= Perturbation
{title2=$F_\varepsilon=F_0+\varepsilon G+o(\varepsilon)$}
A perturbation is a small change in a model, operator, state or parameter. It can be organized as a family $F_\varepsilon=F_0+\varepsilon G+o(\varepsilon)$ in a specified norm. If $F_\varepsilon(z_\varepsilon)=0$ and $DF_0(z_0)$ is invertible, differentiating at zero gives $z'_0=-[DF_0(z_0)]^{-1}G(z_0)$. The <implicit function theorem> justifies this continued solution locally. Singular derivatives at a bifurcation instead require an unfolding or reduced equation; smallness of the change alone does not preserve a degenerate bifurcation.
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