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]

References

1. Sidorov, Nikolai (2011). Lyapunov-Schmidt methods in nonlinear analysis and applications. Springer. ISBN 9789048161508. OCLC 751509629.
2. Golubitsky, Martin; Schaeffer, David G. (1985), "The Hopf Bifurcation", Applied Mathematical Sciences, Springer New York, pp. 337–396, doi:10.1007/978-1-4612-5034-0_8, ISBN 9781461295334
3. 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. doi:10.1016/j.cej.2008.08.025. ISSN 1385-8947.
4. Balakotaiah, Vemuri (March 2004). "Hyperbolic averaged models for describing dispersion effects in chromatographs and reactors". Korean Journal of Chemical Engineering. 21 (2): 318–328. doi:10.1007/bf02705415. ISSN 0256-1115.
5. 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). doi:10.2202/1934-2659.1135. ISSN 1934-2659.

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.