Editing Complex system (section)

==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.