Reversible Limit Cycles for Linear Plus Cubic Homogeneous Polynomial Differential System

• Luping Wang ^[1] ; Yulin Zhao ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 2, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In the present paper, we study the reversible limit cycles for the system x˙ = a1x + a2 y + a3x3 + a4x2 y + a5x y2 + a6 y3, y˙ = b1x + b2 y + b3x3 + b4x2 y + b5x y2 + b6 y3, which is called a linear plus cubic homogeneous polynomial differential system. It is proved by Zhou that the reversible limit cycles of a polynomial differential system are algebraic. We show that the degree of the reversible limit cycle of the above system is at most 6. Moreover, this system has no reversible limit cycles of degree 3 with respect to straight lines passing through the origin in the phase plane. We present a system showing that the above system can have two reversible limit cycles of degree 4.

• Referencias bibliográficas
  • 1. Alberich-Carramiñana, M., Ferragut, A., Llibre, J.: Quadratic planar differential systems with algebraic limit cycles via quadratic plane...
  • 2. Belevec, P.S., Valeeva, R.T.: A class of dynamic systems with an algebraic limit cycle. Kuibyshev. Gos. Ped. Inst. Nauchn. Trudy 232, 5–8...
  • 3. Chavarriga, J., Giacomini, H., Llibre, J.: Uniqueness of algebraic limit cycles for quadratic systems. J. Math. Anal. Appl. 261, 85–99...
  • 4. Chavarriga, J., Llibre, J., Moulin Ollagnier, J.: On a result of Darboux. LMS J. Comput. Math. 4, 197–210 (2001)
  • 5. Chavarriga, J., Llibre, J., Sorolla, J.: Algebraic limit cycles of degrre 4 for quadratic systems. J. Differ. Equ. 200, 206–244 (2004)
  • 6. Ch’in, Y.: On algebraic limit cycles of degree 2 of the differential equation dy/dx = 0≤i+ j≤2ai j xi y j / 0≤i+ j≤2bi j xi...
  • 7. Christopher, C., Llibre, J., Pantazi, C., Zhang, X.: Darboux integrability and invariant algebraic curves for planar polynomial systems....
  • 8. Christopher, C., Llibre, J., Swirszcz, G.: Invariant algebraic curves of large degree for quadratic systems. ´ J. Math. Anal. Appl. 303,...
  • 9. Evdokimenco, R.M.: Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves...
  • 10. Evdokimenco, R.M.: Behavior of integral curves of a dynamic system. Differ. Equ. 9, 1095–1103 (1974)
  • 11. Evdokimenco, R.M.: Investigation in the large of a dynamic system. Differ. Equ. 15, 215–221 (1979)
  • 12. Filiptsov, V.F.: Algebraic limit cycles. Differ. Equ. 9, 983–986 (1973)
  • 13. García-Saldaña, J.D., Gasull, A., Giacomini, H.: A new approach for the study of limit cycles. J. Differ. Equ. 269, 6269–6292 (2020)
  • 14. Giné, J., Llibre, J., Valls, C.: The cubic polynomial differential systems with two circles as algebraic limit cycles. Adv. Nonlinear...
  • 15. Jin, Y.: The qualitative analysis of cubic Kules system, Ph.D. Thesis, Sun Yat-sen University (2024)
  • 16. Liu, D.: The cubic systems with cubic curve solution y2 = ax3 + bx2 + cx + d. Northeast. Math. J. 5, 437–448 (1989)
  • 17. Llibre, J., Swirszcz, G.: Classification of quadratic systems admitting the existence of an algebraic ´ limit cycle. Bull. Sci. Math....
  • 18. Llibre, J., Zhao, Y.: Algebraic limit cycles in polynomial systems of differential equations. J. Phys. A 40, 14207–14222 (2007)
  • 19. Novikov, D., Yakovenko, S.: Redundant Picard-Fuchs system for abelian integrals. J. Differ. Equ. 177, 267–306 (2001)
  • 20. Wang, L., Tian, Y., Zhao, Y.: Algebraic limit cycle of degree 2 for linear type center plus cubic homogeneous polynomial differential...
  • 21. Yablonskii, A.I.: Limit cycles of a certain differential equations. Differ. Equ. 2, 335–344 (1966). ((in Russian))
  • 22. Zhou, H.: The reversible limit cycles of planar polynomial systems, M.S. Thesis, Sun Yat-sen University (2023)