Mr. Integral
Math notes/Differential Equations/Systems and Phase Planes

Nonlinear Systems and Local Linearization — Cheat sheet

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Core rule

J=Df(x∗),J=D\mathbf f(\mathbf x_*), u′≈Ju.\mathbf u'\approx J\mathbf u.

Watch for

Local stability is not a global claim about every starting point. A stable equilibrium may have a limited basin of attraction, and a linear approximation loses accuracy far from its expansion point.