August  2011, 30(3): 641-670. doi: 10.3934/dcds.2011.30.641

Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds

1. 

AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland

2. 

Jagiellonian University, Institute of Computer Science, Łojasiewicza 6, 30-387 Kraków

Received  March 2010 Revised  October 2010 Published  March 2011

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.
Citation: Maciej J. Capiński, Piotr Zgliczyński. Cone conditions and covering relations for topologically normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 641-670. doi: 10.3934/dcds.2011.30.641
References:
[1]

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M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems,, J. of Diff. Equations, 202 (2004), 33.   Google Scholar

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Stephen Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems,", Applied Mathematical Sciences, 105 (1994).   Google Scholar

[11]

P. Zgliczyński, Covering relations, cone conditions and stable manifold theorem,, J. Differential Equations, 246 (2009), 1774.   Google Scholar

show all references

References:
[1]

P. W. Bates, K. Lu and C. Zeng, Approximately invariant manifolds and global dynamics of spike states,, Invent. Math., 174 (2008), 355.  doi: 10.1007/s00222-008-0141-y.  Google Scholar

[2]

M. J. Capiński, Covering relations and the existence of topologically normally hyperbolic invariant sets,, Discrete Contin. Dyn. Syst. Ser A., 23 (2009), 705.   Google Scholar

[3]

M. Chaperon, Stable manifolds and the Perron-Irwin method,, Ergodic Theory Dynam. Systems, 24 (2004), 1359.  doi: 10.1017/S0143385703000701.  Google Scholar

[4]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems,, J. of Diff. Equations, 202 (2004), 33.   Google Scholar

[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.  doi: 10.3934/dcdsb.2006.6.1261.  Google Scholar

[6]

M. Hirsh, "Differential Topology,", Graduate Texts in Mathematics, (1976).   Google Scholar

[7]

M. Hirsh, C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Mathematics, 583 (1977).   Google Scholar

[8]

C. K. R. T. Jones, Geometric singular perturbation theory. Dynamical systems (Montecatini Terme, 1994),, Lecture Notes in Math., 1609 (1995), 44.   Google Scholar

[9]

N. G. Lloyd, "Degree Theory,", Cambridge Tracts in Math., (1978).   Google Scholar

[10]

Stephen Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems,", Applied Mathematical Sciences, 105 (1994).   Google Scholar

[11]

P. Zgliczyński, Covering relations, cone conditions and stable manifold theorem,, J. Differential Equations, 246 (2009), 1774.   Google Scholar

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