Linearisation
Linearisation approximates a nonlinear system near an equilibrium by a linear system.
Small perturbations
Let \(\mathbf{x}^*\) be an equilibrium and write
\[\mathbf{x}=\mathbf{x}^*+\mathbf{u},\]where \(\mathbf{u}\) is a small perturbation.
Linear approximation
For
\[\frac{d\mathbf{x}}{dt}=\mathbf{F}(\mathbf{x}),\]the first-order approximation near \(\mathbf{x}^*\) is
\[\frac{d\mathbf{u}}{dt}\approx J(\mathbf{x}^*)\mathbf{u}.\]Higher-order terms are neglected because they become relatively small close to the equilibrium.
Why it is useful
Linear systems are much easier to analyse. Their eigenvalues often determine the local behaviour of the original nonlinear system.
Key idea. Linearisation does not replace the nonlinear model globally; it provides a local approximation near an equilibrium.