American Institute of Mathematical Sciences

Advanced
September  2008, 10(2&3, September): 295-322. doi: 10.3934/dcdsb.2008.10.295

One dimensional invariant manifolds of Gevrey type in real-analytic maps

I. Baldomá 1, and  Àlex Haro 2,

1. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona

Received  November 2006 Revised  July 2007 Published  June 2008

In this paper we study the basic questions of existence, uniqueness, differentiability, analyticity and computability of one dimensional parabolic manifolds of degenerate fixed points, i.e. invariant manifolds tangent to the eigenspace of 1, which is assumed to be a simple eigenvalue. We use the parameterization method, reducing the dynamics on the parabolic manifold to a polynomial. We prove that the asymptotic expansions of the parabolic manifold are of Gevrey type. Moreover, under suitable hypothesis, we also prove that the asymptotic expansions correspond to a real-analytic parameterization of an invariant curve that goes to the fixed point. The parameterization is Gevrey at the fixed point, hence $C^\infty$.
Keywords: asymptotic series, Gevrey., parabolic points, Invariant manifolds.
Mathematics Subject Classification: Primary: 37D1.
Citation: I. Baldomá, Àlex Haro. One dimensional invariant manifolds of Gevrey type in real-analytic maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 295-322. doi: 10.3934/dcdsb.2008.10.295
[1]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
[2]

Inmaculada Baldomá, Ernest Fontich, Rafael de la Llave, Pau Martín. The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 835-865. doi: 10.3934/dcds.2007.17.835
[3]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35
[4]

P. De Maesschalck. Gevrey normal forms for nilpotent contact points of order two. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 677-688. doi: 10.3934/dcds.2014.34.677
[5]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[6]

Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511
[7]

José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
[8]

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
[9]

George Osipenko. Indestructibility of invariant locally non-unique manifolds. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 203-219. doi: 10.3934/dcds.1996.2.203
[10]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967
[11]

Arturo Echeverría-Enríquez, Alberto Ibort, Miguel C. Muñoz-Lecanda, Narciso Román-Roy. Invariant forms and automorphisms of locally homogeneous multisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (4) : 397-419. doi: 10.3934/jgm.2012.4.397
[12]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Invariant manifolds as pullback attractors of nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 579-596. doi: 10.3934/dcds.2006.15.579
[13]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1
[14]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197
[15]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455
[16]

Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701
[17]

Keith Burns, Eugene Gutkin. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 403-413. doi: 10.3934/dcds.2008.21.403
[18]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[19]

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
[20]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences