The period of the limit cycle bifurcating from a persistent polycycle

• Marín, David ^[1] ; Queiroz, Lucas ^[2] ; Villadelprat, Jordi ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Universidade Estadual Paulista

     Universidade Estadual Paulista

     Brasil

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 69, Nº 2, 2025, págs. 299-318
• Idioma: inglés
• DOI: 10.5565/publmat6922502
• Enlaces
  • Texto completo
• Resumen

  • We consider smooth families of planar polynomial vector fields {Xµ}µ∈Λ, where Λ is an open subset of RN , for which there is a hyperbolic polycycle Γ that is persistent (i. e. , such that none of the separatrix connections is broken along the family). It is well known that in this case the cyclicity of Γ at µ0 is zero unless its graphic number r(µ0) is equal to one. It is also well known that if r(µ0) = 1 (and some generic conditions on the return map are verified), then the cyclicity of Γ at µ0 is one, i. e. , exactly one limit cycle bifurcates from Γ. In this paper we prove that this limit cycle approaches Γ exponentially fast and that its period goes to infinity as 1/|r(µ) − 1| when µ → µ0. Moreover, we prove that if those generic conditions are not satisfied, although the cyclicity may be exactly 1, the behavior of the period of the limit cycle is not determined.

• Referencias bibliográficas
  • A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maıer, Theory of Bifurcations of Dynamic Systems on a Plane, Translated from the...
  • H. Broer, R. Roussarie, and C. Simo, Invariant circles in the Bogdanov–Takens bifurcation for diffeomorphisms, Ergodic Theory Dynam. Systems...
  • F. Dumortier, J. Llibre, and J. C. Artes , Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006....
  • A. Gasull, V. Manosa, and F. Manosas , Stability of certain planar unbounded polycycles, J. Math. Anal. Appl. 269(1) (2002), 332–351. DOI:...
  • A. Gasull, V. Manosa, and J. Villadelprat, On the period of the limit cycles appearing in one-parameter bifurcations, J. Differential Equations...
  • A. Guillamon, R. Prohens, A. E. Teruel, and C. Vich, Estimation of synaptic conductance in the spiking regime for the McKean neuron model,...
  • J. Hale and H. Koc¸ak, Dynamics and Bifurcations, Texts Appl. Math. 3, Springer-Verlag, New York, 1991. DOI: 10.1007/978-1-4612-4426-4
  • L. Hurwicz and M. K. Richter, Implicit functions and diffeomorphisms without C1, in: Advances in Mathematical Economics, Vol. 5, Springer-Verlag,...
  • A. Jacquemard, F. Z. Khechichine-Mourtada, and A. Mourtada, Algorithmes formels appliques à l’etude de la cyclicité d’un polycycle algebrique...
  • D. Marin and J. Villadelprat, Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: local setting, J. Differential...
  • D. Marin and J. Villadelprat, Asymptotic expansion of the Dulac map and time for unfoldings of hyperbolic saddles: general setting, J. Differential...
  • D. Marin and J. Villadelprat, On the cyclicity of Kolmogorov polycycles, Electron. J. Qual. Theory Differ. Equ. 2022, Paper no. 35 (2022),...
  • A. Mourtada, Cyclicit´e finie des polycycles hyperboliques de champs de vecteurs du plan: mise sous forme normale, in: Bifurcations of Planar...
  • A. Mourtada, Bifurcation de cycles limites au voisinage de polycycles hyperboliques et generiques a trois sommets, Ann. Fac. Sci. Toulouse...
  • R. Roussarie, Smoothness property for bifurcation diagrams, Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996), Publ. Mat....
  • J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Etudes Sci. Publ. Math. 43 (1974),...
  • J. Sotomayor, Curvas definidas por equacoes diferenciais no plano, 13o Coloquio Brasileiro de Matematica, Instituto de Matematica Pura e Aplicada,...