-Ning, C.N. Yang at the State Uni ...
* Michel Hénon *
Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Lia ...
*
Norman Packard Norman Harry Packard (born 1954 in Billings, Montana) is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. He is an alumnus of Reed College and the University of California, Santa Cruz. Packard is known f ...
*
Otto Rössler Otto Eberhard Rössler (born 20 May 1940) is a German biochemist known for his work on chaos theory and the theoretical equation known as the Rössler attractor. He is best known to the general public for his involvement in a failed lawsuit to ha ...
*
David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
* Oleksandr Mikolaiovich Sharkovsky * Robert Shaw *
Floris Takens Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attra ...
* 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 Christophe may refer to: People * Christophe (given name), list of people with this name * Christophe (singer) (1945–2020), French singer * Cristophe (hairstylist) (born 1958), Belgian hairstylist * Georges Colomb (1856–1945), French comic str ...
, ''Chaos in Nature'', World Scientific Publishing Company, 2012, . * * * * 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 Predrag Cvitanović (; born April 1, 1946) is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory. Life Cvitanović earned his B.S. from MIT in 1969 and his Ph.D. at Cornel ...
, ''Universality in Chaos'', Adam Hilger 1989, 648 pp. *
Leon Glass Leon Glass (born 1943) is an American scientist who has studied various aspects of the application of mathematical and physical methods to biology, with special interest in vision, cardiac arrhythmia, and genetic networks. Biography Leon Gl ...
and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp. *
James Gleick James Gleick (; born August 1, 1954) is an American author and historian of science whose work has chronicled the cultural impact of modern technology. Recognized for his writing about complex subjects through the techniques of narrative nonficti ...
, '' 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 Edward Norton Lorenz (May 23, 1917 – April 16, 2008) was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and ...
, ''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 Heinz-Otto Peitgen (born April 30, 1945 in Bruch, Nümbrecht near Cologne) is a German mathematician and was President of Jacobs University from January 1, 2013 to December 31, 2013. Peitgen contributed to the study of fractals, chaos theory, an ...
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 Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...
, ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991. *
Clifford A. Pickover Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...
, ''Chaos in Wonderland: Visual Adventures in a Fractal World'', St Martins Pr 1994. *
Ilya Prigogine Viscount Ilya Romanovich Prigogine (; russian: Илья́ Рома́нович Приго́жин; 28 May 2003) was a physical chemist and Nobel laureate noted for his work on dissipative structures, complex systems, and irreversibility. Biogra ...
and
Isabelle Stengers Isabelle Stengers (; ; born 1949) is a Belgian philosopher, noted for her work in the philosophy of science. Trained as a chemist, she has collaborated with Russian-Belgian chemist Ilya Prigogine and French philosopher/sociologist Bruno Latour a ...
, ''Order Out of Chaos'', Bantam 1984. * *
David Ruelle David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
, ''Chance and Chaos'', Princeton University Press 1993. *
Ivars Peterson Ivars Peterson (born 4 December 1948) is an American mathematics writer. Early life Peterson received a B.Sc. in Physics and Chemistry and a B.Ed. in Education from the University of Toronto. Peterson received an M.A. in Journalism from the ...
, ''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 Steven Henry Strogatz (), born August 13, 1959, is an American mathematician and the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. He is known for his work on nonlinear systems, including contributions to the study o ...
, ''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 Florence ( ; it, Firenze ) is a city in Central Italy and the capital city of the Tuscany region. It is the most populated city in Tuscany, with 383,083 inhabitants in 2016, and over 1,520,000 in its metropolitan area.Bilancio demografico an ...
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...

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
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 Étienne Ghys (born 29 December 1954) is a French mathematician. His research focuses mainly on geometry and dynamical systems, though his mathematical interests are broad. He also expresses much interest in the historical development of mathema ...
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 Derek Alexander Muller (born 9 November 1982) is an Australian-Canadian science communicator, filmmaker, and television personality, who is best known for his YouTube channel Veritasium. Muller has also appeared as a correspondent on the Net ...
{{DEFAULTSORT:Chaos Theory Complex systems theory Computational fields of study