Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input.^[1]^[2] Nonlinear problems are of interest to engineers, biologists,^[3]^[4]^[5] physicists,^[6]^[7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature.^[8] Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a 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 variables or 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 solitons, chaos,^[9] and 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 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:

Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.
— Stanisław Ulam^[10]

^ "Explained: Linear and nonlinear systems". MIT News. Retrieved 2018-06-30.
^ "Nonlinear systems, Applied Mathematics - University of Birmingham". www.birmingham.ac.uk. Retrieved 2018-06-30.
^ "Nonlinear Biology", The Nonlinear Universe, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp. 181–276, doi:10.1007/978-3-540-34153-6_7, ISBN 9783540341529
^ Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches". Annals of Biomedical Engineering. 24 (2): 250–268. doi:10.1007/bf02667354. ISSN 0090-6964. PMID 8678357. S2CID 20643206.
^ Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent Croquette; Bensimon, David (2008). "Some nonlinear challenges in biology". Nonlinearity. 21 (8): T131. Bibcode:2008Nonli..21..131M. doi:10.1088/0951-7715/21/8/T03. ISSN 0951-7715. S2CID 119808230.
^ Gintautas, V. (2008). "Resonant forcing of nonlinear systems of differential equations". Chaos. 18 (3): 033118. arXiv:0803.2252. Bibcode:2008Chaos..18c3118G. doi:10.1063/1.2964200. PMID 19045456. S2CID 18345817.
^ Stephenson, C.; et., al. (2017). "Topological properties of a self-assembled electrical network via ab initio calculation". Sci. Rep. 7: 41621. Bibcode:2017NatSR...741621S. doi:10.1038/srep41621. PMC 5290745. PMID 28155863.
^ de Canete, Javier, Cipriano Galindo, and Inmaculada Garcia-Moral (2011). System Engineering and Automation: An Interactive Educational Approach. Berlin: Springer. p. 46. ISBN 978-3642202292. Retrieved 20 January 2018.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Nonlinear Dynamics I: Chaos Archived 2008-02-12 at the Wayback Machine at MIT's OpenCourseWare
^ Campbell, David K. (25 November 2004). "Nonlinear physics: Fresh breather". Nature. 432 (7016): 455–456. Bibcode:2004Natur.432..455C. doi:10.1038/432455a. ISSN 0028-0836. PMID 15565139. S2CID 4403332.

[1] "Explained: Linear and nonlinear systems". MIT News. Retrieved 2018-06-30.

[2] "Nonlinear systems, Applied Mathematics - University of Birmingham". www.birmingham.ac.uk. Retrieved 2018-06-30.

[3] "Nonlinear Biology", The Nonlinear Universe, The Frontiers Collection, Springer Berlin Heidelberg, 2007, pp. 181–276, doi:10.1007/978-3-540-34153-6_7, ISBN 9783540341529

[4] Korenberg, Michael J.; Hunter, Ian W. (March 1996). "The identification of nonlinear biological systems: Volterra kernel approaches". Annals of Biomedical Engineering. 24 (2): 250–268. doi:10.1007/bf02667354. ISSN 0090-6964. PMID 8678357. S2CID 20643206.

[5] Mosconi, Francesco; Julou, Thomas; Desprat, Nicolas; Sinha, Deepak Kumar; Allemand, Jean-François; Vincent Croquette; Bensimon, David (2008). "Some nonlinear challenges in biology". Nonlinearity. 21 (8): T131. Bibcode:2008Nonli..21..131M. doi:10.1088/0951-7715/21/8/T03. ISSN 0951-7715. S2CID 119808230.

[6] Gintautas, V. (2008). "Resonant forcing of nonlinear systems of differential equations". Chaos. 18 (3): 033118. arXiv:0803.2252. Bibcode:2008Chaos..18c3118G. doi:10.1063/1.2964200. PMID 19045456. S2CID 18345817.

[7] Stephenson, C.; et., al. (2017). "Topological properties of a self-assembled electrical network via ab initio calculation". Sci. Rep. 7: 41621. Bibcode:2017NatSR...741621S. doi:10.1038/srep41621. PMC 5290745. PMID 28155863.

[8] Canete, Javier, Cipriano Galindo, and Inmaculada Garcia-Moral (2011). System Engineering and Automation: An Interactive Educational Approach. Berlin: Springer. p. 46. ISBN 978-3642202292. Retrieved 20 January 2018.{{cite book}}: CS1 maint: multiple names: authors list (link)

[9] Nonlinear Dynamics I: Chaos Archived 2008-02-12 at the Wayback Machine at MIT's OpenCourseWare

[10] Campbell, David K. (25 November 2004). "Nonlinear physics: Fresh breather". Nature. 432 (7016): 455–456. Bibcode:2004Natur.432..455C. doi:10.1038/432455a. ISSN 0028-0836. PMID 15565139. S2CID 4403332.

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