Poincaré Compactification for Non-polynomial Vector Fields

• Bravo, José Luis ^[1] ; Fernández, Manuel ^[1] ; Teruel, Antonio E ^[2]
  1. [1] Universidad de Extremadura

     Universidad de Extremadura

     Badajoz, España

     
  2. [2] Universitat de les Illes Balears

     Universitat de les Illes Balears

     Palma de Mallorca, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00386-1
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• Dialnet Métricas: 1 Cita
• Resumen

  • In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

• Referencias bibliográficas
  • 1. Andronov, A.A., Leontovitch, E.A., Gordon y, I.I., Maier, A.G.: Qualitative Theory of Second-Order Dynamic Systems. Wiley, New York (1973)
  • 2. Andronov, A.A., Vitt, A., Khaikin, S.: Theory of Oscillators. Pergamon Press, Oxford (1966)
  • 3. Azamov, A., Suvanov, S., Tilavov, A.: Studying of behavior at infinity of vector fields on Poincaré sphere: revisited. Qual. Theory. Dyn....
  • 4. Coll, B., Gasull, A., Prohens, R.: Differential equations defined by the sum of two quasi-homogeneous vector fields. Can. J. Math. 49(2),...
  • 5. Delgado, J., Lacomba, E.A., Llibre, J., Pérez, E.: Poincaré compactification of Hamiltonian polynomial vector fields. In: Dumas, H.S.,...
  • 6. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical...
  • 7. Elias, U., Gingold, H.: Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification....
  • 8. García, A., Pérez-Chavela, E., Susin, A.: A generalization of the Poincaré compactification. Arch. Ration. Mech. Anal. 179, 285–302 (2006)
  • 9. Li, S., Llibre, J.: Phase portraits of continuous piecewise linear Lienard differential systems with three zones. Chaos Solitons Fractals...
  • 10. Llibre, J., Teruel, A.E.: Introduction to the Qualitative Theory of Differential Systems: Planar, Symmetric and Continuous Piecewise Linear...
  • 11. Martínez-Jeraldo, N., Aguirre, P.: Allee effect acting on the prey species in a Leslie–Gower predation model. Nonlinear Anal. Real World...
  • 12. Matsue, K.: On blow-up solutions of differential equations with Poincaré-type compactifications. SIAM J. Appl. Dyn. Syst. 17–3, 2249–2288...
  • 13. Pessoa, C., Sotomayor, J.: Stable piecewise polynomial vector fields. Electron. J. Differ. Equ. 165, 1–15 (2012)
  • 14. Pessoa, C., Tonon, D.J.: Piecewise smooth vector fields in R3 at infinity. J. Math. Anal. Appl. 427–2, 841–855 (2015)
  • 15. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. Oeuvres T.1. J. Math. Pures Appl. 7, 375–422 (1881)
  • 16. Priyadarshi, A., Banerjee, S., Gakkhar, S.: Geometry of the Poincaré compactification of a fourdimensional food-web system. Appl. Math....
  • 17. Vidal, C., Gómez, P.: An extension of the Poincaré compactification and a geometric interpretation. Proyecciones 22–3, 161–180 (2003)...