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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.22 no.3 Antofagasta Dec. 2003

http://dx.doi.org/10.4067/S0716-09172003000300001 

Proyecciones
Vol. 22, N o 3, pp. 161-180, December 2003.
Universidad Católica del Norte
Antofagasta - Chile

AN EXTENSION OF THE POINCARÉ
COMPACTIFICATION AND A GEOMETRIC
INTERPRETATION

CLAUDIO VIDAL
Universidade Federal de Pernambuco - Brasil.
PEDRO GÓMEZ
Universidad Federal de Paraiba - Brasil

Abstract

Our purpose in this paper is to understand the geometry of the
Poincaré compactifcation and to apply this technique to prove that
there exists a Poincaré compactifcation of vector felds defned by
rational functions and of vector feld that are the quotient of some
power of polynomial. We will give also a global expressions for the
Poincaré vector feld associated. Furthermore, we summarize these
results proving that there exist a Poincaré vector feld for any vector
feld whose rate of growth at infnity of each component is not bigger
than a polynomial growth.

Mathematics Subject Classifcation : 34A34, 34C05, 34A99.

Key Words : Poincaré compactifcation, rational vector feld,
equilibrium solutions.

References

[1] Chazy, J. : Sur l'allure du mouvement dans le problµeme des trois corps quand le temps croit indµefniment. Ann. Ecole Norm. Sup. 39 (3), pp.29-130, (1922).        [ Links ]

[2] Cima, A., and Llibre, J. :. Bounded polynomial vector felds. Trans.Amer. Math. Soc. 318, pp. 557-579, (1990).        [ Links ]

[3] Cors, J., and Llibre, J. : The global flow of the parabolic restricted three body problem. Phd thesis, Universitat Autµonoma de Barcelona, (1994).        [ Links ]

[4] Cors, J., and Llibre, J. : Qualitative study of the parabolic collision restricted three body problem. Contemporary Mathematics. 198, pp. 1-19, (1996).        [ Links ]

[5] Delgado, J. ; Lacomba, E. A.; Llibre, J.; Pérez, E. : Poincaré compactifcation of the Kepler and the collinear three body problem. Seminaron Dynamical Systems (St. Petersburg, 1991), pp. 117{128, Progr. Nonlinear Diferential Equations Appl., 12, BirkhÄ auser, Basel, (1994).        [ Links ]

[6] Delgado, J. ; Lacomba, E. A.; Llibre, J.; Pérez, E. : Poincaré compactifcation of the collinear three body problem. Hamiltonian systems and celestial mechanics (Guanajuato, 1991), 85{100, Adv. Ser. Nonlinear Dynam., 4, World Sci. Publishing, River Edge, NJ, (1993).        [ Links ]

[7] Delgado, J. ; Lacomba, E. A. ; Llibre, J. ; Pérez, E. : Poincaré compactifcation of Hamiltonian polynomial vector fields. Hamiltonian dynamical systems (Cincinnati, OH, 1992), 99{114, IMA Vol. Math. Appl., 63, Springer, New York, (1995).        [ Links ]

[8] González, E.: Generic properties of polynomial vector felds at infnity. Trans. Amer. Math. Soc. 143, pp. 201-222, (1969).        [ Links ]

[9] Hegie, D. : A global regularization of the gravitational n-body problem. Cel. Mech. 10, pp. 217-241, (1974).        [ Links ]

[10] Poincaré, H. : Mémoire sur les courbes défnies par une equationdiférentielle. J. Mathématiques (3), 7, pp. 375-422, (1881).        [ Links ]

[11] Wang, Q. : Qualitative study of n-body problem: Untized momentum transformation and its application restricted isoceles three-body problem with positive energy, Space Dynamics and Celestial Mechanics, K.B. Bhantnagar, ed., pp. 61-69, (1986).        [ Links ]

Received : November 2002.

Claudio Vidal
Departamento de Matemática
Universidade Federal de Pernambuco
Av. Prof. Luiz Freire, s/n
Cidade Universitária
Recife-PE
Brasil
e-mail : claudio@dmat.ufpe.br

and

Pedro Gómez
Departamento de Matemática
Universidade Federal da Paraiba
Cidade Universitária, Jo~ao Pessoa-PB
Brazil
e-mail : venegas@mat.ufpb.br

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