Editing Chaos theory (section)

===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.