Изворни код странице Nelinearni sistem

U [[mathematics|matematici]] i [[science|nauci]], '''nelinearni sistem''' je [[sistem]] u kome promena izlaza nije [[proportionality (mathematics)|proporcionalna]] promeni na ulazu.<ref name="Boeing2016Systems">{{cite journal|author=Boeing, G.|title=Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction|journal=Systems|date=2016|volume=4|issue=4|pages=37|doi=10.3390/systems4040037|arxiv=1608.04416}}</ref><ref>{{Cite news|url=http://news.mit.edu/2010/explained-linear-0226|title=Explained: Linear and nonlinear systems|work=MIT News|access-date=30. 6. 2018}}</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=30. 6. 2018}}</ref> Nelinearni problemi su važni za [[inženjer]]e, [[biologija|biologe]],<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|last=Korenberg|first=Michael J.|last2=Hunter|first2=Ian W.|date=mart 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|issn=0090-6964}}</ref><ref>{{cite journal|last=Mosconi|first=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}}</ref> [[fizičar]]e,<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}}</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> [[matematičar]]e i mnoge druge [[naučnik]]e, jer je većina sistema po svojoj prirodi nelinearna.<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/?id=h8rCQYXGGY8C&pg=PA46&lpg=PA46&dq=most+systems+are+inherently+nonlinear+in+nature#v=onepage&q=most%20systems%20are%20inherently%20nonlinear%20in%20nature&f=false|accessdate=20. 1. 2018}}</ref> Nelinearni [[dynamical system|dinamički sistemi]], koji opisuju promene promenljivih tokom vremena, mogu se činiti haotičnim, nepredvidljivim ili kontraintuitivnim, za razliku od mnogo jednostavnijih [[linear system|linearnih sistema]].

Tipično, ponašanje nelinearnog sistema opisano je u matematici ''nelinearnim sistemom jednačina'', koje su skup istovremenih [[jednačina]] u kojima se nepoznate (ili nepoznate funkcije u slučaju [[Diferencijalna jednačina|diferencijalnih jednačina]]) pojavljuju kao promenljive [[polinom]]a sa stepenom većim od jedan ili u argumentu [[function (mathematics)|funkcije]] koja nije polinom stepena jedan. Drugim rečima, u nelinearnom sistemu jednačina jednačine koje treba rešiti ne mogu se zapisati kao [[linear combination|linearna kombinacija]] nepoznatih [[Променљива (математика)|promenljivih]] ili [[function (mathematics)|funkcija]] koje se pojavljuju u njima. Sistemi se mogu definisati kao nelinearni, bez obzira da li se poznate linearne funkcije pojavljuju u jednačinama. Konkretno, diferencijalna jednačina je ''linearna'' ako je linearna u odnosu na nepoznatu funkciju i njene derivate, čak i ako je nelinearna u pogledu ostalih promenljivih koje se u njoj pojavljuju.

Kako je nelinearne dinamičke jednačine teško rešiti, nelinearni sistemi se obično aproksimiraju linearnim jednačinama ([[linearization |lineararizacija]]). To dobro funkcioniše do neke tačnosti i određenog opsega ulaznih vrednosti, mada se neki zanimljivi fenomeni, poput [[soliton]]a, [[Теорија хаоса|haosa]],<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=12. 2. 2008 }} at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare] {{Wayback|url=http://ocw.mit.edu/OcwWeb/index.htm |date=20081120155025 }}</ref> i [[Singularity (mathematics)|singulariteta]], skrivaju linearizacijom. Iz ovog sledi da se neki aspekti dinamičkog ponašanja nelinearnog sistema mogu činiti kontratuktivnim, nepredvidljivim ili čak haotičnim. Iako takvo haotično ponašanje može da liči na [[Randomness|slučajno]] ponašanje, ono zapravo nije randomno. Na primer, neki aspekti vremenskih prilika izgledaju haotično, pri čemu jednostavne promene u jednom delu sistema proizvode složene efekte širom sistema. Ova nelinearnost je jedan od razloga zašto su precizne dugoročne metereološke prognoze nemoguće sa sadašnjom tehnologijom.

Neki autori koriste termin '''nelinearna nauka''' za izučavanje nelinearnih sistemsa. Drugi to osporavaju, poput [[Stanislav Ulam|Stanislava Ulama]]:
„Korištenje izraza kao što je nelinearna nauka slično je pozivanju na najveći deo zoologije kao na proučavanje [[Логичка негација|ne]]slonovskih životinja.”<ref>{{cite journal|last1=Campbell|first1=David K.|title=Nonlinear physics: Fresh breather|journal=Nature|date=25. 11. 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}}</ref>

== Definicija ==
U [[mathematics|matematici]], [[linear map|linearna mapa]] (ili ''linearna funkcija'') <math>f(x)</math> je ona koja zadovoljava sledeća svojstva:
* Aditivnost ili [[superposition principle|princip superpozicije]]: <math>\textstyle f(x + y) = f(x) + f(y);</math>
* Homogenost: <math>\textstyle f(\alpha x) = \alpha f(x).</math>

Aditivnost podrazumeva homogenost za svako [[rational number|racionalno]] ''α'', i, za [[continuous function|neprekidne funkcije]], za svako [[real number|realno]] ''α''. Za [[complex number|kompleksno]] ''α'', homogenost ne sledi iz aditivnosti. Na primer, [[antilinear map|antilinearna mapa]] je aditivna, ali nije homogena. Uslovi aditivnosti i homogenosti se često kombinuju u principu superpozicije
:<math>f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)</math>

Jednačina napisana kao
:<math>f(x) = C</math>

se naziva '''linearnom''' ako je <math>f(x)</math> linearna mapa (kao što je gore definisanao), a inače '''nonlinearna'''. Jednačina se naziva ''homogenom'' ako je <math>C = 0</math>.

Definicija <math>f(x) = C</math> je veoma generalna u smislu da <math>x</math> može da bude bilo koji senzibilni matematički objekat (broj, vektor, funkcija, etc.), i funkcija <math>f(x)</math> može doslovno da bude bilo koje [[map (mathematics)|mapiranje]], uključujući integraciju ili diferencijaciju sa asociranim ograničenjima (kao što su [[Boundary value problem|granične vrednosti]]). Ako <math>f(x)</math> sadrži [[derivative|diferencijaciju]] u odnosu na <math>x</math>, rezultat će biti [[diferencijalna jednačina]].

== Nelinearne algebrske jednačine ==
{{Main-lat|Algebarska jednačina|Sistem polinomskih jednačina}}

Nelinearne [[algebraic equation|algebarske jednačine]], koje se takođe nazivaju ''polinomskim jednačinama'', definisane su izjednačavanjem [[polinom]]a (stepena većeg od jedan) sa nulom. Na primer,
:<math>x^2 + x - 1 = 0\,.</math>

Za pojedinačnu polinomsku jednačinu, [[Root-finding algorithm|algoritmi nalaženje korena]] se mogu koristiti za nalaženje rešenja jednačine (tj. skupa vrednosti promenljivih koje zadovoljavaju jednačinu). Međutim, sistemi algebarskih jednačina su komplikovaniji; njihovo proučavanje je jedna od motivacija polja [[Алгебарска геометрија|algebarske geometrije]], tegobne grane savremene matematike. Često je teško čak i da se odluči da li određeni algebrski sistem ima kompleksna rešenja (pogledajte [[Hilbert's Nullstellensatz|teoremu nula]]<ref>{{citation|last=Brownawell|first=W. Dale|title=Bounds for the degrees in the Nullstellensatz|journal=[[Ann. of Math.]]|volume=126|year=1987|issue=3|pages=577–591|mr=0916719|doi=10.2307/1971361}}</ref><ref>{{citation| first=János| last=Kollár| title=Sharp Effective Nullstellensatz| journal=[[Journal of the American Mathematical Society]]| volume=1| issue=4| year=1988| pages=963–975| url=http://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Kollar/kollarnullstellen.pdf| doi=10.2307/1990996| mr=0944576| accessdate=15. 08. 2019| archiveurl=https://web.archive.org/web/20140303174253/https://www.math.ucdavis.edu/~deloera/MISC/BIBLIOTECA/trunk/Kollar/kollarnullstellen.pdf| archivedate=03. 03. 2014| url-status=dead}}</ref>). Ipak, u slučaju sistema sa ograničenim brojem složenih rešenja, ovi [[System of polynomial equations|sistemi polinomnih jednačina]] su sada dobro izučeni i postoje efikasne metode za njihovo rešavanje.<ref>{{cite journal |last1= Lazard |first1= D. |title= Thirty years of Polynomial System Solving, and now? |doi= 10.1016/j.jsc.2008.03.004 |journal= Journal of Symbolic Computation |volume= 44 |issue= 3 |pages= 222–231 |year= 2009 |pmid= |pmc=}}</ref>

== Reference ==
{{Reflist|}}

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* L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.
* Arvind Kumar, ''Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature '', National Book Trust, 2003.
* Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.
* [[Edward Lorenz]], ''The Essence of Chaos'', University of Washington Press, 1996.
* Alan Marshall (2002) The Unity of Nature: Wholeness and Disintegration in Ecology and Science, Imperial College Press: London
* David Peak and Michael Frame, ''Chaos Under Control: The Art and Science of Complexity'', Freeman, 1994.
* [[Heinz-Otto Peitgen]] and [[Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
* [[Clifford A. Pickover]], ''Chaos in Wonderland:  Visual Adventures in a Fractal World'', St Martins Pr 1994.
* [[Ilya Prigogine]] and [[Isabelle Stengers]], ''Order Out of Chaos'', Bantam 1984.
* Heinz-Otto Peitgen and P. H. Richter, ''The Beauty of Fractals : Images of Complex Dynamical Systems'', Springer 1986, 211 pp.
* [[David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.
* [[Ivars Peterson]], ''Newton's Clock: Chaos in the Solar System'', Freeman, 1993.
* {{cite book |author1=Ian Roulstone |author2=John Norbury |title=Invisible in the Storm: the role of mathematics in understanding weather |url=https://books.google.com/?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=978-0691152721 }}
* David Ruelle, ''Chaotic Evolution and Strange Attractors'', Cambridge University Press, 1989.
* Manfred Schroeder, ''Fractals, Chaos, and Power Laws'', Freeman, 1991.
* Peter Smith, ''Explaining Chaos'', Cambridge University Press, 1998.
* [[Ian Stewart (mathematician)|Ian Stewart]], ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.
* [[Steven Strogatz]], ''Sync: The emerging science of spontaneous order'', Hyperion, 2003.
* Yoshisuke Ueda, ''The Road To Chaos'', Aerial Pr, 1993.
* M. Mitchell Waldrop, ''Complexity : The Emerging Science at the Edge of Order and Chaos'', Simon & Schuster, 1992.
* Antonio Sawaya, ''Financial Time Series Analysis : Chaos and Neurodynamics Approach'', Lambert, 2012.
{{Литература крај}}

== Spoljašnje veze ==
{{Commons category-lat|Nonlinear system}}
* -{[http://www.dodccrp.org/ Command and Control Research Program (CCRP)]}-
* -{[http://necsi.edu/guide/concepts/linearnonlinear.html New England Complex Systems Institute: Concepts in Complex Systems]}-
* -{[http://ocw.mit.edu/courses/mathematics/18-353j-nonlinear-dynamics-i-chaos-fall-2012/ Nonlinear Dynamics I: Chaos] at [http://ocw.mit.edu/OcwWeb/index.htm MIT's OpenCourseWare] {{Wayback|url=http://ocw.mit.edu/OcwWeb/index.htm |date=20081120155025 }}}-
* -{[http://www.hedengren.net/research/models.htm Nonlinear Model Library] {{Wayback|url=http://www.hedengren.net/research/models.htm |date=20081219003453 }}{{snd}} (in [[MATLAB]]) a Database of Physical Systems}-
* -{[http://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory] {{Wayback|url=http://cnls.lanl.gov/ |date=20081006065436 }}}-

{{Authority control-lat}}
{{СОРТИРАЊЕ:Нелинеарни систем}}

[[Категорија:Нелинеарни системи| ]]
[[Категорија:Динамички системи]]
[[Категорија:Фундаментални концепти физике]]