# Linearization

{{#invoke:Hatnote|hatnote}} In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.

## Linearization of a function

While the concept of local linearity applies the most to points arbitrarily close to $x=a$ , those relatively close work relatively well for linear approximations. The slope $M$ should be, most accurately, the slope of the tangent line at $x=a$ .

The final equation for the linearization of a function at $x=a$ is:

## Uses of linearization

Linearization makes it possible to use tools for studying nonlinear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation

${\frac {d{\mathbf {x} }}{dt}}={\mathbf {F} }({\mathbf {x} },t)$ ,

the linearized system can be written as

${\frac {d{\mathbf {x}}}{dt}}\approx {\mathbf {F}}({\mathbf {x_{0}}},t)+D{\mathbf {F}}({\mathbf {x_{0}}},t)\cdot ({\mathbf {x}}-{\mathbf {x_{0}}})$ ### Stability analysis

In stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium. This is the content of linearization theorem. For time-varying systems, the linearization requires additional justification.

### Microeconomics

In microeconomics, decision rules may be approximated under the state-space approach to linearization. Under this approach, the Euler equations of the utility maximization problem are linearized around the stationary steady state. A unique solution to the resulting system of dynamic equations then is found.