Editing Chaos theory (section)

==Chaotic dynamics==
[[File:Chaos Sensitive Dependence.svg|thumb|The [[Map (mathematics)|map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.]]
In common usage, "chaos" means "a state of disorder".<ref>Definition of {{linktext|chaos}} at [[Wiktionary]];</ref><ref>{{Cite web|url=https://www.dictionary.com/browse/chaos|title=Definition of chaos {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-24}}</ref> 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:<ref>{{cite book|title=A First Course in Dynamics: With a Panorama of Recent Developments|last=Hasselblatt|first=Boris|author2=Anatole Katok|year=2003|publisher=Cambridge University Press|isbn=978-0-521-58750-1}}</ref>

# it must be [[sensitive dependence on initial conditions|sensitive to initial conditions]],
# it must be [[Mixing (mathematics)#Topological mixing|topologically transitive]],
# it must have [[dense set|dense]] [[periodic orbit]]s.

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions.<ref>{{cite book |author=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |year=1999 |isbn=978-1-58488-002-8 |page=137 }}</ref><ref>{{cite book |author=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=978-0-471-68755-9 |page=42 }}</ref> In the discrete-time case, this is true for all [[Continuous function|continuous]] [[Map (mathematics)|maps]] on [[metric space]]s.<ref>{{cite journal | author1=Banks | author2=Brooks | author3=Cairns | author4=Davis | author5=Stacey | title= On Devaney's definition of chaos | journal=The American Mathematical Monthly | volume=99|issue=4|date=1992| pages=332–334 | doi=10.1080/00029890.1992.11995856 }}</ref> 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 [[Interval (mathematics)|intervals]], the second property implies the other two.<ref>{{cite journal |author1=Vellekoop, Michel |author2=Berglund, Raoul |title=On Intervals, Transitivity = Chaos |journal=The American Mathematical Monthly |volume=101 |issue=4 |pages=353–5 |date=April 1994 |jstor=2975629 |doi=10.2307/2975629}}</ref> An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.<ref>{{cite book |author1=Medio, Alfredo |author2=Lines, Marji |title=Nonlinear Dynamics: A Primer |url=https://archive.org/details/nonlineardynamic00medi |url-access=limited |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-55874-7 |page=[https://archive.org/details/nonlineardynamic00medi/page/n175 165] }}</ref>

===Sensitivity to initial conditions===
{{Main|Butterfly effect}}
[[File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the y variable. The initial conditions for ''x'' and ''z'' were kept the same but those for ''y'' were changed between '''1.001''', '''1.0001''' and '''1.00001'''. The values for <math>\rho</math>, <math>\sigma</math> and <math>\beta</math> were '''45.91''', '''16''' and '''4 ''' respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on 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.<ref name=":1" />

Sensitivity to initial conditions is popularly known as the "[[butterfly effect]]", so-called because of the title of a paper given by [[Edward Lorenz]] in 1972 to the [[American Association for the Advancement of Science]] in Washington, D.C., entitled ''Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?''.<ref>{{Cite web|url=http://news.mit.edu/2008/obit-lorenz-0416|title=Edward Lorenz, father of chaos theory and butterfly effect, dies at 90|website=MIT News|date=16 April 2008 |access-date=2019-11-24}}</ref> 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,<ref name="Lorenz"/>{{rp|8}} "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."<ref name="Lorenz"/>{{rp|23}} 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.<ref name="Lorenz"/>{{rp|189–204}} A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |date=2022-05-07 |title=One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models |journal=Atmosphere |language=en |volume=13 |issue=5 |pages=753 |doi=10.3390/atmos13050753 |bibcode=2022Atmos..13..753S |issn=2073-4433|doi-access=free }}</ref>

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.<ref name="RGW">{{cite book |author=Watts, Robert G. |title=Global Warming and the Future of the Earth |url=https://archive.org/details/globalwarmingfut00watt_399 |url-access=limited |publisher=Morgan & Claypool |year=2007 |page=[https://archive.org/details/globalwarmingfut00watt_399/page/n22 17] }}</ref> 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 {{convert|100|C|F}} or fall below {{convert|-130|C|F}} on earth (during the current [[geologic era]]), but we cannot predict exactly which day will have the hottest temperature of the year.

In more mathematical terms, the [[Lyapunov exponent]] measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.<ref>{{Cite web|url=http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html|title=Lyapunov Characteristic Exponent|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-24}}</ref> More specifically, given two starting [[trajectory|trajectories]] in the [[phase space]] that are infinitesimally close, with initial separation <math>\delta \mathbf{Z}_0</math>, the two trajectories end up diverging at a rate given by

:<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>

where <math>t</math> is the time and <math>\lambda</math> 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, coupled with the solution's boundedness, is usually taken as an indication that the system is chaotic.<ref name=":2" />

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, [[Measure (mathematics)|measure-theoretical]] [[Mixing (mathematics)|mixing]] (as discussed in [[ergodicity|ergodic]] theory) and properties of a [[Kolmogorov automorphism|K-system]].<ref name="WerndlCharlotte" />

===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===
[[File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4  x_k  (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.]]
[[File:Chaos Topological Mixing.png|thumb|The map defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> also displays [[topological mixing]]. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.]]

[[Topological mixing]] (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or [[open set]] of its [[phase space]] eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored [[dye]]s or fluids is an example of a chaotic system.

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

===Topological transitivity===
A map <math>f:X \to X</math> is said to be topologically transitive if for any pair of non-empty [[open set]]s <math>U, V \subset X</math>, there exists <math>k > 0</math> such that <math>f^{k}(U) \cap V \neq \emptyset</math>. 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.<ref name="Devaney">{{harvnb|Devaney|2003}}</ref>

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 space|second countable]], [[complete metric space]], then topological transitivity implies the existence of a [[dense set]] of points in ''X'' that have dense orbits.<ref>{{harvnb|Robinson|1995}}</ref>

===Density of periodic orbits===
For a chaotic system to have [[Dense set|dense]] [[periodic orbits]] means that every point in the space is approached arbitrarily closely by periodic orbits.<ref name="Devaney"/> The one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> is one of the simplest systems with density of periodic orbits. For example, <math>\tfrac{5-\sqrt{5}}{8}</math>&nbsp;→ <math>\tfrac{5+\sqrt{5}}{8}</math>&nbsp;→ <math>\tfrac{5-\sqrt{5}}{8}</math> (or approximately 0.3454915&nbsp;→ 0.9045085&nbsp;→ 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]]).<ref>{{harvnb|Alligood|Sauer|Yorke|1997}}</ref>

Sharkovskii's theorem is the basis of the Li and Yorke<ref>{{cite journal|last1=Li |first1=T.Y. |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |author-link=Tien-Yien Li |doi=10.2307/2318254 |issue=10 |author2-link=James A. Yorke |bibcode=1975AmMM...82..985L |url-status=dead |archive-url=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |archive-date=2009-12-29 |jstor=2318254 |citeseerx=10.1.1.329.5038 }}</ref> (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===
[[File:TwoLorenzOrbits.jpg|thumb|right|The [[Lorenz attractor]] displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.]]

Some dynamical systems, like the one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x''),</span> are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an [[attractor]], since then a large set of initial conditions leads to orbits that converge to this chaotic region.<ref>{{cite journal|last1=Strelioff|first1=Christopher|last2=et.|first2=al.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|pages=044101|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|bibcode = 2006PhRvL..96d4101S }}</ref>

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

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

=== Coexisting attractors ===
[[File:Coexisting Attractors.png|thumb|Coexisting chaotic and non-chaotic attractors within the generalized Lorenz model.<ref name=":4" /><ref name=":5" /><ref name=":6" /> There are 128 orbits in different colors, beginning with different initial conditions for dimensionless time between 0.625 and 5 and a heating parameter r = 680. Chaotic orbits recurrently return close to the saddle point at the origin. Nonchaotic orbits eventually approach one of two stable critical points, as shown with large blue dots. Chaotic and nonchaotic orbits occupy different regions of attraction within the phase space.]]

In contrast to single type chaotic solutions, studies using Lorenz models<ref>{{Cite journal |last1=Yorke |first1=James A. |last2=Yorke |first2=Ellen D. |date=1979-09-01 |title=Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model |url=https://doi.org/10.1007/BF01011469 |journal=Journal of Statistical Physics |language=en |volume=21 |issue=3 |pages=263–277 |doi=10.1007/BF01011469 |bibcode=1979JSP....21..263Y |s2cid=12172750 |issn=1572-9613}}</ref><ref>{{Cite book |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=R. A. |last3=Zeng |first3=X. |last4=Baik |first4=J.-J. |last5=Faghih-Naini |first5=S. |last6=Cui |first6=J. |last7=Atlas |first7=R. |last8=Reyes |first8=T. A. L. |title=13th Chaotic Modeling and Simulation International Conference |chapter=Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models |date=2021 |editor-last=Skiadas |editor-first=Christos H. |editor2-last=Dimotikalis |editor2-first=Yiannis |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-70795-8_57 |series=Springer Proceedings in Complexity |language=en |location=Cham |publisher=Springer International Publishing |pages=805–825 |doi=10.1007/978-3-030-70795-8_57 |isbn=978-3-030-70795-8|s2cid=245197840 }}</ref> 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,<ref name=":4" /><ref name=":5" /><ref name=":6">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=Roger Pielke |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Kesarkar |first7=Amit |last8=Zeng |first8=Xiping |last9=Atlas |first9=Robert |date=2022-11-12 |title=The Dual Nature of Chaos and Order in the Atmosphere |journal=Atmosphere |language=en |volume=13 |issue=11 |pages=1892 |doi=10.3390/atmos13111892 |bibcode=2022Atmos..13.1892S |issn=2073-4433|doi-access=free }}</ref> 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===
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|right|[[Bifurcation diagram]] of the [[logistic map]] <span style="white-space: nowrap;">''x'' → ''r'' ''x'' (1 – ''x'').</span> Each vertical slice shows the attractor for a specific value of ''r''. The diagram displays [[Period-doubling bifurcation|period-doubling]] as ''r'' increases, eventually producing chaos. Darker points are visited more frequently.]]

Discrete chaotic systems, such as the [[logistic map]], can exhibit strange attractors whatever their [[dimension]]ality. In contrast, for [[continuous function (topology)|continuous]] dynamical systems, the [[Poincaré–Bendixson theorem]] shows that a strange attractor can only arise in three or more dimensions. [[Dimension (vector space)|Finite-dimensional]] [[linear system]]s are never chaotic; for a dynamical system to display chaotic behavior, it must be either [[nonlinearity|nonlinear]] or infinite-dimensional.

The [[Poincaré–Bendixson theorem]] states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three [[differential equation]]s such as:

: <math> \begin{align}
\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\
\frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\
\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
\end{align} </math>

where <math>x</math>, <math>y</math>, and <math>z</math> make up the [[State space representation|system state]], <math>t</math> is time, and <math>\sigma</math>, <math>\rho</math>, <math>\beta</math> are the system [[parameter]]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 map|Rössler equations]], which have only one nonlinear term out of seven. Sprott<ref>{{cite journal|last=Sprott |first=J.C.|year=1997|title=Simplest dissipative chaotic flow|journal=[[Physics Letters A]]|volume=228|issue=4–5 |pages=271–274|doi=10.1016/S0375-9601(97)00088-1|bibcode = 1997PhLA..228..271S }}</ref> 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<ref>{{cite journal|last1=Fu |first1=Z. |last2=Heidel |first2=J.|year=1997|title=Non-chaotic behaviour in three-dimensional quadratic systems|journal=[[Nonlinearity (journal)|Nonlinearity]]|volume=10|issue=5 |pages=1289–1303|doi=10.1088/0951-7715/10/5/014
|bibcode = 1997Nonli..10.1289F |s2cid=250757113 }}</ref><ref>{{cite journal|last1=Heidel |first1=J. |last2=Fu |first2=Z.|year=1999|title=Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case|journal=Nonlinearity|volume=12|issue=3 |pages=617–633|doi=10.1088/0951-7715/12/3/012|bibcode = 1999Nonli..12..617H |s2cid=250853499 }}</ref> 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 analysis|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 (mathematics)|plane]] cannot be chaotic, two-dimensional continuous systems with [[non-Euclidean geometry]] can still exhibit some chaotic properties.<ref>{{Cite journal |last=Ulcigrai |first=Corinna |date=2021 |title=Slow chaos in surface flows |url=http://link.springer.com/10.1007/s40574-020-00267-0 |journal=Bollettino dell'Unione Matematica Italiana |language=en |volume=14 |issue=1 |pages=231–255 |arxiv=2010.06231 |doi=10.1007/s40574-020-00267-0 |issn=1972-6724}}</ref> Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.<ref>{{cite journal
 |last1=Bonet |first1=J. |last2=Martínez-Giménez |first2=F. |last3=Peris |first3=A.
 |year=2001
 |title=A Banach space which admits no chaotic operator
 |journal=Bulletin of the London Mathematical Society
 |volume=33
 |issue=2 |pages=196–8
 |doi=10.1112/blms/33.2.196
|s2cid=121429354 }}</ref> A theory of linear chaos is being developed in a branch of mathematical analysis known as [[functional analysis]].

The above set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model.<ref>{{Cite journal |last=Shen |first=Bo-Wen |date=2014-05-01 |title=Nonlinear Feedback in a Five-Dimensional Lorenz Model |url=https://journals.ametsoc.org/doi/10.1175/JAS-D-13-0223.1 |journal=Journal of the Atmospheric Sciences |language=en |volume=71 |issue=5 |pages=1701–1723 |doi=10.1175/JAS-D-13-0223.1 |bibcode=2014JAtS...71.1701S |s2cid=123683839 |issn=0022-4928}}</ref> Since 1963, higher-dimensional Lorenz models have been developed in numerous studies<ref>{{Cite journal |last1=Musielak |first1=Dora E. |last2=Musielak |first2=Zdzislaw E. |last3=Kennamer |first3=Kenny S. |date=2005-03-01 |title=The onset of chaos in nonlinear dynamical systems determined with a new fractal technique |url=https://www.worldscientific.com/doi/abs/10.1142/S0218348X0500274X |journal=Fractals |volume=13 |issue=1 |pages=19–31 |doi=10.1142/S0218348X0500274X |issn=0218-348X}}</ref><ref>{{Cite journal |last1=Roy |first1=D. |last2=Musielak |first2=Z. E. |date=2007-05-01 |title=Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations |url=https://www.sciencedirect.com/science/article/pii/S0960077906001937 |journal=Chaos, Solitons & Fractals |language=en |volume=32 |issue=3 |pages=1038–1052 |doi=10.1016/j.chaos.2006.02.013 |bibcode=2007CSF....32.1038R |issn=0960-0779}}</ref><ref name=":4">{{Cite journal |last=Shen |first=Bo-Wen |date=2019-03-01 |title=Aggregated Negative Feedback in a Generalized Lorenz Model |journal=International Journal of Bifurcation and Chaos |volume=29 |issue=3 |pages=1950037–1950091 |doi=10.1142/S0218127419500378 |bibcode=2019IJBC...2950037S |s2cid=132494234 |issn=0218-1274|doi-access=free }}</ref><ref name=":5">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Baik |first4=Jong-Jin |last5=Faghih-Naini |first5=Sara |last6=Cui |first6=Jialin |last7=Atlas |first7=Robert |date=2021-01-01 |title=Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model |journal=Bulletin of the American Meteorological Society |language=EN |volume=102 |issue=1 |pages=E148–E158 |doi=10.1175/BAMS-D-19-0165.1 |bibcode=2021BAMS..102E.148S |s2cid=208369617 |issn=0003-0007|doi-access=free }}</ref> 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<ref name="Moloney, J V 1986">{{cite journal |title=Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis
|journal=Journal of Mathematical Physics|volume=29 |issue=1 |pages=63 |year=1988
|last1= Adachihara |first1=H
|last2= McLaughlin |first2=D W
|last3= Moloney |first3=J V
|last4= Newell |first4=A C
|doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:
<math>\psi_{n+1}(\vec r,t)  = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,

where kernel <math>K(\vec r - \vec r^{,},t)</math> is propagator derived as Green function of a relevant physical system,<ref name="Okulov, A Yu 1988">{{cite book |chapter=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps
|title=Proceedings of the Lebedev Physics Institute |language=ru |editor=N.G. Basov |publisher=Nauka |lccn=88174540 |volume=187 |pages=202–222 |year=1988 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N }}</ref>

<math> f [\psi_{n}(\vec r,t) ] </math> might be logistic map alike <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math> or [[complex map]]. For examples of complex maps the [[Julia set]] <math> f[\psi] = \psi^2</math> or [[Ikeda map]]
<math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> may serve. When wave propagation problems at distance <math>L=ct</math> with wavelength <math>\lambda=2\pi/k</math> are considered the kernel <math>K</math> may have a form of Green function for [[Schrödinger equation]]:.<ref name="Okulov, A Yu 2000">{{cite journal |title=Spatial soliton laser: geometry and stability
|journal=Optics and Spectroscopy|volume=89 |issue=1 |pages=145–147 |year=2000 |last1= Okulov |first1=A Yu|s2cid=122790937|doi=10.1134/BF03356001  |bibcode=2000OptSp..89..131O}}</ref><ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps
|journal=Chaos, Solitons & Fractals|volume=133 |issue=4|page=109638 |year=2020|last1= Okulov |first1=A Yu|doi=10.1016/j.chaos.2020.109638|arxiv=1901.09274|bibcode=2020CSF...13309638O|s2cid=118828500}}</ref>

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