linearization (of a nonlinear system)

Replacing a nonlinear system near an equilibrium with the linear system given by its Jacobian there.

The linearization of $\mathbf{x}'=F(\mathbf{x})$ at an equilibrium $\mathbf{x}^*$ is the linear system $\mathbf{u}'=J\mathbf{u}$ , where $J=DF(\mathbf{x}^*)$ and $\mathbf{u}=\mathbf{x}-\mathbf{x}^*$ -- the local, first-order approximation to the true nonlinear flow near $\mathbf{x}^*$ .