Topological shadowing and the Grobman-Hartman theorem
Keywords
Grobman-Hartman theorem, Hölder regularity, covering relation, isolating segment, cone conditionAbstract
We give geometric proofs for the Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone conditions for ODEs. We establish topological versions of the Grobman-Hartman theorem as the existence of some semiconjugaces.References
V.I. Arnol’d, Supplementary Chapters to the Theory of Ordinary Differential Equations, Nauka, Moscow, 1978.
L. Barreira and C. Valls, Hölder Grobman–Hartman linearization, Discrete Contin. Dyn. Syst. 18 (2007), 187–197.
G. Belitskiı̆, On the Grobman–Hartman Theorem in the class C α , preprint.
G. Belitskiı̆ and V. Rayskin, On the Grobman–Hartman Theorem in α-Hölder class for Banach spaces, preprint.
C. Chicone, Ordinary Differential Equations with Applications, Springer, New York, 1999.
C. Chicone and R. Swanson, Linearization via the Lie derivative, Electron. J. Differential Equations, Art. No. 2, (2000).
C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS, vol. 38, Amer. Math. Soc., Providence, 1978.
D.M. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk SSSR 128 (1959), 880–881.
D.M. Grobman, Topological classification of neighborhoods of a singularity in n-space, Mat. Sb. (N.S.) 56 (1962), no. 98, 77–94.
P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620.
P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2) 5 (1960), 220–241.
P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc. 14 (1963), 568–573.
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and Its Applications, vol. 54, Cambridge University Press, Cambridge, 1995.
J. Moser, On a theorem of Anosov, J. Differential Equations 5 (1969), 411–440.
J. Palis, On the local structure of hyperbolic points in Banach spaces, An. Acad. Brasil. Ciênc. 40 (1968), 263–266.
J. Palis and W. de Melo, Geometric Theory of Dynamical Systems. An Introduction, Springer, New York, 1982.
C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363–367.
R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomuous ordinary differential equations, Nonlinear Anal. 22 (1994), 707–737.
R. Srzednicki, On detection of chaotic dynamics in ordinary differential equations, Nonlinear Anal. 30 (1997), no. 8, 4927–4935.
R. Srzednicki, On geometric detection of periodic solutions and chaos, Proceedings of the Conference “Nonlinear Analysis and Boundary Value Problems”, CISM Udine, October 2–6, 1995, CISM Courses and Lectures, vol. 371, Springer, New York, 1996, pp. 197–209.
R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), 66–82.
T. Ważewski, Sur un principe topologique de l’examen de l’allure asymptotique des intégrales des équations différentielles ordinaires, Ann. Soc. Polon. Math. 20 (1947), 279–313.
K. Wójcik and P. Zgliczyński, Isolating segments, fixed point index and symbolic dynamics, J. Differential Equations 161 (2000), 245–288.
E. Zehnder, Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic Capacities, EMS Textbooks in Mathematics, EMS, Zürich, 2010.
P. Zgliczyński, Covering relations, cone conditions and the stable manifold theorem, J. Differential Equations 246 (2009), 1774–1819.
P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), no. 1, 32–58.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0