Monotonicity of the Period Function in Planar Separable Hamiltonian Systems: An Extension of Chicone’s Criterion

• F. J. S. Nascimento ^[1]
  1. [1] Universidade Federal do Vale do São Francisco

     Universidade Federal do Vale do São Francisco

     Brasil

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
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• Resumen

  • We investigate the monotonicity of the period function associated with planar Hamiltonian systems of the form H(x, y) = F(x) + G(y), where G is an even function.

    We derive explicit sufficient conditions, expressed solely in terms of the functions F, G, and their derivatives, ensuring the monotonicity of the period function associated with a nondegenerate center. Our results extend Chicone’s classical criterion, originally established for potential systems, to a broader class of separable Hamiltonian systems. The effectiveness of the proposed criterion is illustrated through applications including polynomial, trigonometric, hyperbolic and relativistic Hamiltonian systems.

• Referencias bibliográficas
  • 1. Cima, A., Gasull, A., Mañosas, F.: Period function for a class of Hamiltonian systems. J. Diff. Equat. 168(1), 180–199 (2000)
  • 2. Hsu, S.-B.: A remark on the period of the periodic solution in the Lotka-Volterra system. J.Math. Anal. Appl. 95(2), 428–436 (1983)
  • 3. Rothe, F.: The periods of the Lotka-Volterra system. J. für die reine und angewandte Mathematik 355, 129–138 (1985)
  • 4. Schaaf, R.: A class of Hamiltonian systems with increasing periods. J. für die reine und angewandte Mathematik 363, 96–109 (1985)
  • 5. Chow, S.-N., Wang, D.: On the monotonicity of the period function of some second-order equations. Cˇ asopis pro peˇstování matematiky 111(1),...
  • 6. Chicone, C.: The monotonicity of the period function for planar Hamiltonian vector fields. J. Diff. Equat. 69(3), 310–321 (1987)
  • 7. Chicone, C., Dumortier, F.: A quadratic system with a nonmonotonic period function. Proceed. Am. Math. Soc. 102(3), 706–710 (1988)
  • 8. Chicone, C.: Geometric methods for two-point nonlinear boundary value problems. J. Diff. Equat. 72(2), 360–407 (1988)
  • 9. Rothe, F.: Remarks on periods of planar Hamiltonian systems. SIAM J. Math. Anal. 24(1), 129–154 (1993)
  • 10. Chicone, C., Dumortier, F.: Finiteness for critical periods of planar analytic vector fields. Nonlinear Anal. Theory Methods Appl. 20(4),...
  • 11. Coppel, W.A., Gavrilov, L.: The period function of a Hamiltonian quadratic system. Differ. Integral Equ. 6(6), 1357–1365 (1993)
  • 12. Gasull, A., Guillamón, A., Mañosas, V.: The period function for Hamiltonian systems with homogeneous nonlinearities. J. Differ. Equ. 139(2),...
  • 13. Cima, A., Mañosas, F., Villadelprat, J.: Isochronicity for several classes of Hamiltonian systems. J. Diff. Equat. 157(2), 373–413 (1999)
  • 14. Freire, E., Gasull, A., Guillamón, A.: First derivative of the period function with applications. J. Diff. Equat. 204(1), 139–162 (2004)
  • 15. Sabatini, M.: On the period function of x + f (x)x2 + g(x) = 0. J. Diff. Equat. 196(1), 151–168 (2004)
  • 16. Gasull, A., Guillamón, A., Villadelprat, J.: The period function for second-order quadratic ODEs is monotone. Qualitative Theory of Dynamical...
  • 17. Schaaf, R.: Global Solution Branches of Two-Point Boundary Value Problems. Lecture Notes in Mathematics, vol. 1458. Springer, Berlin (1990)
  • 18. Chow, S.-N., Sanders, J.A.: On the number of critical points of the period. J. Diff. Equat. 64(1), 51–66 (1986)
  • 19. Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312(2), 433–486 (1989)
  • 20. Chicone, C.: Bifurcations of nonlinear oscillations and frequency entrainment near resonance. SIAM J. Math. Anal. 23(6), 1577–1608 (1992)
  • 21. Gasull, A., Yu, J.: On the critical periods of perturbed isochronous centers. J. Diff. Equat. 244(3), 696–715 (2008)
  • 22. Urabe, M.: Potential forces which yield periodic motions of a fixed period. J. Math. Mech. 10(4), 569–578 (1961)
  • 23. Urabe, M.: Relations between periods and amplitudes of periodic solutions of ¨x+g(x) = 0. Funkcialaj Ekvacioj 6, 63–88 (1964)
  • 24. Rocha, C.: Realization of period maps of planar Hamiltonian systems. J. Dyn. Diff. Equat. 19(3), 571–591 (2007)
  • 25. Ragazzo, C.: Scalar autonomous second order ordinary differential equations. Qualitative Theory Dynamical Syst. 11(2), 277–415 (2012)
  • 26. Grotta-Ragazzo, C., Nascimento, F.J.S.: Global normalizations for centers of planar vector fields. J. Diff. Equat. 415, 701–721 (2025)
  • 27. Nascimento, F.J.S.: Mechanical normal forms for analytic centers. Qualitative Theory of Dynamical Syst. 25(2), 48 (2026)
  • 28. Sabatini, M.: On the period function of planar systems with unknown normalizers. Proceed. Am. Math. Soc. 134(2), 531–539 (2006)
  • 29. Villadelprat, J., Zhang, X.: The period function of Hamiltonian systems with separable variables. J. Dyn. Diff. Equat. 32(2), 741–767...
  • 30. Whitney, H.: Differentiable even functions. Duke Math. J. 10(1), 159–160 (1943)
  • 31. Babusci, D., Dattoli, G., Quattromini, M., Sabia, E.: Relativistic harmonic oscillator, the associated equations of motion, and algebraic...