Editing Complex system

{{short description|System composed of many interacting components}}
{{Redirect|Complex systems|the journal|Complex Systems (journal){{!}}''Complex Systems'' (journal)}}
{{Complex systems}}
A '''complex system''' is a [[system]] composed of many components which may interact with each other.<ref>{{cite journal
| last1 = Ladyman
| first1 = James
| last2 = Lambert
| first2 = James
| last3 = Wiesner
| first3 = Karoline
| date = 2013
| title = What is a complex system?
| url = https://philsci-archive.pitt.edu/9044/4/LLWultimate.pdf
| journal = European Journal for Philosophy of Science 
| volume = 3
| pages = 33–67
| doi = 10.1007/s13194-012-0056-8
| access-date = 28 July 2024
| quote = [S]pecial issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 6. 'A complex system is literally one in which there are multiple interactions between many different components.' (40, p. 105)}}</ref> Examples of complex systems are Earth's global [[climate]], [[organisms]], the [[human brain]], infrastructure such as power grid, transportation or communication systems, complex [[software]] and electronic systems, social and economic organizations (like [[cities]]), an [[ecosystem]], a living [[Cell (biology)|cell]], and, ultimately, for some authors, the entire [[universe]].<ref>{{cite journal
| last1 = Ladyman
| first1 = James
| last2 = Lambert
| first2 = James
| last3 = Wiesner
| first3 = Karoline
| date = 2013
| title = What is a complex system?
| url = https://philsci-archive.pitt.edu/9044/4/LLWultimate.pdf
| journal = European Journal for Philosophy of Science 
| volume = 3
| pages = 33–67
| doi = 10.1007/s13194-012-0056-8
| access-date = 28 July 2024
| quote = The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [:] 8. 'In recent years the scientific community has coined the rubric 'complex system' to describe phenomena, structure, aggregates, organisms, or problems that share some common theme: (i) They are inherently complicated or intricate ...; (ii) they are rarely completely deterministic; (iii) mathematical models of the system are usually complex and involve non-linear, ill-posed, or chaotic behavior; (iv) the systems are predisposed to unexpected outcomes (so-called emergent behaviour).’ (14, p. 410)}}</ref><ref>Parker, B. R. (2013). ''Chaos in the Cosmos: the Stunning Complexity of the Universe''. Springer.</ref><ref>Bekenstein, J. D. (2003). Information in the holographic universe, ''Scientific American'', ''289''(2), 58-65.</ref>

The behavior of a complex system is intrinsically difficult to model due to the dependencies, competitions, relationships, and other types of interactions between their parts or between a given system and its environment.<ref>{{cite journal
| last1 = Ladyman
| first1 = James
| last2 = Lambert
| first2 = James
| last3 = Wiesner
| first3 = Karoline
| date = 2013
| title = What is a complex system?
| url = https://philsci-archive.pitt.edu/9044/4/LLWultimate.pdf
| journal = European Journal for Philosophy of Science 
| volume = 3
| pages = 33–67
| doi = 10.1007/s13194-012-0056-8
| access-date = 28 July 2024
| quote = The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 3. 'In a general sense, the adjective 'complex' describes a system or component that by design or function or both is difficult to understand and verify. ...complexity is determined by such factors as the number of components and the intricacy of the interfaces between them, the number and intricacy of conditional branches, the degree of nesting, and the types of data structures.'(50, p. 92)}}</ref> Systems that are "[[Complexity|complex]]" have distinct properties that arise from these relationships, such as [[Nonlinear system|nonlinearity]], [[emergence]], [[spontaneous order]], [[Complex adaptive system|adaptation]], and [[Feedback|feedback loops]], among others.<ref>{{cite journal
| last1 = Ladyman
| first1 = James
| last2 = Lambert
| first2 = James
| last3 = Wiesner
| first3 = Karoline
| date = 2013
| title = What is a complex system?
| url = https://philsci-archive.pitt.edu/9044/4/LLWultimate.pdf
| journal = European Journal for Philosophy of Science 
| volume = 3
| pages = 33–67
| doi = 10.1007/s13194-012-0056-8
| access-date = 28 July 2024
| quote = The following quotations (apart from the last one) come from a special issue of Science on 'Complex Systems' featuring many key figures in the field (Science 2 April 1999) [...] [:] 4. 'Complexity theory indicates that large populations of units can self-organize into aggregations that generate pattern, store information, and engage in collective decision-making.' (39, p. 99)}}</ref> Because such systems appear in a wide variety of fields, the commonalities among them have become the topic of their independent area of research. In many cases, it is useful to represent such a system as a network where the nodes represent the components and links represent their interactions.

The term ''complex systems'' often refers to the study of complex systems, which is an approach to science that investigates how relationships between a system's parts give rise to its collective behaviors and how the system interacts and forms relationships with its environment.<ref>{{Cite journal |last=Bar-Yam |first=Yaneer |date=2002 |title=General Features of Complex Systems |url=http://www.eolss.net/sample-chapters/c15/E1-29-01-00.pdf |url-status=live |journal=Encyclopedia of Life Support Systems |archive-url=https://ghostarchive.org/archive/20221009/http://www.eolss.net/sample-chapters/c15/E1-29-01-00.pdf |archive-date=2022-10-09 |access-date=16 September 2014}}</ref> The study of complex systems regards collective, or system-wide, behaviors as the fundamental object of study; for this reason, complex systems can be understood as an alternative paradigm to [[reductionism]], which attempts to explain systems in terms of their constituent parts and the individual interactions between them.

As an interdisciplinary domain, complex systems draw contributions from many different fields, such as the study of [[self-organization]] and critical phenomena from physics, of [[spontaneous order]] from the social sciences, [[Chaos theory|chaos]] from mathematics, [[Complex adaptive system|adaptation]] from biology, and many others. ''Complex systems'' is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines, including [[statistical physics]], [[information theory]], [[Nonlinear system|nonlinear dynamics]], [[anthropology]], [[computer science]], [[meteorology]], [[sociology]], [[economics]], [[psychology]], and [[biology]].

== Types of systems ==
Complex systems can be:

* [[Complex adaptive system|'''Complex adaptive systems''']] which have the capacity to change.
* '''Polycentric systems :''' “where many elements are capable of making mutual adjustments for ordering their relationships with one another within a general system of rules where each element acts with independence of other elements”.<ref>{{Cite book |last=McGinnis |first=Michael Dean |url=https://books.google.com/books?id=iBZ32c7KLWUC&dq=Ostrom,+V.:+Polycentricity%E2%80%94Part+1.+In:+McGinnis,+M.+(ed.)+Polycentricity+and+Local+Public+Economies:+Readings+from+the+Workshop+in+Political+Theory+and+Policy+Analysis,+pp.+52%E2%80%9374.+Ann+Arbor:+University+of+Michigan+Press+(1999)&pg=PR7 |title=Polycentricity and Local Public Economies: Readings from the Workshop in Political Theory and Policy Analysis |date=1999 |publisher=University of Michigan Press |isbn=978-0-472-08622-1 |language=en}}</ref>
* '''Disorganised systems''' involving localized interactions of multiple entities that do not form a coherent whole.<ref>{{Cite journal |last=Weaver |first=Warren |date=1948 |title=Science and Complexity |url=https://www.jstor.org/stable/27826254 |journal=American Scientist |volume=36 |issue=4 |pages=536–544 |jstor=27826254 |pmid=18882675 |issn=0003-0996}}</ref> Disorganised systems are linked to [[Self-organization|self-organisation]] processes.
* '''Hierarchic systems''' which are analyzable into successive sets of subsystems.<ref>{{Cite journal |last=Simon |first=Herbert A. |date=1962 |title=The Architecture of Complexity |url=https://www.jstor.org/stable/985254 |journal=Proceedings of the American Philosophical Society |volume=106 |issue=6 |pages=467–482 |jstor=985254 |issn=0003-049X}}</ref> They can also be called nested or embedded systems.
* '''[[Cybernetic system|Cybernetic systems]]''' involve information feedback loops.

== Key concepts ==

=== Adaptation ===
[[Complex adaptive system]]s are special cases of complex systems that are [[Adaptive system|adaptive]] in that they have the capacity to change and learn from experience.<ref>{{cite book
| last = Holland
| first = John H.
| author-link = John Henry Holland
| date = 2014
| title = Complexity: A Very Short Introduction
| publisher = Oxford University Press
| isbn = 978-0-19-966254-8
| quote = All CAS agents, whatever their particularities, have three levels of activity: {{pb}} 1. Performance (moment-by-moment capabilities) {{pb}} 2. Credit-assignment (rating the usefulness of available capabilities) {{pb}} 3. Rule-discovery (generating new capabilities).}}</ref> Examples of complex adaptive systems include the international [[trade]] markets, social insect and [[ant]] colonies, the [[biosphere]] and the [[ecosystem]], the [[Human brain|brain]] and the [[immune system]], the [[Cell (biology)|cell]] and the developing [[embryo]], cities, [[Manufacturing|manufacturing businesses]] and any human social group-based endeavor in a cultural and [[social system]] such as [[Political party|political parties]] or [[Community|communities]].<ref>{{Cite journal |last1=Skrimizea |first1=Eirini |last2=Haniotou |first2=Helene |last3=Parra |first3=Constanza |year=2019 |title=On the 'complexity turn' in planning: An adaptive rationale to navigate spaces and times of uncertainty |journal=Planning Theory |volume=18 |pages=122–142 |doi=10.1177/1473095218780515 |s2cid=149578797 |doi-access=free}}</ref>

=== Decomposability ===
A system is '''decomposable''' if the parts of the system (subsystems) are independent from each other, for exemple the model of a [[perfect gas]] consider the relations among molecules negligeable.<ref name=":1">{{Cite journal |last=Simon |first=Herbert A. |date=1962 |title=The Architecture of Complexity |url=https://www.jstor.org/stable/985254 |journal=Proceedings of the American Philosophical Society |volume=106 |issue=6 |pages=467–482 |jstor=985254 |issn=0003-049X}}</ref> 

In a '''nearly decomposable''' system, the interactions between subsystems are weak but not negligeable, this is often the case in social systems.<ref name=":1" /> Conceptually, a system is nearly decomposable if the variables composing it can be separated into classes and subclasses, if these variables are independent for many functions but affect each other, and if the whole system is greater than the parts.<ref>{{Cite journal |last=Ostrom |first=Elinor |date=2007 |title=Sustainable Social-Ecological Systems: An Impossibility? |url=http://www.ssrn.com/abstract=997834 |journal=SSRN Electronic Journal |language=en |doi=10.2139/ssrn.997834 |issn=1556-5068|hdl=10535/3826 |hdl-access=free }}</ref>

==Features==
Complex systems may have the following features:<ref>{{Cite book |last=Alan Randall |url=https://books.google.com/books?id=IlHj3fvJzMsC |title=Risk and Precaution |publisher=Cambridge University Press |year=2011 |isbn=9781139494793 |author-link=Alan Randall (economist)}}</ref>

;Complex systems may be open
: Complex systems are usually [[Open system (systems theory)|open systems]] – that is, they exist in a [[thermodynamic]] gradient and dissipate energy. In other words, complex systems are frequently far from energetic [[thermodynamic equilibrium|equilibrium]]: but despite this flux, there may be [[pattern stability]],<ref>{{Cite book |last=Pokrovskii |first=Vladimir |title=Thermodynamics of Complex Systems: Principles and applications. |publisher=IOP Publishing, Bristol, UK. |year=2021 |language=English |bibcode=2020tcsp.book.....P}}</ref> see [[synergetics (Haken)|synergetics]].

;Complex systems may exhibit critical transitions
[[File:Alternative stable states, critical transitions, and the direction of critical slowing down.png|thumb|Graphical representation of alternative stable states and the direction of critical slowing down prior to a critical transition (taken from Lever et al. 2020).<ref name="auto11">{{Cite journal |last1=Lever |first1=J. Jelle |last2=Leemput |first2=Ingrid A. |last3=Weinans |first3=Els |last4=Quax |first4=Rick |last5=Dakos |first5=Vasilis |last6=Nes |first6=Egbert H. |last7=Bascompte |first7=Jordi |last8=Scheffer |first8=Marten |year=2020 |title=Foreseeing the future of mutualistic communities beyond collapse |journal=Ecology Letters |volume=23 |issue=1 |pages=2–15 |doi=10.1111/ele.13401 |pmc=6916369 |pmid=31707763|bibcode=2020EcolL..23....2L }}</ref> Top panels (a) indicate stability landscapes at different conditions. Middle panels (b) indicate the rates of change akin to the slope of the stability landscapes, and bottom panels (c) indicate a recovery from a perturbation towards the system's future state (c.I) and in another direction (c.II).]]
:[[Critical transition]]s are abrupt shifts in the state of [[ecosystem]]s, the [[climate]], financial and economic systems or other complex systems that may occur when changing conditions pass a critical or [[bifurcation theory|bifurcation point]].<ref>{{Cite journal |last1=Scheffer |first1=Marten |last2=Carpenter |first2=Steve |last3=Foley |first3=Jonathan A. |last4=Folke |first4=Carl |last5=Walker |first5=Brian |date=October 2001 |title=Catastrophic shifts in ecosystems |url=https://www.nature.com/articles/35098000 |journal=Nature |language=en |volume=413 |issue=6856 |pages=591–596 |bibcode=2001Natur.413..591S |doi=10.1038/35098000 |issn=1476-4687 |pmid=11595939 |s2cid=8001853|url-access=subscription }}</ref><ref>{{Cite book |last=Scheffer |first=Marten |title=Critical transitions in nature and society |date=26 July 2009 |publisher=Princeton University Press |isbn=978-0691122045}}</ref><ref>{{Cite journal |last1=Scheffer |first1=Marten |last2=Bascompte |first2=Jordi |last3=Brock |first3=William A. |last4=Brovkin |first4=Victor |last5=Carpenter |first5=Stephen R. |last6=Dakos |first6=Vasilis |last7=Held |first7=Hermann |last8=van Nes |first8=Egbert H. |last9=Rietkerk |first9=Max |last10=Sugihara |first10=George |date=September 2009 |title=Early-warning signals for critical transitions |url=https://www.nature.com/articles/nature08227 |journal=Nature |language=en |volume=461 |issue=7260 |pages=53–59 |bibcode=2009Natur.461...53S |doi=10.1038/nature08227 |issn=1476-4687 |pmid=19727193 |s2cid=4001553|url-access=subscription }}</ref><ref>{{Cite journal |last1=Scheffer |first1=Marten |last2=Carpenter |first2=Stephen R. |last3=Lenton |first3=Timothy M. |last4=Bascompte |first4=Jordi |last5=Brock |first5=William |last6=Dakos |first6=Vasilis |last7=Koppel |first7=Johan van de |last8=Leemput |first8=Ingrid A. van de |last9=Levin |first9=Simon A. |last10=Nes |first10=Egbert H. van |last11=Pascual |first11=Mercedes |last12=Vandermeer |first12=John |date=19 October 2012 |title=Anticipating Critical Transitions |url=https://www.science.org/doi/10.1126/science.1225244 |url-status=live |journal=Science |language=en |volume=338 |issue=6105 |pages=344–348 |bibcode=2012Sci...338..344S |doi=10.1126/science.1225244 |issn=0036-8075 |pmid=23087241 |s2cid=4005516 |archive-url=https://web.archive.org/web/20200624023841/https://science.sciencemag.org/content/338/6105/344 |archive-date=24 June 2020 |access-date=10 June 2020 |hdl-access=free |hdl=11370/92048055-b183-4f26-9aea-e98caa7473ce}}</ref> The 'direction of critical slowing down' in a system's state space may be indicative of a system's future state after such transitions when delayed negative feedbacks leading to oscillatory or other complex dynamics are weak.<ref name="auto11" />

;Complex systems may be [[Hierarchy#Nested hierarchy|nested]]
:The components of a complex system may themselves be complex systems. For example, an [[Economics|economy]] is made up of [[organisation]]s, which are made up of [[person|people]], which are made up of [[cell (biology)|cells]] – all of which are complex systems. The arrangement of interactions within complex bipartite networks may be nested as well. More specifically, bipartite ecological and organisational networks of mutually beneficial interactions were found to have a nested structure.<ref>{{Cite journal |last1=Bascompte |first1=J. |last2=Jordano |first2=P. |last3=Melian |first3=C. J. |last4=Olesen |first4=J. M. |date=24 July 2003 |title=The nested assembly of plant-animal mutualistic networks |journal=Proceedings of the National Academy of Sciences |volume=100 |issue=16 |pages=9383–9387 |bibcode=2003PNAS..100.9383B |doi=10.1073/pnas.1633576100 |pmc=170927 |pmid=12881488 |doi-access=free}}</ref><ref>{{Cite journal |last1=Saavedra |first1=Serguei |last2=Reed-Tsochas |first2=Felix |last3=Uzzi |first3=Brian |date=January 2009 |title=A simple model of bipartite cooperation for ecological and organizational networks |journal=Nature |language=en |volume=457 |issue=7228 |pages=463–466 |bibcode=2009Natur.457..463S |doi=10.1038/nature07532 |issn=1476-4687 |pmid=19052545 |s2cid=769167}}</ref> This structure promotes indirect facilitation and a system's capacity to persist under increasingly harsh circumstances as well as the potential for large-scale systemic regime shifts.<ref>{{Cite journal |last1=Bastolla |first1=Ugo |last2=Fortuna |first2=Miguel A. |last3=Pascual-García |first3=Alberto |last4=Ferrera |first4=Antonio |last5=Luque |first5=Bartolo |last6=Bascompte |first6=Jordi |date=April 2009 |title=The architecture of mutualistic networks minimizes competition and increases biodiversity |journal=Nature |language=en |volume=458 |issue=7241 |pages=1018–1020 |bibcode=2009Natur.458.1018B |doi=10.1038/nature07950 |issn=1476-4687 |pmid=19396144 |s2cid=4395634}}</ref><ref>{{Cite journal |last1=Lever |first1=J. Jelle |last2=Nes |first2=Egbert H. van |last3=Scheffer |first3=Marten |last4=Bascompte |first4=Jordi |date=2014 |title=The sudden collapse of pollinator communities |journal=Ecology Letters |language=en |volume=17 |issue=3 |pages=350–359 |doi=10.1111/ele.12236 |issn=1461-0248 |pmid=24386999 |bibcode=2014EcolL..17..350L |hdl-access=free |hdl=10261/91808}}</ref>

;Dynamic network of multiplicity
:As well as [[coupling]] rules, the dynamic [[Biological network|network]] of a complex system is important. [[Small-world network|Small-world]] or [[Scale-free network|scale-free]] networks<ref>{{Cite journal |last=A. L. Barab´asi |first=R. Albert |year=2002 |title=Statistical mechanics of complex networks |journal=Reviews of Modern Physics |volume=74 |issue=1 |pages=47–94 |arxiv=cond-mat/0106096 |bibcode=2002RvMP...74...47A |citeseerx=10.1.1.242.4753 |doi=10.1103/RevModPhys.74.47 |s2cid=60545}}</ref><ref>{{Cite book |last=M. Newman |title=Networks: An Introduction |publisher=Oxford University Press |year=2010 |isbn=978-0-19-920665-0}}</ref> which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human [[Cerebral cortex|cortex]] for example, we see dense local connectivity and a few very long [[axonal|axon]] projections between regions inside the cortex and to other brain regions.

[[File:Gospers glider gun.gif|frame|right|[[Bill Gosper|Gosper's]] [[Gun (cellular automaton)|Glider Gun]] creating "[[Glider (Conway's Life)|gliders]]" in the cellular automaton [[Conway's Game of Life]]<ref>[[Daniel Dennett]] (1995), ''[[Darwin's Dangerous Idea]]'', Penguin Books, London, {{ISBN|978-0-14-016734-4}}, {{ISBN|0-14-016734-X}}</ref>]]
; May produce emergent phenomena
:Complex systems may exhibit behaviors that are [[emergence|emergent]], which is to say that while the results may be sufficiently determined by the activity of the systems' basic constituents, they may have properties that can only be studied at a higher level.  For example, empirical food webs display regular, scale-invariant features across aquatic and terrestrial ecosystems when studied at the level of clustered 'trophic' species.<ref>{{Cite book |last1=Cohen |first1=J.E. |url=https://link.springer.com/book/10.1007/978-3-642-83784-5 |title=Community Food Webs: Data and Theory |last2=Briand |first2=F. |last3=Newman |first3=C.M. |date=1990 |publisher=Springer |isbn=9783642837869 |location=Berlin, Heidelberg, New York |page=308 |doi=10.1007/978-3-642-83784-5}}</ref><ref>{{Cite journal |last1=Briand |first1=F. |last2=Cohen |first2=J.E. |date=1984 |title=Community food webs have scale-invariant structure |journal=Nature |volume=307 |issue=5948 |pages=264–267 |bibcode=1984Natur.307..264B |doi=10.1038/307264a0 |s2cid=4319708}}</ref> Another example is offered by the [[termites]] in a mound which have physiology, biochemistry and biological development at one level of analysis, whereas their [[social behavior]] and mound building is a property that emerges from the collection of termites and needs to be analyzed at a different level.

; Relationships are non-linear
: In practical terms, this means a small perturbation may cause a large effect (see [[butterfly effect]]), a proportional effect, or even no effect at all. In linear systems, the effect is ''always'' directly proportional to cause. See [[nonlinearity]].

; Relationships contain feedback loops
:Both negative ([[Damping ratio|damping]]) and positive (amplifying) [[feedback]] are always found in complex systems. The effects of an element's behavior are fed back in such a way that the element itself is altered.

== History ==
In 1948, Dr. Warren Weaver published an essay on "Science and Complexity",<ref>{{cite journal |last1=Warren |first1=Weaver |title=Science and Complexity |journal=American Scientist |date=Oct 1948 |volume=36 |issue=4 |pages=536–544 |jstor=27826254 |pmid=18882675 |url=https://www.jstor.org/stable/27826254 |access-date=28 October 2023}}</ref> exploring the diversity of problem types by contrasting problems of simplicity, disorganized complexity, and organized complexity. Weaver described these as "problems which involve dealing simultaneously with a sizable number of factors which are interrelated into an organic whole."

While the explicit study of complex systems dates at least to the 1970s,<ref>{{Cite book |last=Vemuri |first=V. |title=Modeling of Complex Systems: An Introduction |date=1978 |publisher=Academic Press |isbn=978-0127165509 |location=New York}}</ref> the first research institute focused on complex systems, the [[Santa Fe Institute]], was founded in 1984.<ref>{{Cite journal |last=Ledford |first=H |year=2015 |title=How to solve the world's biggest problems |journal=Nature |volume=525 |issue=7569 |pages=308–311 |bibcode=2015Natur.525..308L |doi=10.1038/525308a |pmid=26381968 |doi-access=free}}</ref><ref>{{Cite web |title=History |publisher=Santa Fe Institute |url=https://www.santafe.edu/about/history |url-status=dead |archive-url=https://web.archive.org/web/20190403154434/https://www.santafe.edu/about/history |archive-date=2019-04-03 |access-date=2018-05-17 |language=en}}</ref> Early Santa Fe Institute participants included physics Nobel laureates [[Murray Gell-Mann]] and [[Philip Warren Anderson|Philip Anderson]], economics Nobel laureate [[Kenneth Arrow]], and Manhattan Project scientists [[George Cowan]] and [[Herbert L. Anderson|Herb Anderson]].<ref>Waldrop, M. M. (1993). [https://archive.org/details/complexity00mmit Complexity: The emerging science at the edge of order and chaos.] Simon and Schuster.</ref> Today, there are over 50 institutes and research centers focusing on complex systems.{{citation needed|date=April 2019}}

Since the late 1990s, the interest of mathematical physicists in researching economic phenomena has been on the rise. The proliferation of cross-disciplinary research with the application of solutions originated from the physics epistemology has entailed a gradual paradigm shift in the theoretical articulations and methodological approaches in economics, primarily in financial economics. The development has resulted in the emergence of a new branch of discipline, namely "econophysics", which is broadly defined as a cross-discipline that applies statistical physics methodologies which are mostly based on the complex systems theory and the chaos theory for economics analysis.<ref>{{Cite journal |last1=Ho |first1=Y. J. |last2=Ruiz Estrada |first2=M. A |last3=Yap |first3=S. F. |date=2016 |title=The evolution of complex systems theory and the advancement of econophysics methods in the study of stock market crashes |url=https://jurcon.ums.edu.my/ojums/index.php/lbibf/article/view/1320 |journal=Labuan Bulletin of International Business & Finance |volume=14 |pages=68–83}}</ref>

The 2021 [[Nobel Prize in Physics]] was awarded to [[Syukuro Manabe]], [[Klaus Hasselmann]], and [[Giorgio Parisi]] for their work to understand complex systems. Their work was used to create more accurate computer models of the effect of global warming on the Earth's climate.<ref>{{Cite news |date=5 October 2021 |title=Nobel in physics: Climate science breakthroughs earn prize |publisher=BBC News |url=https://www.bbc.co.uk/news/science-environment-58790160}}</ref>

== Applications ==

===Complexity in practice===
The traditional approach to dealing with complexity is to reduce or constrain it. Typically, this involves compartmentalization: dividing a large system into separate parts. Organizations, for instance, divide their work into departments that each deal with separate issues. Engineering systems are often designed using modular components. However, modular designs become susceptible to failure when issues arise that bridge the divisions.

===Complexity of cities===
Jane Jacobs described cities as being a problem in organized complexity in 1961, citing Dr. Weaver's 1948 essay.<ref>{{cite book |last1=Jacobs |first1=Jane |title=The Death and Life of Great American Cities |date=1961 |publisher=Vintage Books |location=New York |pages=428–448}}</ref> As an example, she explains how an abundance of factors interplay into how various urban spaces lead to a diversity of interactions, and how changing those factors can change how the space is used, and how well the space supports the functions of the city. She further illustrates how cities have been severely damaged when approached as a problem in simplicity by replacing organized complexity with simple and predictable spaces, such as Le Corbusier's "Radiant City" and Ebenezer Howard's "Garden City". Since then, others have written at length on the complexity of cities.<ref>{{cite web |title=Cities, scaling, & sustainability |url=https://www.santafe.edu/research/projects/cities-scaling-sustainability |publisher=Santa Fe Institute |access-date=28 October 2023}}</ref>

===Complexity economics===
Over the last decades, within the emerging field of [[complexity economics]], new predictive tools have been developed to explain economic growth. Such is the case with the models built by the [[Santa Fe Institute]] in 1989 and the more recent [[economic complexity index]] (ECI), introduced by the [[MIT]] physicist [[Cesar A. Hidalgo]] and the [[Harvard]] economist [[Ricardo Hausmann]].

[[Recurrence quantification analysis]] has been employed to detect the characteristic of [[business cycles]] and [[economic development]]. To this end, Orlando et al.<ref>{{Cite journal |last1=Orlando |first1=Giuseppe |last2=Zimatore |first2=Giovanna |date=18 December 2017 |title=RQA correlations on real business cycles time series |journal=Indian Academy of Sciences – Conference Series |volume=1 |issue=1 |pages=35–41 |doi=10.29195/iascs.01.01.0009 |doi-access=free}}</ref> developed the so-called recurrence quantification correlation index (RQCI) to test correlations of RQA on a sample signal and then investigated the application to business time series. The said index has been proven to detect hidden changes in time series. Further, Orlando et al.,<ref>{{Cite journal |last1=Orlando |first1=Giuseppe |last2=Zimatore |first2=Giovanna |date=1 May 2018 |title=Recurrence quantification analysis of business cycles |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077918300924 |journal=Chaos, Solitons & Fractals |language=en |volume=110 |pages=82–94 |bibcode=2018CSF...110...82O |doi=10.1016/j.chaos.2018.02.032 |issn=0960-0779 |s2cid=85526993|url-access=subscription }}</ref> over an extensive dataset, shown that recurrence quantification analysis may help in anticipating transitions from laminar (i.e. regular) to turbulent (i.e. chaotic) phases such as USA GDP in 1949, 1953, etc. Last but not least, it has been demonstrated that recurrence quantification analysis can detect differences between macroeconomic variables and highlight hidden features of economic dynamics.

=== Complexity and education ===
Focusing on issues of student persistence with their studies, Forsman, Moll and Linder explore the "viability of using complexity science as a frame to extend methodological applications for physics education research", finding that "framing a social network analysis within a complexity science perspective offers a new and powerful applicability across a broad range of PER topics".<ref>{{Cite journal |last1=Forsman |first1=Jonas |last2=Moll |first2=Rachel |last3=Linder |first3=Cedric |date=2014 |title=Extending the theoretical framing for physics education research: An illustrative application of complexity science |journal=Physical Review Special Topics - Physics Education Research |volume=10 |issue=2 |pages=020122 |bibcode=2014PRPER..10b0122F |doi=10.1103/PhysRevSTPER.10.020122 |doi-access=free |hdl=10613/2583|hdl-access=free }}</ref>

=== Complexity in healthcare research and practice ===

Healthcare systems are prime examples of complex systems, characterized by interactions among diverse stakeholders, such as patients, providers, policymakers, and researchers, across various sectors like health, government, community, and education. These systems demonstrate properties like non-linearity, emergence, adaptation, and feedback loops.<ref name=":0">{{Cite journal |last1=Kitson |first1=Alison |last2=Brook |first2=Alan |last3=Harvey |first3=Gill |last4=Jordan |first4=Zoe |last5=Marshall |first5=Rhianon |last6=O’Shea |first6=Rebekah |last7=Wilson |first7=David |date=2018-03-01 |title=Using Complexity and Network Concepts to Inform Healthcare Knowledge Translation |url=https://www.ijhpm.com/article_3385.html |journal=International Journal of Health Policy and Management |language=en |volume=7 |issue=3 |pages=231–243 |doi=10.15171/ijhpm.2017.79 |issn=2322-5939 |pmc=5890068 |pmid=29524952}}</ref> Complexity science in healthcare frames [[knowledge translation]] as a dynamic and interconnected network of processes—problem identification, knowledge creation, synthesis, implementation, and evaluation—rather than a linear or cyclical sequence. Such approaches emphasize the importance of understanding and leveraging the interactions within and between these processes and stakeholders to optimize the creation and movement of knowledge. By acknowledging the complex, adaptive nature of healthcare systems, [[Complexity Science|complexity science]] advocates for continuous stakeholder engagement, [[transdisciplinary]] collaboration, and flexible strategies to effectively translate research into practice.<ref name=":0" />

=== Complexity and biology ===
Complexity science has been applied to living organisms, and in particular to biological systems. Within the emerging field of [[fractal physiology]], bodily signals, such as heart rate or brain activity, are characterized using [[entropy]] or fractal indices. The goal is often to assess the state and the health of the underlying system, and diagnose potential disorders and illnesses.{{citation needed|date=July 2024}}

===Complexity and chaos theory===
Complex systems theory is related to [[chaos theory]], which in turn has its origins more than a century ago in the work of the French mathematician [[Henri Poincaré]]. Chaos is sometimes viewed as extremely complicated information, rather than as an absence of order.<ref>Hayles, N. K. (1991). ''[https://books.google.com/books?id=9g9QDwAAQBAJ&pg=PR7 Chaos Bound: Orderly Disorder in Contemporary Literature and Science]''. Cornell University Press, Ithaca, NY.</ref> Chaotic systems remain deterministic, though their long-term behavior can be difficult to predict with any accuracy. With perfect knowledge of the initial conditions and the relevant equations describing the chaotic system's behavior, one can theoretically make perfectly accurate predictions of the system, though in practice this is impossible to do with arbitrary accuracy.

The emergence of complex systems theory shows a domain between deterministic order and randomness which is complex.<ref name="PC98">[[Paul Cilliers|Cilliers, P.]] (1998). ''Complexity and Postmodernism: Understanding Complex Systems'', Routledge, London.</ref> This is referred to as the "[[edge of chaos]]".<ref>[[Per Bak]] (1996). ''How Nature Works: The Science of Self-Organized Criticality'', Copernicus, New York, U.S.</ref>

[[File:Lorenz attractor yb.svg|thumb|right|200px|A plot of the [[Lorenz attractor]]]]

When one analyzes complex systems, sensitivity to initial conditions, for example, is not an issue as important as it is within chaos theory, in which it prevails. As stated by Colander,<ref>Colander, D. (2000). ''The Complexity Vision and the Teaching of Economics'', E. Elgar, Northampton, Massachusetts.</ref> the study of complexity is the opposite of the study of chaos. Complexity is about how a huge number of extremely complicated and dynamic sets of relationships can generate some simple behavioral patterns, whereas chaotic behavior, in the sense of deterministic chaos, is the result of a relatively small number of non-linear interactions.<ref name="PC98" /> For recent examples in economics and business see Stoop et al.<ref>{{Cite journal |last1=Stoop |first1=Ruedi |last2=Orlando |first2=Giuseppe |last3=Bufalo |first3=Michele |last4=Della Rossa |first4=Fabio |date=2022-11-18 |title=Exploiting deterministic features in apparently stochastic data |journal=Scientific Reports |language=en |volume=12 |issue=1 |pages=19843 |bibcode=2022NatSR..1219843S |doi=10.1038/s41598-022-23212-x |issn=2045-2322 |pmc=9674651 |pmid=36400910}}</ref> who discussed [[Android (operating system)|Android]]'s market position, Orlando<ref>{{Cite journal |last=Orlando |first=Giuseppe |date=2022-06-01 |title=Simulating heterogeneous corporate dynamics via the Rulkov map |url=https://www.sciencedirect.com/science/article/pii/S0954349X22000121 |journal=Structural Change and Economic Dynamics |language=en |volume=61 |pages=32–42 |doi=10.1016/j.strueco.2022.02.003 |issn=0954-349X|url-access=subscription }}</ref> who explained the corporate dynamics in terms of mutual synchronization and chaos regularization of bursts in a group of chaotically bursting cells and Orlando et al.<ref>{{Cite journal |last1=Orlando |first1=Giuseppe |last2=Bufalo |first2=Michele |last3=Stoop |first3=Ruedi |date=2022-02-01 |title=Financial markets' deterministic aspects modeled by a low-dimensional equation |journal=Scientific Reports |language=en |volume=12 |issue=1 |pages=1693 |bibcode=2022NatSR..12.1693O |doi=10.1038/s41598-022-05765-z |issn=2045-2322 |pmc=8807815 |pmid=35105929}}</ref> who modelled financial data (Financial Stress Index, swap and equity, emerging and developed, corporate and government, short and long maturity) with a low-dimensional deterministic model.

Therefore, the main difference between chaotic systems and complex systems is their history.<ref>Buchanan, M. (2000). ''Ubiquity : Why catastrophes happen'', three river press, New-York.</ref> Chaotic systems do not rely on their history as complex ones do. Chaotic behavior pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'.{{clarify|date=September 2011}} On the other hand, complex systems evolve far from equilibrium at the edge of chaos. They evolve at a critical state built up by a history of irreversible and unexpected events, which physicist [[Murray Gell-Mann]] called "an accumulation of frozen accidents".<ref>Gell-Mann, M. (1995). What is Complexity?  Complexity 1/1, 16-19</ref> In a sense chaotic systems can be regarded as a subset of complex systems distinguished precisely by this absence of historical dependence. Many real complex systems are, in practice and over long but finite periods, robust. However, they do possess the potential for radical qualitative change of kind whilst retaining systemic integrity. Metamorphosis serves as perhaps more than a metaphor for such transformations.

{{clear left}}

===Complexity and network science===
A complex system is usually composed of many components and their interactions. Such a system can be represented by a network where nodes represent the components and links represent their interactions.<ref name="DorogovtsevMendes2003">{{Cite book |last1=Dorogovtsev |first1=S.N. |title=Evolution of Networks |last2=Mendes |first2=J.F.F. |year=2003 |isbn=9780198515906 |volume=51 |pages=1079 |doi=10.1093/acprof:oso/9780198515906.001.0001 |arxiv=cond-mat/0106144}}</ref><ref name="Newman2010">{{Cite book |last=Newman |first=Mark |url=https://cds.cern.ch/record/1281254 |title=Networks |year=2010 |isbn=9780199206650 |doi=10.1093/acprof:oso/9780199206650.001.0001}}{{Dead link|date=July 2020 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> For example, the [[Internet]] can be represented as a network composed of nodes (computers) and links (direct connections between computers). Other examples of complex networks include social networks, financial institution interdependencies,<ref>{{Cite journal |last1=Battiston |first1=Stefano |last2=Caldarelli |first2=Guido |last3=May |first3=Robert M. |last4=Roukny |first4=tarik |last5=Stiglitz |first5=Joseph E. |date=2016-09-06 |title=The price of complexity in financial networks |journal=Proceedings of the National Academy of Sciences |language=en |volume=113 |issue=36 |pages=10031–10036 |bibcode=2016PNAS..11310031B |doi=10.1073/pnas.1521573113 |pmc=5018742 |pmid=27555583 |doi-access=free}}</ref> airline networks,<ref name="BarratBarthelemy2004">{{Cite journal |last1=Barrat |first1=A. |last2=Barthelemy |first2=M. |last3=Pastor-Satorras |first3=R. |last4=Vespignani |first4=A. |year=2004 |title=The architecture of complex weighted networks |journal=Proceedings of the National Academy of Sciences |volume=101 |issue=11 |pages=3747–3752 |arxiv=cond-mat/0311416 |bibcode=2004PNAS..101.3747B |doi=10.1073/pnas.0400087101 |issn=0027-8424 |pmc=374315 |pmid=15007165 |doi-access=free}}</ref> and biological networks.

== Notable scholars ==
<!--Entries in this list should be "notable" with a sourced Wikipedia article.-->
{{columns-list|colwidth=20em|
* [[Robert McCormick Adams Jr.|Robert McCormick Adams]]
* [[Christopher Alexander]]
* [[Philip Warren Anderson|Philip Anderson]]
* [[Kenneth Arrow]]
* [[Robert Axelrod (political scientist)|Robert Axelrod]]
* [[W. Brian Arthur]]
* [[Per Bak]]
* [[Béla H. Bánáthy]]
* [[Niklas Luhmann]]
* [[Albert-László Barabási|Albert-Laszlo Barabasi]]
* [[Yaneer Bar-Yam]]
* [[Gregory Bateson]]
* [[Ludwig von Bertalanffy]]
* [[Alexander Bogdanov]]
* [[Samuel Bowles (economist)|Samuel Bowles]]
* [[Guido Caldarelli]]
* [[Paul Cilliers]]
* [[Walter Clemens, Jr.]]
* [[James P. Crutchfield]]
* [[Chris Danforth]]
* [[Peter Sheridan Dodds]]
* [[Brian J. Enquist|Brian Enquist]]
* [[Joshua M. Epstein|Joshua Epstein]]
* [[J. Doyne Farmer|Doyne Farmer]]
* [[Jay Forrester]]
* [[Nigel R. Franks]]
* [[Murray Gell-Mann]]
* [[Carlos Gershenson]]
* [[Nigel Goldenfeld]]
* [[Vittorio Guidano]]
* [[Hermann Haken]]
* [[James Hartle]]
* [[Friedrich Hayek|F. A. Hayek]]
* [[Dirk Helbing]]
* [[John Henry Holland|John Holland]]
* [[Alfred Hübler|Alfred Hubler]]
* [[Arthur Iberall]]
* Johannes Jaeger
* [[Stuart Kauffman]]
* [[J. A. Scott Kelso]]
* [[David Krakauer (scientist)|David Krakauer]]
* [[Simon A. Levin]]
* [[Ellen Levy]]
* [[Robert May, Baron May of Oxford|Robert May]]
* [[Donella Meadows]]
* [[José Fernando Mendes]]
* [[Melanie Mitchell]]
* [[Cris Moore]]
* [[Yamir Moreno]]
* [[Edgar Morin]]
* [[Harold J. Morowitz|Harold Morowitz]]
* [[Adilson E. Motter]]
* [[Scott E. Page|Scott Page]]
* [[Luciano Pietronero]]
* [[David Pines]]
* [[Vladimir Pokrovskii]]
* [[William T. Powers]]
* [[Ilya Prigogine]]
* [[Steen Rasmussen (physicist)|Steen Rasmussen]]
* [[Sidney Redner]]
* [[Jerry Sabloff]]
* [[Cosma Shalizi]]
* [[Herbert A. Simon|Herbert Simon]]
* [[Dave Snowden]]
* [[Sergei Starostin]]
* [[Steven Strogatz]]
* [[Stefan Thurner]]
* [[Alessandro Vespignani]]
* [[Andreas Wagner]]
* [[Duncan J. Watts|Duncan Watts]]
* [[Geoffrey West]]
* [[Stephen Wolfram]]
* [[David Wolpert]]
}}

== See also ==
{{Portal|Systems science}}
{|
|- style="vertical-align:top"
|style="padding-right:2em"|
* [[Biological organisation]]
* [[Chaos theory]]
* [[Cognitive model#Dynamical systems|Cognitive modeling]]
* [[Cognitive science]]
* [[Complex (disambiguation)]]
* [[Complex adaptive system]]
* [[Complex networks]]
* [[Complexity]]
* [[Complexity (disambiguation)]]
* [[Complexity economics]]
* [[Complexity Science Hub Vienna]]
* [[Cybernetics]]
|style="padding-right:2em"|
* [[Decision engineering]]
* [[Dissipative system]]
* [[Dual-phase evolution]]
* [[Dynamical system]]
* [[Dynamical systems theory]]
* [[Emergence]]
* [[Enterprise systems engineering]]
* [[Fractal]]
* [[Fractal physiology]]
* [[Generative sciences]]
* [[Hierarchy theory]]
|style="padding-right:2em"|
* [[Homeokinetics]]
* [[Interdependent networks]]
* [[Invisible hand]]
* [[Mixed reality]]
* [[Multi-agent system]]
* [[Network science]]
* [[Neuroscience and intelligence]]
* [[:fr:Noogenèse]]
* [[Nonlinearity]]
* [[Pattern-oriented modeling]]
* [[Percolation]]
* [[Percolation theory]]
|
* [[Process architecture]]
* [[Self-organization]]
* [[Sociology and complexity science]]
* [[System accident]]
* [[System dynamics]]
* [[System equivalence]]
*[[Systems engineering]] 
* [[Systems theory]]
* [[Systems theory in anthropology]]
* [[Tektology]]
* [[Ultra-large-scale systems]]
* [[VUCA|Volatility, uncertainty, complexity and ambiguity]]
|}

== References ==
{{Reflist}}

== Further reading ==
* [https://web.archive.org/web/20200424112123/https://complexityexplained.github.io/ Complexity Explained].
* [[Luis Amaral|L.A.N. Amaral]] and J.M. Ottino, [http://amaral-lab.org/media/publication_pdfs/Amaral-2004-Eur.Phys.J.B-38-147.pdf ''Complex networks – augmenting the framework for the study of complex system''], 2004.
* {{Cite journal |last1=Chu |first1=D. |last2=Strand |first2=R. |last3=Fjelland |first3=R. |year=2003 |title=Theories of complexity |journal=Complexity |volume=8 |issue=3 |pages=19–30 |bibcode=2003Cmplx...8c..19C |doi=10.1002/cplx.10059}}
* [[Walter Clemens, Jr.]], [https://web.archive.org/web/20150219221633/http://www.sunypress.edu/p-5782-complexity-science-and-world-af.aspx ''Complexity Science and World Affairs''], SUNY Press, 2013.
* {{Cite journal |last=Gell-Mann |first=Murray |year=1995 |title=Let's Call It Plectics |journal=Complexity |volume=1 |issue=5 |pages=3–5 |bibcode=1996Cmplx...1e...3G |doi=10.1002/cplx.6130010502 |doi-access=free}}
* A. Gogolin, A. Nersesyan and A. Tsvelik, [https://web.archive.org/web/20070715195144/http://www.cmth.bnl.gov/~tsvelik/theory.html ''Theory of strongly correlated systems ''], Cambridge University Press, 1999.
* [[Nigel Goldenfeld]] and Leo P. Kadanoff, [http://guava.physics.uiuc.edu/~nigel/articles/complexity.html ''Simple Lessons from Complexity''] {{Webarchive|url=https://web.archive.org/web/20170928102108/http://guava.physics.uiuc.edu/~nigel/articles/complexity.html |date=2017-09-28 }}, 1999
* Kelly, K. (1995). [http://www.kk.org/outofcontrol/contents.php ''Out of Control''], Perseus Books Group.
* {{Cite book |last1=Orlando |first1=Giuseppe Orlando |url=https://link.springer.com/book/10.1007/978-3-030-70982-2#editorsandaffiliations |title=Nonlinearities in Economics |last2=Pisarchick |first2=Alexander |last3=Stoop |first3=Ruedi |year=2021 |isbn=978-3-030-70981-5 |series=Dynamic Modeling and Econometrics in Economics and Finance |volume=29 |language=en-gb |doi=10.1007/978-3-030-70982-2 |s2cid=239756912}}
* Syed M. Mehmud (2011), [https://web.archive.org/web/20120426052819/http://predictivemodeler.com/sitecontent/book/Ch06_Applications/Actuarial/HEC_Model/Healthcare%20Exchange%20Complexity%20Model%20-%20Report%20-%20Aug2011.pdf ''A Healthcare Exchange Complexity Model'']
* [https://web.archive.org/web/20110220054920/http://www.oeaw.ac.at/byzanz/repository/Preiser_WorkingPapers_Calculating_I.pdf Preiser-Kapeller, Johannes, "Calculating Byzantium. Social Network Analysis and Complexity Sciences as tools for the exploration of medieval social dynamics". August 2010]
* {{Cite journal |last=Donald Snooks |first=Graeme |year=2008 |title=A general theory of complex living systems: Exploring the demand side of dynamics |journal=Complexity |volume=13 |issue=6 |pages=12–20 |bibcode=2008Cmplx..13f..12S |doi=10.1002/cplx.20225 |doi-access=free}}
* [[Stefan Thurner]], Peter Klimek, Rudolf Hanel: ''Introduction to the Theory of Complex Systems'', Oxford University Press, 2018, {{ISBN|978-0198821939}}
* [https://web.archive.org/web/20200728201713/https://sfi-edu.s3.amazonaws.com/sfi-edu/production/uploads/publication/2016/10/31/Bulletin_Fall_2014_FINAL.pdf SFI @30, Foundations & Frontiers] (2014).

== External links ==
{{Commons category|Complex systems}}
{{Wiktionary|complex systems}}
* {{Cite web |title=The Open Agent-Based Modeling Consortium |url=http://www.openabm.org}}
* {{Cite web |title=Complexity Science Focus |url=http://www.complexity.ecs.soton.ac.uk/ |url-status=dead |archive-url=https://web.archive.org/web/20171205062624/http://www.complexity.ecs.soton.ac.uk/ |archive-date=2017-12-05 |access-date=2017-09-22}}
* {{Cite web |title=Santa Fe Institute |url=http://www.santafe.edu/}}
* {{Cite web |title=The Center for the Study of Complex Systems, Univ. of Michigan Ann Arbor |url=http://www.lsa.umich.edu/cscs/ |access-date=2017-09-22 |archive-date=2017-12-13 |archive-url=https://web.archive.org/web/20171213132925/https://lsa.umich.edu/cscs |url-status=dead }}
* {{Cite web |title=INDECS |url=http://indecs.eu/}} (Interdisciplinary Description of Complex Systems)
* {{Cite web |title=Introduction to Complexity – Free online course by Melanie Mitchell |url=http://www.complexityexplorer.org/courses/89-introduction-to-complexity |url-status=dead |archive-url=https://web.archive.org/web/20180830073728/https://www.complexityexplorer.org/courses/89-introduction-to-complexity |archive-date=2018-08-30 |access-date=2018-08-29}}
* {{Cite web |last=Jessie Henshaw |date=October 24, 2013 |title=Complex Systems |url=http://www.eoearth.org/view/article/51cbed507896bb431f69154d/?topic=51cbfc79f702fc2ba8129ed7 |publisher=[[Encyclopedia of Earth]]}}
* [http://www.scholarpedia.org/article/Complex_Systems Complex systems] in scholarpedia.
* [http://cssociety.org Complex Systems Society]
* [https://web.archive.org/web/20080723135438/http://www.complexsystems.net.au/ (Australian) Complex systems research network.]
* [https://web.archive.org/web/20091130204009/http://informatics.indiana.edu/rocha/complex/csm.html Complex Systems Modeling] based on [[Luis M. Rocha]], 1999.
* [https://web.archive.org/web/20110722075059/http://www.crm.cat/ComplexSystems_Lines/defaultsistemescomplexos.htm CRM Complex systems research group]
* [https://web.archive.org/web/20110430200327/http://www.ccsr.uiuc.edu/ The Center for Complex Systems Research, Univ. of Illinois at Urbana-Champaign]
* [http://ifisc.uib.es/ Institute for Cross-Disciplinary Physics and Complex Systems (IFISC)]

{{Complex systems topics}}
{{Authority control}}

[[Category:Complex dynamics]]
[[Category:Complex systems theory| ]]
[[Category:Mathematical modeling]]