The period function for Hamiltonian systems with homogeneous nonlinearities. (English) Zbl 0891.34033

Authors’ abstract: The paper deals with Hamiltonian systems with homogeneous nonlinearities. We prove that such systems have no isochronous centers, that the period annulus of any of its centres is either bounded or the whole plane and that the period function associated to the origin has at most one critical point.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Andronov, A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G., Theory of Bifurcations of Dynamic Systems on a Plane (1967), Wiley: Wiley New York Toronto
[2] Artés, J. C.; Llibre, J., Quadratic Hamiltonian vector fields, J. Differential Equations, 107, 80-95 (1994) · Zbl 0791.34048
[3] Carbonell, M.; Llibre, J., Limit cycles of a class of polynomial systems, Proc. Roy. Soc. Edinburgh A, 109, 187-199 (1988) · Zbl 0664.34039
[4] Cherkas, L. A., Number of limit cycles of an aotonomous second-order system, Differential Equations, 5, 666-668 (1976) · Zbl 0365.34039
[5] Chicone, C.; Jacobs, M., Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312, 433-486 (1989) · Zbl 0678.58027
[6] Christopher, C. J.; Devlin, J., Isochronous centres in planar polynomial systems, SIAM J. Math. Anal., 28, 162-177 (1997) · Zbl 0881.34057
[7] Cima, A.; Gasull, A.; Mañosas, F., On polynomial Hamiltonian planar vector fields, J. Differential Equations, 106, 367-383 (1993) · Zbl 0792.34026
[8] Coll, B.; Gasull, A.; Prohens, R., Differential equations defined by the sum of two quasi-homogeneous vector fields, Can. J. Math., 49, 212-231 (1997) · Zbl 0990.34030
[9] Collins, C. B., The period function of some polynomial systems of arbitrary degree, Differential Integral Equations, 9, 251-266 (1996) · Zbl 0849.34025
[10] Conti, R., Centers of quadratic systems, Ricerche di Mat. Suppl., 36, 117-126 (1987) · Zbl 0685.34024
[11] Conti, R., Uniformly isochronous centres of polynomial systems in \(R^2\), Lecture Notes in Pure and Applied Math (1994), Dekker: Dekker New York, p. 21-31 · Zbl 0795.34021
[12] Coppel, W. A.; Gavrilov, L., The period of a Hamiltonian quadratic system, Differential Integral Equations, 6, 799-841 (1993) · Zbl 0780.34023
[13] Devlin, J., Coexisting isochronous and nonisochronous centres, Bull. London Math. Soc., 28, 495-500 (1996) · Zbl 0853.34032
[15] Gasull, A.; Guillamon, A.; Mañosa, V., An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. (1997) · Zbl 0891.34033
[17] Loud, W. S., Behavior of the period of solutions of certain plane autonomous systems near centres, Contrib. Differential Equations, 3, 21-36 (1964) · Zbl 0139.04301
[19] Pleshkan, I. I., A new method of investigating the isochronicity of a system of two differential equations, Differential Equations, 5, 796-802 (1969) · Zbl 0252.34034
[20] Schuman, B., Sur la forme normale de Birkhoff et les centres isochrones, C. R. Acad. Sci. Paris I, 322, 21-24 (1996) · Zbl 0837.34041
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