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
- [math]\displaystyle{ f(x,\lambda)=0 \, }[/math]
be the given nonlinear equation, [math]\displaystyle{ X,\Lambda, }[/math] and [math]\displaystyle{ Y }[/math] are Banach spaces ([math]\displaystyle{ \Lambda }[/math] is the parameter space). [math]\displaystyle{ f(x,\lambda) }[/math] is the [math]\displaystyle{ C^p }[/math]-map from a neighborhood of some point [math]\displaystyle{ (x_0,\lambda_0)\in X\times \Lambda }[/math] to [math]\displaystyle{ Y }[/math] and the equation is satisfied at this point
- [math]\displaystyle{ f(x_0,\lambda_0)=0. }[/math]
For the case when the linear operator [math]\displaystyle{ f_x(x,\lambda) }[/math] is invertible, the implicit function theorem assures that there exists a solution [math]\displaystyle{ x(\lambda) }[/math] satisfying the equation [math]\displaystyle{ f(x(\lambda),\lambda)=0 }[/math] at least locally close to [math]\displaystyle{ \lambda_0 }[/math].
In the opposite case, when the linear operator [math]\displaystyle{ f_x(x,\lambda) }[/math] is non-invertible, the Lyapunov–Schmidt reduction can be applied in the following way.
Assumptions
One assumes that the operator [math]\displaystyle{ f_x(x,\lambda) }[/math] is a Fredholm operator.
[math]\displaystyle{ \ker f_x (x_0,\lambda_0)=X_1 }[/math] and [math]\displaystyle{ X_1 }[/math] has finite dimension.
The range of this operator [math]\displaystyle{ \mathrm{ran} f_x (x_0,\lambda_0)=Y_1 }[/math] has finite co-dimension and is a closed subspace in [math]\displaystyle{ Y }[/math].
Without loss of generality, one can assume that [math]\displaystyle{ (x_0,\lambda_0)=(0,0). }[/math]
Lyapunov–Schmidt construction
Let us split [math]\displaystyle{ Y }[/math] into the direct product [math]\displaystyle{ Y= Y_1 \oplus Y_2 }[/math], where [math]\displaystyle{ \dim Y_2 \lt \infty }[/math].
Let [math]\displaystyle{ Q }[/math] be the projection operator onto [math]\displaystyle{ Y_1 }[/math].
Consider also the direct product [math]\displaystyle{ X= X_1 \oplus X_2 }[/math].
Applying the operators [math]\displaystyle{ Q }[/math] and [math]\displaystyle{ I-Q }[/math] to the original equation, one obtains the equivalent system
- [math]\displaystyle{ Qf(x,\lambda)=0 \, }[/math]
- [math]\displaystyle{ (I-Q)f(x,\lambda)=0 \, }[/math]
Let [math]\displaystyle{ x_1\in X_1 }[/math] and [math]\displaystyle{ x_2 \in X_2 }[/math], then the first equation
- [math]\displaystyle{ Qf(x_1+x_2,\lambda)=0 \, }[/math]
can be solved with respect to [math]\displaystyle{ x_2 }[/math] by applying the implicit function theorem to the operator
- [math]\displaystyle{ Qf(x_1+x_2,\lambda): \quad X_2\times(X_1\times\Lambda)\to Y_1 \, }[/math]
(now the conditions of the implicit function theorem are fulfilled).
Thus, there exists a unique solution [math]\displaystyle{ x_2(x_1,\lambda) }[/math] satisfying
- [math]\displaystyle{ Qf(x_1+x_2(x_1,\lambda),\lambda)=0. \, }[/math]
Now substituting [math]\displaystyle{ x_2(x_1,\lambda) }[/math] into the second equation, one obtains the final finite-dimensional equation
- [math]\displaystyle{ (I-Q)f(x_1+x_2(x_1,\lambda),\lambda)=0. \, }[/math]
Indeed, the last equation is now finite-dimensional, since the range of [math]\displaystyle{ (I-Q) }[/math] is finite-dimensional. This equation is now to be solved with respect to [math]\displaystyle{ x_1 }[/math], which is finite-dimensional, and parameters :[math]\displaystyle{ \lambda }[/math]
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
- ↑ 1.0 1.1 Sidorov, Nikolai (2011). Lyapunov-Schmidt methods in nonlinear analysis and applications. Springer. ISBN 9789048161508. OCLC 751509629.
- ↑ 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
- ↑ 3.0 3.1 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.
- ↑ 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.
- ↑ 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.
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