Linearization: Difference between revisions

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Linearization makes it possible to use tools for studying [[linear system]]s 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

:<math>\frac{d\boldmathbf{x}}{dt} = \boldmathbf{F}(\boldmathbf{x},t)</math>,

the linearized system can be written as

:<math>\frac{d\boldmathbf{x}}{dt} \approx \boldmathbf{F}(\boldmathbf{x_0},t) + D\boldmathbf{F}(\boldmathbf{x_0},t) \cdot (\boldmathbf{x} - \boldmathbf{x_0})</math>

where <math>\boldmathbf{x_0}</math> is the point of interest and <math>D\boldmathbf{F}(\boldmathbf{x_0})</math> is the [[Jacobian matrix and determinant|Jacobian]] of <math>\boldmathbf{F}(\boldmathbf{x})</math> evaluated at <math>\boldmathbf{x_0}</math>.

===Stability analysis===