---
tag: CC666JT89X
kind: definition
name: Lyapunov–Schmidt reduction
msc: [47J15, 34C23]
defines: MAVAK2WWXK
uses: [06GZ956657, 0S23E6YE83, 033DA1WE8Y, 0BXBADR94C, 0VW90BBSJ4]
sources:
  - url: https://en.wikipedia.org/wiki/Lyapunov%E2%80%93Schmidt_reduction
---

## Definition

Let $X$, $Y$ and $\Lambda$ be [[Banach spaces|06GZ956657]], let $U \subseteq X \times \Lambda$ be an open neighbourhood of $(0, 0)$, and let $F \colon U \to Y$ be [[continuously differentiable|0VW90BBSJ4]] with $F(0, 0) = 0$.
Let $L = D_x F(0, 0) \colon X \to Y$, and assume that $L$ is a Fredholm operator:

1. the [[kernel|0S23E6YE83]] $N = \ker L$ has finite dimension;
2. the range $R = L(X)$ is closed and has a complement of finite dimension in $Y$.

Choose closed subspaces $X_1$ and $Y_1$ with $X = N \oplus X_1$ and $Y = R \oplus Y_1$, and let $Q \colon Y \to R$ be the continuous [[projection|033DA1WE8Y]] onto $R$ along $Y_1$.
The restriction of $QL$ to $X_1$ is an isomorphism onto $R$.
Write $x = v + w$ with $v \in N$ and $w \in X_1$.
By the [[implicit function theorem|0BXBADR94C]], the equation $Q F(v + w, \lambda) = 0$ has a unique solution $w = W(v, \lambda)$ near $0$.
Here $(v, \lambda)$ is near $(0, 0)$, and $W(0, 0) = 0$.
The *Lyapunov–Schmidt reduction* of $F = 0$ at $(0, 0)$ is the *bifurcation equation*
$$(I - Q)\, F\big(v + W(v, \lambda), \lambda\big) = 0,$$
for $v \in N$ and $\lambda$ near $0$.
Its values lie in $Y_1$.

## Remarks

Near $(0, 0)$, the solutions of $F(x, \lambda) = 0$ are exactly the points $x = v + W(v, \lambda)$ with $(v, \lambda)$ a solution of the bifurcation equation.
Since $N$ and $Y_1$ are finite-dimensional, this is a system of $\dim Y_1$ equations in $\dim N$ unknowns.

Example: $X = Y = \R^2$, $\Lambda = \R$, $F(x, y, \lambda) = (\lambda x - x y,\ y - x^2)$.
Here $L = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$, so $N$ is the $x$-axis and $R$ is the $y$-axis; take $X_1 = R$ and $Y_1 = N$.
The equation $Q F = 0$ is $y - x^2 = 0$, so $W(x, \lambda) = x^2$.
The bifurcation equation is $\lambda x - x^3 = 0$, which has a pitchfork bifurcation at $\lambda = 0$.

Non-example: on the sequence space $\ell^2$, the operator $L(x_1, x_2, \ldots) = (x_1, x_2/2, x_3/3, \ldots)$ has trivial kernel, but its range is dense and not closed, so clause 2 fails.