Cone conditions and covering relations fortopologically normally hyperbolic invariant manifolds

Maciej J. Capiński; Piotr Zgliczyński

doi:10.3934/dcds.2011.30.641

Article Contents

2011, Volume 30, Issue 3: 641-670. Doi: 10.3934/dcds.2011.30.641

This issue Previous Article Discrete gradient fields on infinite complexes Next Article Homeomorphisms of the annulus with a transitive lift II

Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds

Maciej J. Capiński^1, and
Piotr Zgliczyński^2,

1.
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków
2.
Jagiellonian University, Institute of Computer Science, Łojasiewicza 6, 30-387 Kraków

Received: February 28, 2010

Revised: September 30, 2010

Published: August 2011

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Abstract

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. We provide conditions which imply the existence of the manifold within an investigated region of the phase space. The required assumptions are formulated in a way which allows for rigorous computer assisted verification. We apply our method to obtain an invariant manifold within an explicit range of parameters for the rotating Hénon map.

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Mathematics Subject Classification: Primary: 34D10, 34D35; Secondary: 37C25.

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References

[1]	P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states, Invent. Math., 174 (2008), 355-433.doi: 10.1007/s00222-008-0141-y.
[2]	M. J. Capiński, Covering relations and the existence of topologically normally hyperbolic invariant sets, Discrete Contin. Dyn. Syst. Ser A., 23 (2009), 705-725.
[3]	M. Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems, 24 (2004), 1359-1394.doi: 10.1017/S0143385703000701.
[4]	M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. of Diff. Equations, 202 (2004), 33-58.
[5]	A. Haro and R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 1261-1300.doi: 10.3934/dcdsb.2006.6.1261.
[6]	M. Hirsh, "Differential Topology," Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.
[7]	M. Hirsh, C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[8]	C. K. R. T. Jones, Geometric singular perturbation theory. Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Math., 1609, Springer, Berlin, 1995, 44-118.
[9]	N. G. Lloyd, "Degree Theory," Cambridge Tracts in Math., No. 73, Cambridge Univ. Press, London, 1978.
[10]	Stephen Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," Applied Mathematical Sciences, 105, Springer-Verlag, New York, 1994. pp x+193.
[11]	P. Zgliczyński, Covering relations, cone conditions and stable manifold theorem, J. Differential Equations, 246 (2009), 1774-1819.