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August  2017, 37(8): 4159-4190. doi: 10.3934/dcds.2017177

Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

Inmaculada Baldomá 1, , Ernest Fontich 2, and  Pau Martín 3,

1. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal 647,08028 Barcelona, Spain
2. 

Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585,08007, Barcelona, Spain
3. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Ed. C3, Jordi Girona 1-3,08034 Barcelona, Spain

Received  December 2016 Revised  March 2017 Published  April 2017

Fund Project: I.B and P.M. have been partially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan Grant 2014SGR504. The work of E.F. has been partially supported by the Spanish Government grants MTM2013-41168P and MTM2016-80117-P and the Catalan Government grant 2014SGR-1145.

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to 1. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.

Keywords: Gevrey class, parabolic points, invariant manifolds.
Mathematics Subject Classification: Primary: 37D10.
Citation: Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
References:
[1]

M. Abate, Fatou flowers and parabolic curves, in Complex Analysis and Geometry, vol. 144of Springer Proc. Math. Stat., Springer, Tokyo, 2015, 1-39. doi: 10.1007/978-4-431-55744-9_1.

[2]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equalto identity, J. Differential Equations, 197 (2004), 45-72.  doi: 10.1016/j.jde.2003.07.005.

[3]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (Ⅱ).approximations by sums of homogeneous functions, 2015, Preprint available at https://arxiv.org/abs/1603.02535.

[4]

I. Baldomá and A. Haro, One dimensional invariant manifolds of Gevrey type in real-analyticmaps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 295-322.  doi: 10.3934/dcdsb.2008.10.295.

[5]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parameterization method for onedimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst., 17 (2007), 835-865.  doi: 10.3934/dcds.2007.17.835.

[6]

W. Balser, From Divergent Power Series to Analytic Functions, vol. 1582 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994, Theory and application of multisummable power series. doi: 10.1007/BFb0073564.

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅰ. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.  doi: 10.1512/iumj.2003.52.2245.

[8]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.

[9]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vols. Ⅰ, Ⅱ, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, Based, in part, on notes left by Harry Bateman.

[10]

M. Guardia, P. Martín, L. Sabbagh and T. M. Seara, Oscillatory orbits in the restricted elliptic planar three body problem, Disc. and Cont. Dyn. Sys. A, 37 (2017), 229-256.  doi: 10.3934/dcds.2017009.

[11]

M. Hakim, Analytic transformations of ($\mathbb{C}$p, 0) tangent to the identity, Duke Math. J., 92 (1998), 403-428.  doi: 10.1215/S0012-7094-98-09212-2.

[12]

À. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds, vol. 195 of Applied Mathematical Sciences, Springer, [Cham], 2016, From rigorous results to effective computations. doi: 10.1007/978-3-319-29662-3.

[13]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,133-163.

[14]

M. C. Irwin, On the stable manifold theorem, Bull. London Math. Soc., 2 (1970), 196-198.  doi: 10.1112/blms/2.2.196.

[15]

M. C. Irwin, A new proof of the pseudostable manifold theorem, J. London Math. Soc. (2), 21 (1980), 557-566.  doi: 10.1112/jlms/s2-21.3.557.

[16]

R. Martínez and C. Simó, On the regularity of the infinity manifolds: the case of sitnikov problem and some global aspects of the dynamics, 2009, URL https://www.fields.utoronto.

[17]

R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion, Regul. Chaotic Dyn., 19 (2014), 745-765.  doi: 10.1134/S1560354714060112.

[18]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88.  doi: 10.1016/0022-0396(73)90077-6.

[19]

K. Meyer and G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer-Verlag, New York, (1992).  doi: 10.1007/978-1-4757-4073-8.

[20]

J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, N. J., 1973, With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J, Annals of Mathematics Studies, No. 77.

show all references

References:
[1]

M. Abate, Fatou flowers and parabolic curves, in Complex Analysis and Geometry, vol. 144of Springer Proc. Math. Stat., Springer, Tokyo, 2015, 1-39. doi: 10.1007/978-4-431-55744-9_1.

[2]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equalto identity, J. Differential Equations, 197 (2004), 45-72.  doi: 10.1016/j.jde.2003.07.005.

[3]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (Ⅱ).approximations by sums of homogeneous functions, 2015, Preprint available at https://arxiv.org/abs/1603.02535.

[4]

I. Baldomá and A. Haro, One dimensional invariant manifolds of Gevrey type in real-analyticmaps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 295-322.  doi: 10.3934/dcdsb.2008.10.295.

[5]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parameterization method for onedimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst., 17 (2007), 835-865.  doi: 10.3934/dcds.2007.17.835.

[6]

W. Balser, From Divergent Power Series to Analytic Functions, vol. 1582 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1994, Theory and application of multisummable power series. doi: 10.1007/BFb0073564.

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅰ. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328.  doi: 10.1512/iumj.2003.52.2245.

[8]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅲ. Overview and applications, J. Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.

[9]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vols. Ⅰ, Ⅱ, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953, Based, in part, on notes left by Harry Bateman.

[10]

M. Guardia, P. Martín, L. Sabbagh and T. M. Seara, Oscillatory orbits in the restricted elliptic planar three body problem, Disc. and Cont. Dyn. Sys. A, 37 (2017), 229-256.  doi: 10.3934/dcds.2017009.

[11]

M. Hakim, Analytic transformations of ($\mathbb{C}$p, 0) tangent to the identity, Duke Math. J., 92 (1998), 403-428.  doi: 10.1215/S0012-7094-98-09212-2.

[12]

À. Haro, M. Canadell, J.-L. Figueras, A. Luque and J.-M. Mondelo, The Parameterization Method for Invariant Manifolds, vol. 195 of Applied Mathematical Sciences, Springer, [Cham], 2016, From rigorous results to effective computations. doi: 10.1007/978-3-319-29662-3.

[13]

M. W. Hirsch and C. C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,133-163.

[14]

M. C. Irwin, On the stable manifold theorem, Bull. London Math. Soc., 2 (1970), 196-198.  doi: 10.1112/blms/2.2.196.

[15]

M. C. Irwin, A new proof of the pseudostable manifold theorem, J. London Math. Soc. (2), 21 (1980), 557-566.  doi: 10.1112/jlms/s2-21.3.557.

[16]

R. Martínez and C. Simó, On the regularity of the infinity manifolds: the case of sitnikov problem and some global aspects of the dynamics, 2009, URL https://www.fields.utoronto.

[17]

R. Martínez and C. Simó, Invariant manifolds at infinity of the RTBP and the boundaries of bounded motion, Regul. Chaotic Dyn., 19 (2014), 745-765.  doi: 10.1134/S1560354714060112.

[18]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88.  doi: 10.1016/0022-0396(73)90077-6.

[19]

K. Meyer and G. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer-Verlag, New York, (1992).  doi: 10.1007/978-1-4757-4073-8.

[20]

J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, N. J., 1973, With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J, Annals of Mathematics Studies, No. 77.

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