Geodesic
Differential-algebraic equations in the theory of invariant manifolds for singular equations
Lobachevskii journal of mathematics, Tome 20 (2005), pp. 77-89.

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Analogs of Grobman-Hartman theorem on stable and unstable manifolds solutions for differential equations in Banach spaces with degenerate Fredholm operator at the derivative are proved. Jordan chains tools and the implicit operator theorem are used. In contrast to the usual evolution equation here the central manifold appears even for the case of spectrum absence on the imaginary axis. If on the imaginary axis there is only a finite number of spectrum points, then the original nonlinear equation is reduced to two differential–algebraic systems on the center manifold.
Keywords: pseudoparabolic singular differential equations, Grobman–Hartman theorem, center manifold, differential-algebraic systems. 
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B. Karasözen; I. V. Konopleva; B. V. Loginov. Differential-algebraic equations in the theory of invariant manifolds for singular equations. Lobachevskii journal of mathematics, Tome 20 (2005), pp. 77-89. http://geodesic.mathdoc.fr/item/LJM_2005_20_a5/

[1] Barenblatt G. I., Zheltov Yu. P., Kochina I. N., “On the principal conceptions of the filtration theory in jointing media”, Appl. Math. Mech., 24:5 (1960), 58–73 | MR

[2] Carr J., Applications of Centre Manifold Theory, Appl. Math. Sci., 35, Springer, Berlin, 1981 | MR | Zbl

[3] Falaleev M. V., Romanova O. A., Sidorov N. A., “Generalized Jordan sets in the theory of singular partial differential-operator equations”, Proc. p. II ICCS 2003, Lecture Notes on Computer Science, 2658, Springer, Berlin, 2003, 523–533 | MR

[4] Govaerts W. J. F., Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelfia, 2000 | MR

[5] Gurel O. (Ed.), Bifurcation Theory and its Applications in Scientific Disciplines, Annals of the New York Acad. Sci., 316, New York, 1979 | MR

[6] Hale J., Introduction to dynamic bifurcation. Bifurcation Theory and Appl., Lecture Notes in Math., 1057, Springer, Berlin, 1984 | MR

[7] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., 840, Springer, Berlin, 1981 | MR | Zbl

[8] Joseph D. J., Stability of Fluid Motions, Springer, Berlin, 1976

[9] Karasözen B., Konopleva I., Loginov B., “Invariant manifolds and Grobman–Hartman theorem for equations with degenerate operator at the derivative”, Proc. p. II ICCS 2003, Lecture Notes on Computer Science, 2658, Springer, Berlin, 2003, 533–543 | MR

[10] Kuznetsov Yu. A., Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995 | MR | Zbl

[11] Loginov B., Rousak Yu., “Generalized Jordan structure in the problem the stability of bifurcating solutions”, Nonlinear Analalysis Theory, Math. Appl., 17:3 (1991), 219–232 | DOI | MR | Zbl

[12] Loginov B., “Branching equation in the root subspace”, Nonlinear Analysis Theory Math. Appl., 32:3 (1998), 439–448 | DOI | MR | Zbl

[13] Loginov B., Konopleva I., “Symmetry of resolving systems for differential equations with Fredholm operator at the derivative”, Proc. Int. Conf. MOGRAN-2000, USATU, Ufa, 2000, 116–119

[14] Marsden J. E., McCracken M., The Hopf Bifurcation and its Applications, Springer, New-York, 1976 | MR

[15] Oskolkov A. P., “Initial-boundary value problems for equations of Kelvin–Foight and Oldroidt fluids”, Proc. Steklov Math. Inst. AN SSSR, 179, Nauka, Moscow, 1988, 126–164 | MR

[16] Ruelle D., “Bifurcations in the presence of a symmetry group”, Arch. Rat. Mech. Anal., 51 (1973), 136–152 | DOI | MR | Zbl

[17] Sidorov N., Blagodatskaya E., “Differential equations with a Fredholm operator in the main differential statement”, Soviet Math. Dokl., 44:1 (1992), 302–305 | MR

[18] Sviridiuk G., “Phase spaces of semilinear Sobolev-type equations with relatively strong sectorial operator”, Algebra i Analiz, 6:5 (1994), 252–272 (Russian)

[19] Vainberg M., Trenogin V., Branching Theory of Solutions of Nonlinear Equations, Wolters Noordorf, Leyden, 1974

[20] Vanderbauwhede A., Iooss G., “Center manifold theory in infinite dimensions”, Dynamics Reported, N.S., 1 (1992), 125–163 | MR | Zbl

[21] Volevich L., Shirikyan A., “Local dynamics for high-order semilinear hyperbolic equations”, Izv. Math., 64:3 (2000), 439–485 | DOI | MR | Zbl

[22] Whitham G. B., Linear and Nonlinear Waves, Wiley-Int. Publ., New York, 1974 | MR | Zbl