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]

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

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

[1]

Maciej J. Capiński. Covering relations and the existence of topologically normally hyperbolic invariant sets. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 705-725. doi: 10.3934/dcds.2009.23.705

[2]

Henk Broer, Aaron Hagen, Gert Vegter. Numerical approximation of normally hyperbolic invariant manifolds. Conference Publications, 2003, 2003 (Special) : 133-140. doi: 10.3934/proc.2003.2003.133

[3]

Amadeu Delshams, Marian Gidea, Pablo Roldán. Transition map and shadowing lemma for normally hyperbolic invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1089-1112. doi: 10.3934/dcds.2013.33.1089

[4]

Maciej J. Capiński, Piotr Zgliczyński. Covering relations and non-autonomous perturbations of ODEs. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 281-293. doi: 10.3934/dcds.2006.14.281

[5]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923

[6]

Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339

[7]

Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407

[8]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91

[9]

Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469

[10]

Brent Everitt, John Ratcliffe and Steven Tschantz. The smallest hyperbolic 6-manifolds. Electronic Research Announcements, 2005, 11: 40-46.

[11]

Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64

[12]

Carlos Arnoldo Morales. Strong stable manifolds for sectional-hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 553-560. doi: 10.3934/dcds.2007.17.553

[13]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451

[14]

Walter D. Neumann and Jun Yang. Invariants from triangulations of hyperbolic 3-manifolds. Electronic Research Announcements, 1995, 1: 72-79.

[15]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Tori with hyperbolic dynamics in 3-manifolds. Journal of Modern Dynamics, 2011, 5 (1) : 185-202. doi: 10.3934/jmd.2011.5.185

[16]

Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018

[17]

Andy Hammerlindl, Rafael Potrie, Mario Shannon. Seifert manifolds admitting partially hyperbolic diffeomorphisms. Journal of Modern Dynamics, 2018, 12: 193-222. doi: 10.3934/jmd.2018008

[18]

Vladimir V. Chepyzhov, Anna Kostianko, Sergey Zelik. Inertial manifolds for the hyperbolic relaxation of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1115-1142. doi: 10.3934/dcdsb.2019009

[19]

Peter Brune, Björn Schmalfuss. Inertial manifolds for stochastic pde with dynamical boundary conditions. Communications on Pure & Applied Analysis, 2011, 10 (3) : 831-846. doi: 10.3934/cpaa.2011.10.831

[20]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]