Nonlinear system: Difference between revisions

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{{hatnote group|

{{About|"nonlinearity" in mathematics, physics and other sciences|video and film editing|Non-linear editing system|other uses|Nonlinearity (disambiguation)}}

{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}

}}

{{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 |url-access=subscription}}</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|pagesarticle-number=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|pagesarticle-number=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.

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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 [[negation|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>}}

==Definition==

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The definition <math>f(x) = C</math> is very general in that <math>x</math> can be any sensible mathematical object (number, vector, function, etc.), and the function <math>f(x)</math> can literally be any [[map (mathematics)|mapping]], including integration or differentiation with associated constraints (such as [[boundary values]]). If <math>f(x)</math> contains [[derivative|differentiation]] with respect to <math>x</math>, the result will be a [[differential equation]].

==Nonlinear algebraicsystems of equations==

A nonlinear system of equations consists of a set of equations in several variables such that at least one of them is not a [[linear equation]].

{{Main|Algebraic equation|System of polynomial equations}}

Nonlinear [[algebraic equation]]s, which are also called ''[[polynomial equation]]s'', are defined by equating [[polynomial]]s (of degree greater than one) to zero. For example,

:<math>x^2 + x - 1 = 0\,.</math>

For a single equation of the form <math>f(x)=0,</math> many methods have been designed; see [[Root-finding algorithm]]. In the case where {{mvar|f}} is a [[polynomial]], one has a ''[[polynomial equation]]'' such as

For a single polynomial equation, [[root-finding algorithm]]s can be used to find solutions to the equation (i.e., sets of values for the variables that satisfy the equation). However, systems of algebraic equations are more complicated; their study is one motivation for the field of [[algebraic geometry]], a difficult branch of modern mathematics. It is even difficult to decide whether a given algebraic system has complex solutions (see [[Hilbert's Nullstellensatz]]). Nevertheless, in the case of the systems with a finite number of complex solutions, these [[systems of polynomial equations]] are now well understood and efficient methods exist for solving them.<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 |doi-access= free }}</ref>

<math>x^2 + x - 1 = 0.</math> The general root-finding algorithms apply to polynomial roots, but, generally they do not find all the roots, and when they fail to find a root, this does not imply that there is no roots. Specific methods for polynomials allow finding all roots or the [[real number|real]] roots; see [[real-root isolation]].

Solving [[systems of polynomial equations]], that is finding the common zeros of a set of several polynomials in several variables is a difficult problem for which elaborate algorithms have been designed, such as [[Gröbner base]] algorithms.<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 |doi-access= free }}</ref>

For the general case of system of equations formed by equating to zero several [[differentiable function]]s, the main method is [[Newton's method#Systems of equations|Newton's method]] and its variants. Generally they may provide a solution, but do not provide any information on the number of solutions.

==Nonlinear recurrence relations==

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==Nonlinear differential equations==

{{Redirect|Nonlinear dynamics|the journal|Nonlinear Dynamics (journal)}}

A [[simultaneous equations|system]] of [[differential equation]]s is said to be nonlinear if it is not a [[system of linear equations]]. Problems involving nonlinear differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples of nonlinear differential equations are the [[Navier–Stokes equations]] in fluid dynamics and the [[Lotka–Volterra equations]] in biology.

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:<math>\frac{d u}{d x} = -u^2</math>

has <math>u=\frac{1}{x+C}</math> as a general solution (and also the special solution <math>u = 0,</math> as a particular solution, corresponding to the limit of the general solution when ''C'' tends to infinity). The equation is nonlinear because it may be written as

:<math>\frac{du}{d x} + u^2=0</math>

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*[[Bifurcation theory]]

*[[Perturbation theory|Perturbation]] methods (can be applied to algebraic equations too)

*Existence of solutions of Finite-Duration,<ref>{{cite book |author = Vardia T. Haimo |title = 1985 24th IEEE Conference on Decision and Control |chapter = Finite Time Differential Equations |year = 1985 |pages = 1729–1733 |doi = 10.1109/CDC.1985.268832 |s2cid = 45426376 |chapter-url=https://ieeexplore.ieee.org/document/4048613}}</ref> which can happen under specific conditions for some non-linear ordinary differential equations.

===Partial differential equations===

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[[File:PendulumLinearizations.png|thumb|Linearizations of a pendulum|right|200px]]

A classic, extensively studied nonlinear problem is the dynamics of a frictionless [[pendulum (mathematics)|pendulum]] under the influence of [[gravity]]. Using [[Lagrangian mechanics]], it may be shown<ref>[httphttps://www.damtp.cam.ac.uk/user/tong/dynamics.html David Tong: Lectures on Classical Dynamics]</ref> that the motion of a pendulum can be described by the [[dimensionless]] nonlinear equation

:<math>\frac{d^2 \theta}{d t^2} + \sin(\theta) = 0</math>

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| isbn= 978-0-387-98489-6

}}

*{{cite book |last1=Sastry |first1=Shankar |title=Nonlinear systems: analysis, stability, and control |series=Interdisciplinary Applied Mathematics |date=2009 |volume=10 |publisher=Springer |___location=New York Berlin Heidelberg |isbn=978-0-387-98513-8 |edition=Nachdr. |doi=10.1007/978-1-4757-3108-8 |url=https://link.springer.com/book/10.1007/978-1-4757-3108-8 |language=en}}

*{{cite book |last1=Orlando |first1=Giuseppe |title=Nonlinearities in economics: an interdisciplinary approach to economic dynamics, growth and cycles |series=Dynamic Modeling and Econometrics in Economics and Finance |date=2021 |volume=29 |publisher=Springer International Publishing AG |___location=Cham |doi=10.1007/978-3-030-70982-2 |isbn=978-3-030-70981-5 |url=https://link.springer.com/book/10.1007/978-3-030-70982-2#editorsandaffiliations |language=en}}

{{Refend}}

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*[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]

*[httphttps://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]

*[http://www.hedengren.net/research/models.htm Nonlinear Model Library]{{snd}} (in [[MATLAB]]) a Database of Physical Systems

*[httphttps://cnls.lanl.gov/ The Center for Nonlinear Studies at Los Alamos National Laboratory]

{{Systems science}}

{{Differential equations topics}}

{{Complex systems topics}}