Lyapunov–Schmidt reduction

Problem setup

Let

: f(x,\lambda)=0 ,

be the given nonlinear equation, X,\Lambda, and Y are Banach spaces (\Lambda is the parameter space). f(x,\lambda) is the C^p -map from a neighborhood of some point (x_0,\lambda_0)\in X\times \Lambda to Y and the equation is satisfied at this point

: f(x_0,\lambda_0)=0.

For the case when the linear operator f_x(x,\lambda) is invertible, the implicit function theorem assures that there exists a solution x(\lambda) satisfying the equation f(x(\lambda),\lambda)=0 at least locally close to \lambda_0 .

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

Assumptions

One assumes that the operator f_x(x,\lambda) is a Fredholm operator.

\ker f_x (x_0,\lambda_0)=X_1 and X_1 has finite dimension.

The range of this operator \mathrm{ran} f_x (x_0,\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,\lambda_0)=(0,0).

Lyapunov–Schmidt construction

Let us split Y into the direct product Y= Y_1 \oplus Y_2 , where \dim Y_2 .

Let Q be the projection operator onto Y_1 .

Consider also the direct product X= X_1 \oplus X_2 .

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

: Qf(x,\lambda)=0 ,

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

Let x_1\in X_1 and x_2 \in X_2 , then the first equation

: Qf(x_1+x_2,\lambda)=0 ,

can be solved with respect to x_2 by applying the implicit function theorem to the operator

: 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 x_2(x_1,\lambda) satisfying

: Qf(x_1+x_2(x_1,\lambda),\lambda)=0. ,

Now substituting x_2(x_1,\lambda) into the second equation, one obtains the final finite-dimensional equation

: (I-Q)f(x_1+x_2(x_1,\lambda),\lambda)=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 : \lambda

Applications

Lyapunov–Schmidt reduction has been used in economics, natural sciences, and engineering often in combination with bifurcation theory, perturbation theory, and regularization. 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.

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.

References

1. Sidorov, Nikolai. (2011). "Lyapunov-Schmidt methods in nonlinear analysis and applications". Springer.
2. Golubitsky, Martin. (1985). "The Hopf Bifurcation". Springer New York.
3. Gupta, Ankur. (January 2009). "Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors". Chemical Engineering Journal.
4. Balakotaiah, Vemuri. (March 2004). "Hyperbolic averaged models for describing dispersion effects in chromatographs and reactors". Korean Journal of Chemical Engineering.
5. Gupta, Ankur. (2008-01-19). "Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions". Chemical Product and Process Modeling.