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October  2021, 41(10): 4667-4704. doi: 10.3934/dcds.2021053

Differentiable invariant manifolds of nilpotent parabolic points

Clara Cufí-Cabré 1, and  Ernest Fontich 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), Barcelona Graduate School of Mathematics (BGSMath), 08193 Bellaterra, Spain
2. 

Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Barcelona Graduate School of Mathematics (BGSMath), Gran Via 585. 08007 Barcelona, Spain

Received  October 2020 Revised  February 2021 Published  October 2021 Early access  April 2021

Fund Project: The first author has been partially supported by the Spanish Government grants MTM2016-77278-P (MINECO/ FEDER, UE), PID2019-104658GB-I00 (MICINN/FEDER, UE) and BES-2017-081570, and by the Catalan Government grant 2017-SGR-1617. The second author has been partially supported by the Spanish Government grants MTM2016-80117-P (MINECO/ FEDER, UE) and PID2019-104851GB-I00 (MICINN/FEDER, UE) and by the Catalan Government grant 2017-SGR-1374

We consider a map $ F $ of class $ C^r $ with a fixed point of parabolic type whose differential is not diagonalizable, and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $ F $, there exist invariant curves of class $ C^r $ away from the fixed point, and that they are analytic when $ F $ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of $ F $ on them.

Keywords: Parabolic point, invariant manifold, parameterization method.
Mathematics Subject Classification: Primary: 37D10; Secondary: 37C25.
Citation: Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4667-4704. doi: 10.3934/dcds.2021053
References:
[1]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations 197 (2004), no. 1, 45-72. doi: 10.1016/j.jde.2003.07.005.

[2]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst. 17 (2007), no. 4,835-865. doi: 10.3934/dcds.2007.17.835.

[3]

I. Baldomá, E. Fontich and P. Martín, Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points, Discrete Contin. Dyn. Syst. 37 (2017), no. 8, 4159-4190. doi: 10.3934/dcds.2017177.

[4]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters, J. Differential Equations 268 (2020), no. 9, 5516 -5573. doi: 10.1016/j.jde.2019.11.100.

[5]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions, J. Differential Equations 268 (2020), no. 9, 5574-5627. doi: 10.1016/j.jde.2019.11.099.

[6]

R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funct. Anal. Appl. 9 (1975), 144-145.

[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), no. 2,283-328. doi: 10.1512/iumj.2003.52.2245.

[8]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds II: Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2003), no. 2,329-360. doi: 10.1512/iumj.2003.52.2407.

[9]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds III: overview and applications, J. Differential Equations 218 (2005), no. 2,444-515. doi: 10.1016/j.jde.2004.12.003.

[10]

J. Casasayas, E. Fontich and A. Nunes, Invariant manifolds for a class of parabolic points, Nonlinearity 5 (1992), no. 5, 1193-1210. doi: 10.1088/0951-7715/5/5/008.

[11]

S. Craig, F. Diacu, E. A. Lacomba and E. Pérez, On the anisotropic Manev problem, J. Math. Phys. 40 (1999), no. 3, 1359-1375. doi: 10.1063/1.532807.

[12]

E. Fontich, Stable curves asymptotic to a degenerate fixed point, Nonlinear Anal. 35 (1999), no. 6, Ser. A: Theory Methods, 711-733. doi: 10.1016/S0362-546X(98)00004-2.

[13]

E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity 13 (2000), no. 5, 1561-1593. doi: 10.1088/0951-7715/13/5/309.

[14]

V. J. García-Garrido, M. Agaoglou and S. Wiggins, Exploring isomerization dynamics on a potential energy surface with an index-2 saddle using Lagrangian descriptors, Commun. Nonlinear Sci. Numer. Simul. 89 (2020), 105-331. doi: 10.1016/j.cnsns.2020.105331.

[15]

R. Guantes, F. Borondo and S. Miret-Artés, Periodic orbits and the homoclinic tangle in atom-surface chaotic scattering, Phys. Rev. E 56 (1997), 378-389. doi: 10.1103/PhysRevE.56.378.

[16]

M. Guardia, P. Martín and T. M-Seara, Oscillatory motions for the restricted planar circular three body problem, Invent. Math. 203 (2016), no. 2,417-492. doi: 10.1007/s00222-015-0591-y.

[17]

M. Guardia, P. Martín, T. M-Seara and L. Sabbagh, Oscillatory orbits in the restricted elliptic planar three body problem, Discrete Contin. Dyn. Syst. 37 (2017), no. 1,229-256. doi: 10.3934/dcds.2017009.

[18]

À. Haro, M. Canadell, J. Ll. Figueras, A. Luque and J. M. Mondelo, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Applied Mathematical Sciences, 195, Springer, 2016. doi: 10.1007/978-3-319-29662-3.

[19]

W. T. Jamieson and O. Merino, Local dynamics of planar maps with a non-isolated fixed point exhibiting 1-1 resonance, Adv. Difference Equ., (2018), Paper No. 142, 22 pp. doi: 10.1186/s13662-018-1595-x.

[20]

L. M. Lerman and J. D. Meiss, Mixed dynamics in a parabolic standard map, Phys. D, 315 (2016), 58-71.  doi: 10.1016/j.physd.2015.09.003.

[21]

J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann., 248 (1980), 153-184.  doi: 10.1007/BF01421955.

[22]

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.

[23]

J. Moser, Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J.; Annals of Mathematics Studies, 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973.

[24]

Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.

[25]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650. 

[26]

F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble), 23 (1973), 163-195.  doi: 10.5802/aif.467.

[27]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100.

[28]

S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbb{C}, 0) \to (\mathbb{C}, 0)$, Funktsional. Anal. i Prilozhen., 15 (1981), 1-17.  doi: 10.1007/BF01082373.

[29]

W. Zhang and W. Zhang, On invariant manifolds and invariant foliations without a spectral gap, Adv. Math., 303 (2016), 549-610.  doi: 10.1016/j.aim.2016.08.027.

show all references

References:
[1]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations 197 (2004), no. 1, 45-72. doi: 10.1016/j.jde.2003.07.005.

[2]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parameterization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Contin. Dyn. Syst. 17 (2007), no. 4,835-865. doi: 10.3934/dcds.2007.17.835.

[3]

I. Baldomá, E. Fontich and P. Martín, Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points, Discrete Contin. Dyn. Syst. 37 (2017), no. 8, 4159-4190. doi: 10.3934/dcds.2017177.

[4]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (I). Existence and dependence on parameters, J. Differential Equations 268 (2020), no. 9, 5516 -5573. doi: 10.1016/j.jde.2019.11.100.

[5]

I. Baldomá, E. Fontich and P. Martín, Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions, J. Differential Equations 268 (2020), no. 9, 5574-5627. doi: 10.1016/j.jde.2019.11.099.

[6]

R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funct. Anal. Appl. 9 (1975), 144-145.

[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), no. 2,283-328. doi: 10.1512/iumj.2003.52.2245.

[8]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds II: Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2003), no. 2,329-360. doi: 10.1512/iumj.2003.52.2407.

[9]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds III: overview and applications, J. Differential Equations 218 (2005), no. 2,444-515. doi: 10.1016/j.jde.2004.12.003.

[10]

J. Casasayas, E. Fontich and A. Nunes, Invariant manifolds for a class of parabolic points, Nonlinearity 5 (1992), no. 5, 1193-1210. doi: 10.1088/0951-7715/5/5/008.

[11]

S. Craig, F. Diacu, E. A. Lacomba and E. Pérez, On the anisotropic Manev problem, J. Math. Phys. 40 (1999), no. 3, 1359-1375. doi: 10.1063/1.532807.

[12]

E. Fontich, Stable curves asymptotic to a degenerate fixed point, Nonlinear Anal. 35 (1999), no. 6, Ser. A: Theory Methods, 711-733. doi: 10.1016/S0362-546X(98)00004-2.

[13]

E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity 13 (2000), no. 5, 1561-1593. doi: 10.1088/0951-7715/13/5/309.

[14]

V. J. García-Garrido, M. Agaoglou and S. Wiggins, Exploring isomerization dynamics on a potential energy surface with an index-2 saddle using Lagrangian descriptors, Commun. Nonlinear Sci. Numer. Simul. 89 (2020), 105-331. doi: 10.1016/j.cnsns.2020.105331.

[15]

R. Guantes, F. Borondo and S. Miret-Artés, Periodic orbits and the homoclinic tangle in atom-surface chaotic scattering, Phys. Rev. E 56 (1997), 378-389. doi: 10.1103/PhysRevE.56.378.

[16]

M. Guardia, P. Martín and T. M-Seara, Oscillatory motions for the restricted planar circular three body problem, Invent. Math. 203 (2016), no. 2,417-492. doi: 10.1007/s00222-015-0591-y.

[17]

M. Guardia, P. Martín, T. M-Seara and L. Sabbagh, Oscillatory orbits in the restricted elliptic planar three body problem, Discrete Contin. Dyn. Syst. 37 (2017), no. 1,229-256. doi: 10.3934/dcds.2017009.

[18]

À. Haro, M. Canadell, J. Ll. Figueras, A. Luque and J. M. Mondelo, The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Applied Mathematical Sciences, 195, Springer, 2016. doi: 10.1007/978-3-319-29662-3.

[19]

W. T. Jamieson and O. Merino, Local dynamics of planar maps with a non-isolated fixed point exhibiting 1-1 resonance, Adv. Difference Equ., (2018), Paper No. 142, 22 pp. doi: 10.1186/s13662-018-1595-x.

[20]

L. M. Lerman and J. D. Meiss, Mixed dynamics in a parabolic standard map, Phys. D, 315 (2016), 58-71.  doi: 10.1016/j.physd.2015.09.003.

[21]

J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann., 248 (1980), 153-184.  doi: 10.1007/BF01421955.

[22]

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.

[23]

J. Moser, Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J.; Annals of Mathematics Studies, 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973.

[24]

Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.

[25]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650. 

[26]

F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble), 23 (1973), 163-195.  doi: 10.5802/aif.467.

[27]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100.

[28]

S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbb{C}, 0) \to (\mathbb{C}, 0)$, Funktsional. Anal. i Prilozhen., 15 (1981), 1-17.  doi: 10.1007/BF01082373.

[29]

W. Zhang and W. Zhang, On invariant manifolds and invariant foliations without a spectral gap, Adv. Math., 303 (2016), 549-610.  doi: 10.1016/j.aim.2016.08.027.

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