An extension of the poincaré compactification and a geometric interpretation

Authors

  • Claudio Vidal Universidade Federal de Pernambuco.
  • Pedro Gómez Universidad Federal de Paraiba.

DOI:

https://doi.org/10.4067/S0716-09172003000300001

Keywords:

Poincaré compactification, Rational vector field, Equilibrium solutions.

Abstract

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

Author Biographies

Claudio Vidal, Universidade Federal de Pernambuco.

Departamento de Matemática.

Pedro Gómez, Universidad Federal de Paraiba.

Departamento de Matemática.

References

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

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

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

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

[5] Delgado, J. ; Lacomba, E. A.; Llibre, J.; Pérez, E. : Poincaré compactification of the Kepler and the collinear three body problem. Seminar on Dynamical Systems (St. Petersburg, 1991), pp. 117–128, Progr. Nonlinear Differential Equations Appl., 12, Birkhäuser, Basel, (1994).

[6] Delgado, J. ; Lacomba, E. A.; Llibre, J.; Pérez, E. : Poincaré compactification 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).

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

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

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

[10] Poincaré, H. : Mémoire sur les courbes d´efinies par une equation diff´erentielle. J. Mathématiques (3), 7, pp. 375-422, (1881).

[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).

Published

2017-04-24

How to Cite

[1]
C. Vidal and P. Gómez, “An extension of the poincaré compactification and a geometric interpretation”, Proyecciones (Antofagasta, On line), vol. 22, no. 3, pp. 161-180, Apr. 2017.

Issue

Section

Artículos

Similar Articles

You may also start an advanced similarity search for this article.