The period function for second-order quadratic odes is monotone

Qualitative Theory of Dynamical Systems

Volumn 4, Issue 2, 2004, Pages 329-352

  Gasull, Armengol ^a   Guillamon, Antoni ^b   Villadelprat, Jordi ^c  

^a UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^c UNIVERSITAT ROVIRA I VIRGILI   (Spain)

Author keywords

Period function; Second order quadratic odes

Indexed keywords

EID: 84896693429     PISSN: 15755460     EISSN: 16623592     Source Type: Journal    
DOI: 10.1007/BF02970864     Document Type: Article

References (18)

1. J. Chavarriga and M. Sabatini. A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999), 1-70.
2. L.A. Cherkas. Number of limit cycles of an autonomous second-order system. Differ. Eq. 5 (1976), 666-668.
3. C. Chicone. The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310-321.
4. C. Chicone. Geometric methods for two-point nonlinear boundary value problems, J. Differential Equations 72 (1988), 360-407.
5. C. Chicone. Review in MathSciNet, ref. 94h:58072.
6. C. Chicone and M. Jacobs. Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), 433-486.
7. W.A. Coppel and L. Gavrilov. The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993), 1357-1365.
8. E. Freire, A. Gasull and A. Guillamon. First derivative of the period function with applications, To appear in J. Differential Equations (2004).
9. W.S. Loud. Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations 3 (1964), 21-36.
10. V. Lunkevich and K. Sibirskii. Integrals of a general quadratic differential system in cases of a center, Differential Equations 18 (1982), 563-568.
11. I.I. Pleshkan. A new method of investigating the isochronicity of a system of two differential equations, J. Differential Equations, 5 (1969) 796-802.
12. C. Rousseau and B. Toni. Local bifurcations of critical periods in the reduced Kukles system, Canad. J. Math. 49 (1997), 338-358.
13. M. Sabatini. Characterizing isochronous centres by Lie brackets, Diff. Eq. Dyn. Sys., 5, 91-99 (1997).
14. D. Schlomiuk. Algebraic and geometric aspects of the theory of polynomial vector fields, in "Bifurcations and periodic orbits of vector fields" (Montreal, PQ, 1992), 429-467, Kluwer Acad. Publ., Dordrecht, 1993.
15. M. Urabe. Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961), 569-578.
16. M. Villarini. Regularity properties of the period function near a centre of a planar vector field, Nonlinear Analysis T.M.A. 19 (1992), 787-803.
17. E. P. Volokitin and V. V. Ivanov. Isochronicity and commutation of polynomial vector fields, Sibirsk. Mat. Zh. 40 (1999), 30-48. (translation in Siberian Math. J. 40 (1999), 23-38.)
18. J. Waldvogel. The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986), 178-184.