Lyapunov–Schmidt reduction

In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after Aleksandr Lyapunov and Erhard Schmidt.

Problem setup

Let

${\displaystyle f(x,\lambda )=0\,}$

be the given nonlinear equation, ${\displaystyle X,\Lambda ,}$ and ${\displaystyle Y}$ are Banach spaces (${\displaystyle \Lambda }$ is the parameter space). ${\displaystyle f(x,\lambda )}$ is the ${\displaystyle C^{p}}$-map from a neighborhood of some point ${\displaystyle (x_{0},\lambda _{0})\in X\times \Lambda }$ to ${\displaystyle Y}$ and the equation is satisfied at this point

${\displaystyle f(x_{0},\lambda _{0})=0.}$

For the case when the linear operator ${\displaystyle f_{x}(x,\lambda )}$ is invertible, the implicit function theorem assures that there exists a solution ${\displaystyle x(\lambda )}$ satisfying the equation ${\displaystyle f(x(\lambda ),\lambda )=0}$ at least locally close to ${\displaystyle \lambda _{0}}$.

In the opposite case, when the linear operator ${\displaystyle f_{x}(x,\lambda )}$ is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following way.

Assumptions

One assumes that the operator ${\displaystyle f_{x}(x,\lambda )}$ is a Fredholm operator.

${\displaystyle \ker f_{x}(x_{0},\lambda _{0})=X_{1}}$ and ${\displaystyle X_{1}}$ has finite dimension.

The range of this operator ${\displaystyle \mathrm {ran} f_{x}(x_{0},\lambda _{0})=Y_{1}}$ has finite co-dimension and is a closed subspace in ${\displaystyle Y}$.

Without loss of generality, one can assume that ${\displaystyle (x_{0},\lambda _{0})=(0,0).}$

Lyapunov–Schmidt construction

Let us split ${\displaystyle Y}$ into the direct product ${\displaystyle Y=Y_{1}\oplus Y_{2}}$, where ${\displaystyle \dim Y_{2}<\infty }$.

Let ${\displaystyle Q}$ be the projection operator onto ${\displaystyle Y_{1}}$.

Consider also the direct product ${\displaystyle X=X_{1}\oplus X_{2}}$.

Applying the operators ${\displaystyle Q}$ and ${\displaystyle I-Q}$ to the original equation, one obtains the equivalent system

${\displaystyle Qf(x,\lambda )=0\,}$

${\displaystyle (I-Q)f(x,\lambda )=0\,}$

Let ${\displaystyle x_{1}\in X_{1}}$ and ${\displaystyle x_{2}\in X_{2}}$, then the first equation

${\displaystyle Qf(x_{1}+x_{2},\lambda )=0\,}$

can be solved with respect to ${\displaystyle x_{2}}$ by applying the implicit function theorem to the operator

${\displaystyle Qf(x_{1}+x_{2},\lambda ):\quad X_{2}\times (X_{1}\times \Lambda )\to Y_{1}\,}$

(now the conditions of the implicit function theorem are fulfilled).

Thus, there exists a unique solution ${\displaystyle x_{2}(x_{1},\lambda )}$ satisfying

${\displaystyle Qf(x_{1}+x_{2}(x_{1},\lambda ),\lambda )=0.\,}$

Now substituting ${\displaystyle x_{2}(x_{1},\lambda )}$ into the second equation, one obtains the final finite-dimensional equation

${\displaystyle (I-Q)f(x_{1}+x_{2}(x_{1},\lambda ),\lambda )=0.\,}$

Indeed, the last equation is now finite-dimensional, since the range of ${\displaystyle (I-Q)}$ is finite-dimensional. This equation is now to be solved with respect to ${\displaystyle x_{1}}$, which is finite-dimensional, and parameters :${\displaystyle \lambda }$

Applications

Lyapunov–Schmidt reduction has been used in economics, natural sciences, and engineering^[1] often in combination with bifurcation theory, perturbation theory, and regularization.^[1][2][3] LS reduction is often used to rigorously regularize partial differential equation models in chemical engineering resulting in models that are easier to simulate numerically but still retain all the parameters of the original model.^[3][4][5]