# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021053

## Differentiable invariant manifolds of nilpotent parabolic points

 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  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.

Citation: Clara Cufí-Cabré, Ernest Fontich. Differentiable invariant manifolds of nilpotent parabolic points. Discrete & Continuous Dynamical Systems, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.  Google Scholar [25] K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650.   Google Scholar [26] F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble), 23 (1973), 163-195.  doi: 10.5802/aif.467.  Google Scholar [27] F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971.  Google Scholar [25] K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650.   Google Scholar [26] F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble), 23 (1973), 163-195.  doi: 10.5802/aif.467.  Google Scholar [27] F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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