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Edge of chaos
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==Adaptation== [[Adaptation]] plays a vital role for all living organisms and systems. All of them are constantly changing their inner properties to better fit in the current environment.<ref>{{cite book|last1=Strogatz|first1=Steven|title=Nonlinear dynamics and Chaos|date=1994|publisher=[[Westview Press]]}}</ref> The most important instruments for the [[adaptation]] are the [[adaptive systems|self-adjusting parameters]] inherent for many natural systems. The prominent feature of systems with self-adjusting parameters is an ability to avoid [[chaos theory|chaos]]. The name for this phenomenon is ''"Adaptation to the edge of chaos"''. Adaptation to the edge of chaos refers to the idea that many [[complex adaptive systems]] (CASs) seem to intuitively evolve toward a regime near the boundary between chaos and order.<ref>{{cite book|last1=Kauffman|first1=S.A.|title=The Origins of Order Self-Organization and Selection in Evolution|url=https://archive.org/details/originsoforderse0000kauf|url-access=registration|date=1993|publisher=[[Oxford University Press]]|location=New York|isbn=9780195079517}}</ref> Physics has shown that edge of chaos is the optimal settings for control of a system.<ref>{{cite journal|last1=Pierre|first1=D.|last2=et.|first2=al.|title=A theory for adaptation and competition applied to logistic map dynamics|journal=Physica D|date=1994|volume=75|issue=1β3|pages=343β360|bibcode=1994PhyD...75..343P|doi=10.1016/0167-2789(94)90292-5}}</ref> It is also an optional setting that can influence the ability of a physical system to perform primitive functions for computation.<ref>{{cite journal|last1=Langton|first1=C.A.|title=Computation at the edge of chaos|journal=Physica D|date=1990|volume=42|issue=1β3|pages=12|doi=10.1016/0167-2789(90)90064-v|bibcode=1990PhyD...42...12L|osti=7264125 |url=https://zenodo.org/record/1258375}}</ref> In CAS, [[coevolution]] generally occurs near the edge of chaos, and a balance should be maintained between flexibility and stability to avoid structural failure.<ref name=CAS-T-29>{{cite web |url=http://www.faculty.umb.edu/david_levy/complex00.pdf|title=Applications and Limitations of Complexity Theory in Organization Theory and Strategy |publisher=umb.edu|last=L. Levy|first=David |access-date=23 August 2020}}</ref><ref name=CAS-T-30>{{cite web |url=http://www.strategy-business.com/article/15099?gko=d48d4|title=Between Chaos and Order: What Complexity Theory Can Teach Business|publisher=strategy-business.com|last=Berreby|first=David|date=1 April 1996|access-date=23 August 2020}}</ref><ref name=CAS-T-35>{{cite web |url=https://forestbioproducts.umaine.edu/wp-content/uploads/sites/202/2010/10/Porter.Coev-Proofs.pdf|title=Coevolution as a research framework for organizations and the natural environment|publisher=University of Maine|last=B. Porter|first=Terry|access-date=23 August 2020}}</ref><ref name=CAS-T-37>{{cite web |url=https://grantome.com/grant/NSF/DBI-9201536|title=Coevolution in Complex Adaptive Systems|publisher=Santa Fe Institute|last=Kauffman|first=Stuart |date=15 January 1992|access-date=24 August 2020}}</ref> As a response to coping with turbulent environments, CAS bring out [[flexibility]], creativity,<ref name=CAS-T-36>{{cite web |url=https://www.researchgate.net/publication/328890717|title=The Order-Chaos Dynamic of Creativity|publisher=University of New Brunswick|last=A Lambert|first=Philip|date=June 2018|access-date=24 August 2020}}</ref> [[agility]], [[anti-fragility]], and innovation near the edge of chaos, provided these systems are sufficiently decentralized and non-hierarchical.<ref name=CAS-T-35/><ref name=CAS-T-30/><ref name=CAS-T-29/> Because of the importance of adaptation in many natural systems, adaptation to the edge of the chaos takes a prominent position in many scientific researches. Physicists demonstrated that adaptation to state at the boundary of chaos and order occurs in population of [[cellular automata]] rules which optimize the performance evolving with a [[genetic algorithm]].<ref>{{cite journal|last1=Packard|first1=N.H.|title=Adaptation toward the edge of chaos|journal=Dynamic Patterns in Complex Systems|date=1988|pages=293β301}}</ref><ref>{{cite journal|last1=Mitchell|first1=M.|last2=Hraber|first2=P.|last3=Crutchfield|first3=J.|title=Revisiting the edge of chaos: Evolving cellular automata to perform computations|journal=Complex Systems|date=1993|volume=7|issue=2|pages=89β130|arxiv=adap-org/9303003|bibcode=1993adap.org..3003M}}</ref> Another example of this phenomenon is the [[self-organized criticality]] in [[avalanche]] and earthquake models.<ref>{{cite journal|last1=Bak|first1=P.|last2=Tang|first2=C.|last3=Wiesenfeld|first3=K.|title=Self-organized criticality|journal=Physical Review A|date=1988|volume=38|issue=1|pages=364β374|doi=10.1103/PhysRevA.38.364|bibcode=1988PhRvA..38..364B|pmid=9900174}}</ref> The simplest model for chaotic dynamics is the [[logistic map]]. Self-adjusting logistic map dynamics exhibit adaptation to the edge of chaos.<ref>{{cite journal|last1=Melby|first1=P.|last2=et.|first2=al.|title=Adaptation to the edge of chaos in the self-adjusting logistic map.|journal=Phys. Rev. Lett.|date=2000|doi=10.1103/PhysRevLett.84.5991|arxiv=nlin/0007006|bibcode=2000PhRvL..84.5991M|volume=84|issue=26|pages=5991β5993|pmid=10991106}}</ref> Theoretical analysis allowed prediction of the location of the narrow parameter regime near the boundary to which the system evolves.<ref>{{cite journal|last1=Baym|first1=M.|last2=et.|first2=al.|title=Conserved quantities and adaptation to the edge of chaos|journal=Physical Review E|date=2006|volume=73|issue=5|pages=056210|doi=10.1103/PhysRevE.73.056210|pmid=16803029 |bibcode=2006PhRvE..73e6210B}}</ref>
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