Global Asymptotic Stability in Cubic Systems with a Nilpotent Point at the Origin and with An Invariant Straight Line

• Érika Diz-Pita ^[1] ; M. Victoria Otero-Espinar ^[1] ; Claudia Valls ^[2]
  1. [1] Universidade de Santiago de Compostela

     Universidade de Santiago de Compostela

     Santiago de Compostela, España

     
  2. [2] Universidade de Lisboa

     Universidade de Lisboa

     Socorro, Portugal

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper we are concerned with the characterization of globally asymptotic stable planar polynomial differential systems. Such systems are very important in the qualitative theory of differential equations because all their trajectories are defined for all positive time and the ω-limit of any point is a unique equilibrium point. The characterization of quadratic planar polynomial differential systems that are globally asymptotic stable is known. Here, we characterize all planar cubic polynomial differential systems with a nilpotent point at the origin, without quadratic terms and with an invariant straight line through the origin, that are globally asymptotically stable.

• Referencias bibliográficas
  • 1. Dickson, R.J., Perko, L.M.: Bounded quadratic systems in the plane. J. Differential Equations 7, 251– 273 (1970)
  • 2. Diz-Pita, E., Llibre, J., Otero-Espinar, M.V.: Global phase portraits of a predator–prey system Electron. J. Qual. Theory Differ. Equ 16,...
  • 3. Dumortier, F., Herssens, C., Perko, L.: Local bifurcations and a survey of bounded quadratic systems. J. Differential Equations 165, 430–467...
  • 4. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative theory of planar differential systems. UniversiText, Springer-Verlag, New York (2006)
  • 5. Gasull, A., Llibre, J., Sotomayor, J.: Global asymptotic stability of differential equations in the plane. J. Differential Equations 91(2),...
  • 6. Kuang, Y., Beretta, E.: Global qualitative analysis of a ratio-dependent predator–prey system. J. Math. Biol. 36, 389–406 (1998)
  • 7. Li, T., Guo, Y.: Optimal control strategy of an online game addiction model with incomplete recovery. J. Optim. Theory Appl. 195(3), 780–807...
  • 8. Llibre, J., Valls, C.: Global asymptotic stability, preprint, (2025)
  • 9. Llibre, J. Zhang, X.: The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems Proceedings of...
  • 10. Ma, Z.P., Han, J.Z.: Global asymptotical stability and Hopf bifurcation for a three-species LotkaVolterra food web model. Math. Methods...
  • 11. Markus, L., Yamabe, H.: Global stability criteria for differential systems. Osaka J. Math. 12, 305–317 (1960)
  • 12. Perko, L.: Differential Equations and Dynamical Systems, Texts in Applied Mathematics 7, 3rd edn. Springer-Verlag, New York (2001)
  • 13. Rodrigues, H.M., Teixeira, M.A., Gameiro, M.: On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications...
  • 14. Xu, J., Geng, Y.: Global stability for a heroin epidemic model in a critical case. Appl. Math. Lett. 121, 107493 (2021). (6 pp)
  • 15. Yang, W., Li, X.: Global asymptotical stability for a diffusive predator-prey model with ratio-dependent Holling type III functional response....
  • 16. Yang, Y.R., Li, X.V., Mu, K.W.: Global stability of wavefronts for an asymmetric infectious disease system with nonlocality. J. Math....