Lyapunov–Schmidt reduction

Let

:$f(x,\backslash lambda)=0\; \backslash ,$

be the given nonlinear equation, $X,\backslash Lambda,$ and $Y$ are

Banach spaces ($\backslash Lambda$ is the parameter space). $f(x,\backslash lambda)$ is the

$C^p$-map from a neighborhood of some point $(x\_0,\backslash lambda\_0)\backslash in\; X\backslash times\; \backslash Lambda$ to

$Y$ and the equation is satisfied at this point

:$f(x\_0,\backslash lambda\_0)=0.$

For the case when the linear operator $f\_x(x,\backslash lambda)$ is invertible, the implicit function theorem assures that there exists

a solution $x(\backslash lambda)$ satisfying the equation $f(x(\backslash lambda),\backslash lambda)=0$ at least locally close to $\backslash lambda\_0$.

In the opposite case, when the linear operator $f\_x(x,\backslash lambda)$ is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following

way.

One assumes that the operator $f\_x(x,\backslash lambda)$ is a Fredholm operator.

$\backslash ker\; f\_x\; (x\_0,\backslash lambda\_0)=X\_1$ and $X\_1$ has finite dimension.

The range of this operator $\backslash mathrm\{ran\}\; f\_x\; (x\_0,\backslash lambda\_0)=Y\_1$ has finite co-dimension and

is a closed subspace in $Y$.

Without loss of generality, one can assume that $(x\_0,\backslash lambda\_0)=(0,0).$

Let us split $Y$ into the direct product $Y=\; Y\_1\; \backslash oplus\; Y\_2$, where .

Let $Q$ be the projection operator onto $Y\_1$.

Consider also the direct product $X=\; X\_1\; \backslash oplus\; X\_2$.

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

:$Qf(x,\backslash lambda)=0\; \backslash ,$

:$(I-Q)f(x,\backslash lambda)=0\; \backslash ,$

Let $x\_1\backslash in\; X\_1$ and $x\_2\; \backslash in\; X\_2$, then the first equation

:$Qf(x\_1+x\_2,\backslash lambda)=0\; \backslash ,$

can be solved with respect to $x\_2$ by applying the implicit function theorem to the operator

:$Qf(x\_1+x\_2,\backslash lambda):\; \backslash quad\; X\_2\backslash times(X\_1\backslash times\backslash Lambda)\backslash to\; Y\_1\; \backslash ,$

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

Thus, there exists a unique solution $x\_2(x\_1,\backslash lambda)$ satisfying

:$Qf(x\_1+x\_2(x\_1,\backslash lambda),\backslash lambda)=0.\; \backslash ,$

Now substituting $x\_2(x\_1,\backslash lambda)$ into the second equation, one obtains the final finite-dimensional equation

:$(I-Q)f(x\_1+x\_2(x\_1,\backslash lambda),\backslash lambda)=0.\; \backslash ,$

Indeed, the last equation is now finite-dimensional, since the range of $(I-Q)$ is finite-dimensional. This equation is now to be solved with respect to $x\_1$, which is finite-dimensional, and parameters :$\backslash lambda$

Lyapunov–Schmidt reduction has been used in economics, natural sciences, and engineering{{Cite book|title=Lyapunov-Schmidt methods in nonlinear analysis and applications|first=Nikolai|last=Sidorov|date=2011|publisher=Springer|isbn=9789048161508|oclc=751509629}} often in combination with bifurcation theory, perturbation theory, and regularization.{{Citation|last=Golubitsky|first=Martin|title=The Hopf Bifurcation|date=1985|work=Applied Mathematical Sciences|pages=337–396|publisher=Springer New York|isbn=9781461295334|last2=Schaeffer|first2=David G.|doi=10.1007/978-1-4612-5034-0_8}}{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=January 2009|title=Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors|journal=Chemical Engineering Journal|volume=145|issue=3|pages=399–411|doi=10.1016/j.cej.2008.08.025|issn=1385-8947}} 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.{{Cite journal|last=Balakotaiah|first=Vemuri|date=March 2004|title=Hyperbolic averaged models for describing dispersion effects in chromatographs and reactors|journal=Korean Journal of Chemical Engineering|volume=21|issue=2|pages=318–328|doi=10.1007/bf02705415|issn=0256-1115}}{{Cite journal|last=Gupta|first=Ankur|last2=Chakraborty|first2=Saikat|date=2008-01-19|title=Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions|journal=Chemical Product and Process Modeling|volume=3|issue=2|doi=10.2202/1934-2659.1135|issn=1934-2659}}

{{Reflist}}

• Louis Nirenberg, Topics in nonlinear functional analysis, New York Univ. Lecture Notes, 1974.
• Aleksandr Lyapunov, Sur les figures d’équilibre peu différents des ellipsoides d’une masse liquide homogène douée d’un mouvement de rotation, Zap. Akad. Nauk St. Petersburg (1906), 1–225.
• Aleksandr Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse 2 (1907), 203–474.
• Erhard Schmidt, Zur Theory der linearen und nichtlinearen Integralgleichungen, 3 Teil, Math. Annalen 65 (1908), 370–399.

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Category:Functional analysis