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Article

Kruck, Amina ; Reitmann, Volker
Upper Hausdorff dimension estimates for invariant sets of evolutionary systems on Hilbert manifolds. (English). In: Mikula, Karol (ed.): Proceedings of Equadiff 14. Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017. Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, Bratislava, 2017. pp. 247-254
MSC: 35B40, 35K57
Keywords:
Hilbert manifold, Hausdorff dimension, singular value
Summary:
We prove a generalization of the Douady-Oesterlé theorem on the upper bound of the Hausdorff dimension of an invariant set of a smooth map on an infinite dimensional manifold. It is assumed that the linearization of this map is a noncompact linear operator. A similar estimate is given for the Hausdorff dimension of an invariant set of a dynamical system generated by a differential equation on a Hilbert manifold.
References:
[1] Boichenko, V. A., Leonov, G. A., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Wiesbaden:Vieweg-Teubner Verlag, 2005. MR 2381409
[2] Chueshov, I. D.: Introduction to the Theory of Infinite-Dimensional Dissipative Systems. ACTA. Kharkov, 1999 (in Russian). English translation: Acta, Kharkov (2002) (see http://www.emis.de/monographs/Chueshov) MR 1632228
[3] Doering, C. R., Gibbon, J. D., Holm, D. D., Nicolaenko, B.: Exact Lyapunov Dimension of the Universal Attractor for the Complex Ginzburg-Landau Equation. Phys. Rev. Lett. 59, Iss. 26-28 (1987), pp. 2911–2914. DOI 10.1103/PhysRevLett.59.2911
[4] Douady, A., Oesterle, J.: Dimension de Hausdorff des attracteurs. Comptes Rendus del’Academie des Sciences Paris Serie A. 290 (1980), pp. 1135-1138. MR 0585918
[5] Ghidaglia, M., Temam, R.: Attractors for damped nonlinear hyperbolic equations. J. Math. Pures Appl., 66 (1987), pp. 273–319. MR 0913856
[6] Henon, M. A.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50 (1976), 2. pp. 69–77. DOI 10.1007/BF01608556 | MR 0422932
[7] Kruck, A. V., Malykh, A. E., Reitmann, V.: Upper Hausdorff dimension estimates and stratification for invariant sets of evolutionary systems on Hilbert manifolds. Differential Equations, 2017 (to appear). MR 3804278
[8] Lang, S.: Differential and Riemannian Manifolds. Springer, New York, 1995. MR 1335233
[9] Leonov, G. A., Reitmann, V., Smirnova, V. B.: Non-local Methods for Pendulum-like Feedback Systems. Teubner-Texte zur Mathematik, Bd. 132, B.G. Teubner Stuttgart-Leipzig, 1992. MR 1216519
[10] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York-Berlin: Springer-Verlag, 1988. MR 0953967
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