Editing Nonlinear system (section)

{{Short description|System where changes of output are not proportional to changes of input}}
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{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}
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{{Complex systems}}
In [[mathematics]] and [[science]], a '''nonlinear system''' (or a '''non-linear system''') is a [[system]] in which the change of the output is not [[proportionality (mathematics)|proportional]] to the change of the input.<ref>{{Cite news|url=https://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=2018-06-30}}</ref><ref>{{Cite web|url=https://www.birmingham.ac.uk/research/activity/mathematics/applied-maths/nonlinear-systems.aspx|title=Nonlinear systems, Applied Mathematics - University of Birmingham|website=www.birmingham.ac.uk|language=en-gb|access-date=2018-06-30}}</ref> Nonlinear problems are of interest to [[engineer]]s, [[biologist]]s,<ref>{{Citation|date=2007|pages=181–276|publisher=Springer Berlin Heidelberg|language=en|doi=10.1007/978-3-540-34153-6_7|isbn=9783540341529|title = The Nonlinear Universe|series = The Frontiers Collection|chapter = Nonlinear Biology}}</ref><ref>{{cite journal|last1=Korenberg|first1=Michael J.|last2=Hunter|first2=Ian W.|date=March 1996|title=The identification of nonlinear biological systems: Volterra kernel approaches|journal=Annals of Biomedical Engineering|language=en|volume=24|issue=2|pages=250–268|doi=10.1007/bf02667354|pmid=8678357|s2cid=20643206|issn=0090-6964}}</ref><ref>{{cite journal|last1=Mosconi|first1=Francesco|last2=Julou|first2=Thomas|last3=Desprat|first3=Nicolas|last4=Sinha|first4=Deepak Kumar|last5=Allemand|first5=Jean-François|last6=Vincent Croquette|last7=Bensimon|first7=David|date=2008|title=Some nonlinear challenges in biology|url=http://stacks.iop.org/0951-7715/21/i=8/a=T03|journal=Nonlinearity|language=en|volume=21|issue=8|pages=T131|doi=10.1088/0951-7715/21/8/T03|issn=0951-7715|bibcode=2008Nonli..21..131M|s2cid=119808230 }}</ref> [[physicist]]s,<ref>{{cite journal|last1=Gintautas|first1=V.|title=Resonant forcing of nonlinear systems of differential equations|journal=Chaos|date=2008|volume=18|issue=3|pages=033118|doi=10.1063/1.2964200|pmid=19045456|arxiv=0803.2252|bibcode=2008Chaos..18c3118G|s2cid=18345817}}</ref><ref>{{cite journal|last1=Stephenson|first1=C.|last2=et.|first2=al.|title=Topological properties of a self-assembled electrical network via ab initio calculation|journal=Sci. Rep.|volume=7|pages=41621|date=2017|doi=10.1038/srep41621|pmid=28155863|pmc=5290745|bibcode=2017NatSR...741621S}}</ref> [[mathematician]]s, and many other [[scientist]]s since most systems are inherently nonlinear in nature.<ref>{{cite book|last1=de Canete|first1=Javier, Cipriano Galindo, and Inmaculada Garcia-Moral|title=System Engineering and Automation: An Interactive Educational Approach|date=2011|publisher=Springer|location=Berlin|isbn=978-3642202292|page=46|url=https://books.google.com/books?id=h8rCQYXGGY8C&q=most+systems+are+inherently+nonlinear+in+nature&pg=PA46|access-date=20 January 2018}}</ref> Nonlinear [[dynamical system]]s, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler [[linear system]]s.

Typically, the behavior of a nonlinear system is described in mathematics by a '''nonlinear system of equations''', which is a set of simultaneous [[equation]]s in which the unknowns (or the unknown functions in the case of [[differential equation]]s) appear as variables of a [[polynomial]] of degree higher than one or in the argument of a [[function (mathematics)|function]] which is not a polynomial of degree one.
In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a [[linear combination]] of the unknown [[variable (mathematics)|variables]] or [[function (mathematics)|functions]] that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential equation is ''linear'' if it is linear in terms of the unknown function and its derivatives, even if nonlinear in terms of the other variables appearing in it.

As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations ([[linearization]]). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as [[soliton]]s, [[chaos theory|chaos]],<ref>[http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm Nonlinear Dynamics I: Chaos] {{webarchive|url=https://web.archive.org/web/20080212045134/http://ocw.mit.edu/OcwWeb/Earth--Atmospheric--and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm |date=2008-02-12}} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare]</ref> and [[mathematical singularity|singularities]] are hidden by linearization. It follows that some aspects of the dynamic behavior of a nonlinear system can appear to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble [[randomness|random]] behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology.

Some authors use the term '''nonlinear science''' for the study of nonlinear systems. This term is disputed by others:

{{quote|Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.|[[Stanisław Ulam]]<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25 November 2004|volume=432|issue=7016|pages=455–456|doi=10.1038/432455a|pmid=15565139|url=https://zenodo.org/record/1134179|language=en|issn=0028-0836|bibcode=2004Natur.432..455C|s2cid=4403332}}</ref>}}