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{{About||the linearization of a partial order|Linear extension|the linearization in concurrent computing|Linearizability}}
In [[mathematics]], '''linearization''' is finding the [[linear approximation]] to a [[function (mathematics)|function]] at a given point. The linear approximation of a function is the first order [[Taylor expansion]] around the point of interest. In the study of [[dynamical system]]s, linearization is a method for assessing the local [[stability theory|stability]] of an [[equilibrium point]] of a [[system]] of [[nonlinear]]{{disambiguation needed}} [[differential equation]]s or discrete [[dynamical system]]s.<ref>[http://www.scholarpedia.org/article/Siegel_disks/Linearization The linearization problem in complex dimension one dynamical systems at Scholarpedia]</ref> This method is used in fields such as [[engineering]], [[physics]], [[economics]], and [[ecology]].
==Linearization of a function==
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===Stability analysis===
In [[stability theory|stability]] analysis of [[Autonomous system (mathematics)|autonomous systems]], one can use the [[eigenvalue]]s of the [[Jacobian matrix and determinant|Jacobian matrix]] evaluated at a [[hyperbolic equilibrium point]] to determine the nature of that equilibrium. This is the content of the [[linearization theorem]]. For time-varying systems, the linearization requires additional justification.<ref>{{cite journal |first=G. A. |last=Leonov |first2=N. V. |last2=Kuznetsov |title=Time-Varying Linearization and the Perron effects |journal=[[International Journal of Bifurcation and Chaos]] |volume=17 |issue=4 |year=2007 |pages=1079–1107 |doi=10.1142/S0218127407017732 |bibcode=2007IJBC...17.1079L }}</ref>
===Microeconomics===
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