Abelian Integrals and Non-generic Turning Points

• Renato Huzak ^[1] ; David Rojas ^[2]
  1. [1] University of Hasselt

     University of Hasselt

     Arrondissement Hasselt, Bélgica

     
  2. [2] Universitat de Girona

     Universitat de Girona

     Gerona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 3, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper we initiate the study of the Chebyshev property of Abelian integrals generated by a non-generic turning point in planar slow-fast systems. Such Abelian integrals generalize the Abelian integrals produced by a slow-fast Hopf point (or generic turning point), introduced in Dumortier et al. (Discrete Contin Dyn Syst Ser S 2(4):723–781, 2009), and play an important role in studying the number of limit cycles born from the non-generic turning point.

• Referencias bibliográficas
  • 1. Chow, S., Li, C., Wang, D.: Uniqueness of periodic orbits of some vector fields with codimension two singularities. J. Differ. Equ. 77(2),...
  • 2. De Maesschalck, P., Dumortier, F.: Time analysis and entry-exit relation near planar turning points. J. Differ. Equ. 215(2), 225–267 (2005)
  • 3. De Maesschalck, P., Dumortier, F.: Slow-fast Bogdanov–Takens bifurcations. J. Differ. Equ. 250(2), 1000–1025 (2011)
  • 4. Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Amer. Math. Soc., 121(577):x+100, (1996). With an appendix by...
  • 5. Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)
  • 6. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
  • 7. Figueras, J.-L., Tucker, W., Villadelprat, J.: Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals....
  • 8. Huzak, R.: Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations. Discrete Contin. Dyn. Syst. 36(1), 171–215...
  • 9. Huzak, R.: Canard explosion near non-Liénard type slow-fast Hopf point. J. Dyn. Differ. Equ. 31(2), 683–709 (2019)
  • 10. Huzak, R., Rojas, D.: Period function of planar turning points. Electron. J. Qual. Theory Differ. Equ., pages Paper No. 16, 21 (2021)
  • 11. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
  • 12. Li, C., Liu, C.: A proof of a Dumortier-Roussarie’s conjecture. Discrete Contin. Dyn. Syst. Ser. S. (2022). https://doi.org/10.3934/dcdss.2022095
  • 13. Liu, C., Li, C., Llibre, J.: The cyclicity of the period annulus of a reversible quadratic system. Proc. R. Soc. Edinb. Sect. A, 1–10...
  • 14. Marín, D., Villadelprat, J.: On the chebyshev property of certain abelian integrals near a polycycle. Qual. Theory Dyn. Syst. 17, 261–270...
  • 15. Xenophontos, C.: A formula for the nth derivative of the quotient of two functions. arXiv:2110.09292 [math.GM], pp. 1–3, (2021)