Lyapunov–Schmidt reduction explained

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

f(x, λ ) =0

be the given nonlinear equation,

X, Λ , and Y areBanach spaces (Λ is the parameter space). f(x, λ ) is theC ^{p} -map from a neighborhood of some point (x _{0,λ} _{0)\in} X x Λ toY and the equation is satisfied at this pointf(x _{0,λ} _{0)=0.}

For the case when the linear operator

f _{x(x,λ)} is invertible, the implicit function theorem assures that there existsa solution x( λ ) satisfying the equation f(x( λ ), λ ) =0 at least locally close to λ _{0} .

In the opposite case, when the linear operator

f _{x(x,λ)} is non-invertible, the Lyapunov–Schmidt reduction can be applied in the followingway.

Assumptions

One assumes that the operator

f _{x(x,λ)} is a Fredholm operator.\ker f _{x} (x _{0,λ} _{0)=X} _{1} and X _{1} has finite dimension.

The range of this operator

ran f _{x} (x _{0,λ} _{0)=Y} _{1} has finite co-dimension andis a closed subspace in Y .

Without loss of generality, one can assume that

(x _{0,λ} _{0)=(0,0).}

Lyapunov–Schmidt construction

Let us split

Y into the direct product Y = Y _{1} ⊕ Y _{2} , where \dim Y _{2} < inf ty .

Let

Q be the projection operator onto Y _{1} .

Consider also the direct product

X = X _{1} ⊕ X _{2} .

Applying the operators

Q and I - Q to the original equation, one obtains the equivalent systemQf(x, λ ) =0 (I - Q)f(x, λ ) =0

Let

x _{1\in} X _{1} and x _{2} \in X _{2} , then the first equationQf(x _{1+x} _{2,λ)=0}

can be solved with respect to

x _{2} by applying the implicit function theorem to the operatorQf(x _{1+x} _{2,λ):} X _{2 x (X} _{1 x Λ)\to} Y _{1}

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

Thus, there exists a unique solution

x _{2(x} _{1,λ)} satisfyingQf(x _{1+x} _{2(x} _{1,λ),λ)=0.}

Now substituting

x _{2(x} _{1,λ)} into the second equation, one obtains the final finite-dimensional equation(I - Q)f(x _{1+x} _{2(x} _{1,λ),λ)=0.}

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 :λ

Applications

Lyapunov–Schmidt reduction has been used in economics, natural sciences, and engineering^[1] often in combination with bifurcation theory, perturbation theory, and regularization.^[2] 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.^{[2] [3] [4]}

References

Bibliography

• 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.

Notes and References

1. Book: Sidorov, Nikolai. Lyapunov-Schmidt methods in nonlinear analysis and applications. 2011. Springer. 9789048161508. 751509629.
2. Gupta. Ankur. Chakraborty. Saikat. January 2009. Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors. Chemical Engineering Journal. 145. 3. 399–411. 10.1016/j.cej.2008.08.025. 1385-8947.
3. Balakotaiah. Vemuri. March 2004. Hyperbolic averaged models for describing dispersion effects in chromatographs and reactors. Korean Journal of Chemical Engineering. 21. 2. 318–328. 10.1007/bf02705415. 0256-1115.
4. Gupta. Ankur. Chakraborty. Saikat. 2008-01-19. Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions. Chemical Product and Process Modeling. 3. 2. 10.2202/1934-2659.1135. 1934-2659.