The easiest polynomial differential systems in R^3 having an invariant cylinder

• Jaume Llibre ^[1] ; Leonardo Serantola ^[2]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Departamento de Matemática, Ibilce–UNESP 15054-000 São José do Rio Preto, Brasil
• Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 40, Nº 1, 2025, págs. 121-141
• Idioma: inglés
• DOI: 10.17398/2605-5686.40.1.121
• Enlaces
  • Texto completo (pdf)
• Resumen

  • This paper answers the following two questions: What are the easiest polynomial differential systems in R3 having an invariant hyperbolic, parabolic or elliptic cylinder?, and for such polynomial differential systems what are their phase portraits on such invariant cylinders?

• Referencias bibliográficas
  • J. Benny, S. Jana, S. Sarkar, Invariant circles and phase portraits of cubic vector fields on the sphere, Qual. Theory Dyn. Syst. 23 (3) (2024),...
  • M.I. Camacho, Geometric properties of homogeneous vector fields of degree two in R3 , Trans. Amer. Math. Soc. 268 (1981), 79 – 101.
  • W.A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293 – 304.
  • G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges), Bull. Sci. Math. Astro.,...
  • F. Dumortier, J. Llibre, J.C. Artés, “ Qualitative theory of planar differential systems ”, Springer-Verlag, Berlin, 2006.
  • M.W. Hirsch, C.C. Pugh, M. Shub, “ Invariant manifolds ”, Lecture Notes in Math., Vol. 583, Springer-Verlag, Berlin-New York, 1977.
  • J.P. Jouanolou, “ Equations de Pfaff algébriques ”, Lectures Notes in Math., 708, Springer, Berlin, 1979.
  • J. Llibre, Integrability of polynomial differential systems, in “ Handbook of differential equations ” (edited by A. Cañada, P. Drabek and...
  • J. Llibre, R. Ramı́rez, V. Ramı́rez, N. Sadovskaia,, The 16th Hilbert problem restricted to circular algebraic limit cycles, J. Differential...
  • J.E. Massetti, Attractive invariant circles à la Chenciner, Regul. Chaotic Dyn. 28 (4-5) (2023), 447 – 467.
  • M. Messias, M.R. Cândido, Analytical results on the existence of periodic orbits and canard-type invariant torus in a simple dissipative oscillator,...
  • C. Pessoa, J. Llibre, Homogeneous polynomial vector fields of degree 2 on the 2-dimensional sphere, Extracta Math. 21 (2006), 167 – 190.
  • C. Pessoa, J. Llibre, Phase portraits for quadratic homogeneous polynomial vector fields on S2 , Rend. Circ. Mat. Palermo (2) 58 (2009), 361...
  • J. Sotomayor, “Lições de equações diferenciais ordinárias”, Projeto Euclides, 11, Instituto de Matemática Pura e Aplicada, Rio de Janeiro,...
  • X. Yang, J. Xu, S. Jiang, Persistence of degenerate lower dimensional invariant tori with prescribed frequencies in reversible systems, J....
  • J. Zhong, Y. Liang, Invariant tori and heteroclinic invariant ellipsoids of a generalized Hopf-Langford system, Internat. J. Bifur. Chaos...