Linearization in mathematics and its applications in general 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. This method is used in fields such as engineering, physics, economics, and ecology.

The analysis of linear functions is well defined, but most representations of actual systems are nonlinear. Linearization allows us to use tools for studying linear 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

$\backslash frac\{d\backslash bold\{x\}\}\{dt\}\; =\; \backslash bold\{F\}(\backslash bold\{x\},t)$,

the linearized system can be written as

$\backslash frac\{d\backslash bold\{x\}\}\{dt\}\; =\; D\backslash bold\{F\}(\backslash bold\{x\_0\},t)\; \backslash cdot\; \backslash bold\{x\}$

where $\backslash bold\{x\_0\}$ is the point of interest and $D\backslash bold\{F\}(\backslash bold\{x\_0\})$ is the Jacobian of $\backslash bold\{F\}(\backslash bold\{x\})$ evaluated at $\backslash bold\{x\_0\}$.

In stability analysis, one can use the eigenvalues of the Jacobian matrix evaluated at an equilibrium point to determine the nature of that equilibrium. If all the eigenvalues are positive, the equilibrium is unstable; if they are all negative the equilibrium is stable; and is the values are of mixed signs, the equilibrium is a saddle point. Any complex eigenvalues will appear in complex conjugate pairs and indicate spiral (or circular if the real components are zero) around the equilibrium.

In introductory calculus courses the linearization function is usually in the form:

$\backslash bold\{F\}\{\backslash bold\{(x)\}\}\; \backslash approx\; \backslash bold\{L\}\{\backslash bold\{(x)\}\}\; =\; \backslash bold\{F\}\{\backslash bold\{(a)\}\}\; +\; \backslash bold\{F\text{'}\}\{\backslash bold\{(a)\}\}(x\; -\; a)$

$\backslash bold\{F\text{'}\}\{\backslash bold\{(a)\}\}$ is the derivative of the function $\backslash bold\{F\}\{\backslash bold\{(x)\}\}$ at a.

'a' is a point near the value 'x' that one wishes to solve for a given function of 'x'. Usually the function would be hard or impossible to solve for at 'x' (for example 1/x, where x equals 2.08), but there is often a number near x that is easy to solve for (2 would be a good number to choose for the previous example as 1/2 is quite easy to solve). By using that number as the point 'a', they can find the linearization of that function at 'a', and use it to approximate F(x).

Note: The closer the value of 'x' is to 'a', and vice versa, the closer the linearization is to the actual function. If the value for 'x' differs too much from 'a' the linearization will be too inaccurate to be of any use.

CalculusDynamical systems

Lineariseren | Линеаризация | Лінеаризація