Differentiable invariant manifolds of nilpotent parabolic points

Clara Cufí Cabré; Ernest Fontich

doi:10.3934/dcds.2021053

Differentiable invariant manifolds of nilpotent parabolic points

Clara Cufí Cabré, Ernest Fontich

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Resum

We consider a map F of class Cr 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 Cr 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.

Idioma original	English
Pàgines (de-a)	4667-4704
Nombre de pàgines	38
Revista	Discrete and Continuous Dynamical Systems- Series A
Volum	41
Número	10
DOIs	https://doi.org/10.3934/dcds.2021053
Estat de la publicació	Acceptat en premsa - 2021

Accés al document

10.3934/dcds.2021053

https://ddd.uab.cat/record/257123

STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY
Llibre Salo, J. (Investigador/a principal), Torregrosa i Arús, J. (Investigador/a Principal 2), Cufi Cabre, C. (Col.laborador/a), Carvalho, Y. R. (Col.laborador/a), Cousillas, G. (Col.laborador/a), Diz Pita, É. (Col.laborador/a), Ferragut Amengual, A. M. (Col.laborador/a), GOUVEIA, L. F. D. S. (Col.laborador/a), Iván Sánchez Sánchez (Col.laborador/a), Jiménez Ruíz, J. J. (Col.laborador/a), RODERO, A. L. (Col.laborador/a), Ramírez, V. (Col.laborador/a), Artes Ferragud, J. C. (Investigador/a), Caubergh , M. M. (Investigador/a), Cima Mollet, A. M. (Investigador/a), Corbera Subirana, M. (Investigador/a), Cors Iglesias, J. M. (Investigador/a), Gasull Embid, A. (Investigador/a), Pantazi ., C. (Investigador/a), Soler Villanueva, J. (Investigador/a), Garrido Pelaez, J. M. (Col.laborador/a) & Valls Anglés, C. (Col.laborador/a)
1/06/20 → 29/02/24
Projecte: Projectes i Ajuts a la Recerca
Algunos aspectos de la dinámica local y global de los sistemas dinámicos contínuos y discretos: Integrabilidad, Periodicidad y Bifurcaciones
Llibre Salo, J. (Investigador/a principal), Torregrosa i Arús, J. (Investigador/a Principal 2), Buzzi, C. A. (Col.laborador/a), Candido , M. R. (Col.laborador/a), Carvalho, Y. R. (Col.laborador/a), Caubergh , M. M. (Col.laborador/a), Cornelio, J. L. (Col.laborador/a), Cousillas, G. (Col.laborador/a), Cufi Cabre, C. (Col.laborador/a), Diz Pita, É. (Col.laborador/a), Ferragut Amengual, A. M. (Col.laborador/a), Ferragut Amengual, A. (Col.laborador/a), Geyer, A. (Col.laborador/a), Giacomini, H. J. (Col.laborador/a), Gouveia, L. F. (Col.laborador/a), Jiménez Ruíz, J. J. (Col.laborador/a), Lembarki, F. (Col.laborador/a), Martínez, Y. P. (Col.laborador/a), Pérez González, S. (Col.laborador/a), Ramírez, V. (Col.laborador/a), Saleta Reig, O. (Col.laborador/a), Sanchez Sanchez, I. (Col.laborador/a), Santos, R. A. T. (Col.laborador/a), Schlomiuk, D. (Col.laborador/a), Valls Angles, C. (Col.laborador/a), Vulpe, N. (Col.laborador/a), Zhang, X. (Col.laborador/a), da Cruz, L. P. C. (Col.laborador/a), de Almedia Silva Menezes, L. (Col.laborador/a), Artes Ferragud, J. C. (Investigador/a), Caubergh , M. M. (Investigador/a), Cima Mollet, A. M. (Investigador/a), Corbera Subirana, M. (Investigador/a), Cors Iglesias, J. M. (Investigador/a), Gasull Embid, A. (Investigador/a), Pantazi, C. (Investigador/a) & Soler Villanueva, J. (Investigador/a)
Ministerio de Economía y Competitividad (MINECO)
30/12/16 → 31/12/20
Projecte: Projectes i Ajuts a la Recerca

Com citar-ho

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title = "Differentiable invariant manifolds of nilpotent parabolic points",

abstract = "We consider a map F of class Cr 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 Cr 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",

author = "{Cuf{\'i} Cabr{\'e}}, Clara and Ernest Fontich",

year = "2021",

doi = "10.3934/dcds.2021053",

language = "English",

volume = "41",

pages = "4667--4704",

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T1 - Differentiable invariant manifolds of nilpotent parabolic points

AU - Cufí Cabré, Clara

AU - Fontich, Ernest

PY - 2021

Y1 - 2021

N2 - We consider a map F of class Cr 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 Cr 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.

AB - We consider a map F of class Cr 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 Cr 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.

KW - Parabolic point

KW - Invariant manifold

KW - Parameterization method

U2 - 10.3934/dcds.2021053

DO - 10.3934/dcds.2021053

M3 - Article

SN - 1553-5231

VL - 41

SP - 4667

EP - 4704

JO - Discrete and Continuous Dynamical Systems- Series A

JF - Discrete and Continuous Dynamical Systems- Series A

IS - 10

ER -

Differentiable invariant manifolds of nilpotent parabolic points

Resum

Accés al document

Projectes

STUDY OF CONTINUOUS AND DISCRETE DYNAMIC SYSTEMS WITH EMPHASIS ON THEIR BIFURCATIONS, PERIODIC ORBITS AND INTEGRABILITY

Algunos aspectos de la dinámica local y global de los sistemas dinámicos contínuos y discretos: Integrabilidad, Periodicidad y Bifurcaciones

Com citar-ho