Perturbation theory ($u(ϵ)\sim u_0+ϵu_1+\cdots$)

Perturbation theory develops approximate solutions in a small parameter around a simpler reference problem. A power-series asymptotic expansion gives equations for successive correction coefficients; a matched asymptotic expansion handles cases where one expansion is not uniformly valid.

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.