Perturbation ($F_{\epsilon }=F_0+\epsilon G+o(\epsilon )$)

A perturbation is a small change in a model, operator, state or parameter. It can be organized as a family $F_{\epsilon }=F_0+\epsilon G+o(\epsilon )$ in a specified norm. If $F_{\epsilon }(z_{\epsilon })=0$ and $D{F}_0(z_0)$ is invertible, differentiating at zero gives $z_0^{\prime }=-[D{F}_0(z_0){]}^{-1}G(z_0)$. 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.