Monotonicity and Convexity of period in a Nonhomogeneous Cubic System

• Zhirong He ^[1] ; Weinian Zhang ^[1] ; Xiaoxiao Zheng ^[1]
  1. [1] Sichuan University

     Sichuan University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper is based on the work [JDE, 2024, 396, 147-171], which gives that the general cubic polynomial differential system has an isochronous global center at the origin if and only if the system can be reduced to the form x˙ = −y+ax2, y˙ = x−2axy+2a2x3.

    In this paper we discuss the monotonicity and the convexity of period function for the system x˙ = −y+ax2, y˙ = x−2axy+bx3, a more general form with a free coefficient b of the cubic term. We prove that the period function is strictly decreasing for b > 2a2, isochronous for b = 2a2 and strictly increasing for b < 2a2. Furthermore, we prove that the period function is convex for b = 2a2.

• Referencias bibliográficas
  • 1. Bonorino, L.P., Brietzke, E.H.M., Lukaszczyk, J.P., Taschetto, C.A.: Properties of the period function for some Hamiltonian systems and...
  • 2. Chicone, C.: The monotonicity of the period function for planar Hamiltonian vector fields. J. Differential Eq. 69, 310–321 (1987)
  • 3. Chicone, C.: Geometric methods for nonlinear two point boundary value problems. J. Differential Eq. 72, 360–407 (1988)
  • 4. Chicone, C.: Bifurcations of nonlinear oscillations and frequency entrainment near resonance. SIAM J. Math. Anal. 23, 1577–1608 (1992)
  • 5. Chicone, C., Dumortier, F.: Finiteness for critical periods of planar analytic vector fields. Nonlinear Anal. 20, 315–335 (1993)
  • 6. Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)
  • 7. Chow, S.-N., Sanders, J.: On the number of critical points of periods. J. Differential Eq. 64, 51–66 (1986)
  • 8. Coppel,W.A., Gavrilov, L.: The period function of a Hamiltonian quadratic system. Differential Integral Eq. 6, 1357–1365 (1993)
  • 9. Gasull, A., Guillamon, A., Mañosa, V., Mañosas, F.: The period function for Hamiltonian systems with homogeneous nonlinearities. J. Differential...
  • 10. Gasull, A., Guillamon, A., Villadelprat, J.: The period function for second-order quadratic ODEs is monotone. Qual. Theory Dyn. Syst....
  • 11. Gavrilov, L.: Remark on the number of critical points of the period. J. Differential Eq. 101, 58–65 (1993)
  • 12. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations. Dynamical Systems and Bifurcations of Vector Fields, Springer, New York (1983)
  • 13. Hale, J. K.: Ordinary Differential Equations (2nd Edition), Robert T. Krieger Publ., (1980)
  • 14. Hsu, S.B.: A remark on the period of the periodic solution in the Lotka-Volterra system. J. Math. Anal. Appl. 95, 428–436 (1983)
  • 15. Iliev, I.D.: Perturbations of quadratic centers. Bull. Sci. Math. 122, 107–161 (1998)
  • 16. Li, C., Lu, K.: The period function of hyperelliptic Hamiltonians of degree 5 with real critical points. Nonlinearity 21, 465–483 (2008)
  • 17. Li, Y., Shi, S., Zhang, J.: Isochronous global center of linear plus homogeneous polynomial systems and cubic systems. J. Differential...
  • 18. Long, T., Liu, C., Wang, S.: The period function of quadratic generalized Lotka-Volterra systems without complex invariant lines. J. Differential...
  • 19. Roussarie, R.: Bifurcation of Planar Vector Fields and Hilbert Sixteen Problem. Birkhäuser Verlag, Basel, Prog. Math. (1998)
  • 20. Sabatini, M.: Period function’s convexity for Hamiltonian centers with separable variables. Ann. Polon. Math. 85, 153–163 (2005)
  • 21. Schaaf, R.: A class of Hamiltonian system with increasing periods. J. Reine Angew. Math. 363, 96–109 (1985)
  • 22. Schaaf, R.: Global solution branches of two point boundary problems. Springer, Berlin (1990)
  • 23. Schlomiuk, D.: Algebrais particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc. 338, 799–841 (1993)
  • 24. Smoller, J., Wasserman, A.: Global bifurcation of steady-state solutions. J. Differential Eq. 39, 269–291 (1981)
  • 25. Villadelprat, J., Zhang, X.: The period function of Hamiltonian systems with separable variables. J. Dyn. Differential Eq. 32, 741–767...
  • 26. Waldvogel, J.: The period in the Lotka-Volterra system is monotonic. J. Math. Anal. Appl. 114, 178–184 (1986)
  • 27. Wu, K.: On the monotonicity of the period function of reversible centers. J. Appl. Anal. Comput. 2, 205–212 (2012)
  • 28. Zhao, Y.: The monotonicity of period function for codimension four quadratic system Q4. J. Differential Eq. 185, 370–387 (2002)
  • 29. Zoladek, H.: On a certain generalization of Bautin’s theorem. Nonlinearity 7, 273–279 (1994)
  • 30. Zoladek, H.: Quadratic systems with centers and their perturbations. J. Differential Eq. 109, 223–273 (1994)