![Double-compound-pendulum](https://upload.wikimedia.org/wikipedia/commons/4/45/Double-compound-pendulum.gif)
Chaos theory is an
interdisciplinary area of
scientific study and branch of
mathematics focused on underlying patterns and deterministic
laws
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s that are highly sensitive to
initial conditions
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, 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: ''Texas'', ''Tejas'') is a state in the South Central region of the United States. At 268,596 square miles (695,662 km2), and with more than 29.1 million residents in 2020, it is the second-largest U.S. state by ...
.
Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in
numerical computation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are
deterministic, meaning that their future behavior follows a unique evolution
and is fully determined by their initial conditions, with no
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...
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, 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 no ...
, 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 be ...
, sociology
Sociology is a social science that focuses on society, human social behavior, patterns of social relationships, social interaction, and aspects of culture associated with everyday life. It uses various methods of empirical investigation an ...
, environmental science, 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 practical disciplines (includi ...
, engineering
Engineering is the use of 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 range of more speciali ...
, economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, 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 overl ...
, and pandemic crisis management
Crisis management is the process by which an organization deals with a disruptive and unexpected event that threatens to harm the organization or its stakeholders. The study of crisis management originated with large-scale industrial and envir ...
. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos
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,
# 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 set ...
s. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.
If attention is restricted to intervals
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 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 , the two trajectories end up diverging at a rate given by
:
where is the time and 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 ballist ...
.
Non-periodicity
A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior.
Topological mixing
Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set
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 su ...
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 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 su ...
s , there exists such that . Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point ''x'' and a region ''V'', there exists a point ''y'' near ''x'' whose orbit passes through ''V''. This implies that it is impossible to decompose the system into two open sets.
An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if ''X'' is a second countable
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 ra ...
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 defined by ''x'' → 4 ''x'' (1 – ''x'') is one of the simplest systems with density of periodic orbits. For example, → → (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem
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 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 o ...
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 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, can exhibit strange attractors whatever their dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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 ...
, is well visible with map proposed as a toy
model for discrete laser dynamics:
,
where stands for electric field amplitude, is laser gain as bifurcation parameter. The gradual increase of at interval _with_qualitatively_the_same_
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:
:
where
,
, and
make up the
system state,
is time, and
,
,
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 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 g ...
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, norm, topology, etc.) and the linear functions defined o ...
.
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:
,
where kernel
is propagator derived as Green function of a relevant physical system,
might be logistic map alike