Quadratic Isochronous centers commute
Sabatini, M.
Applicationes Mathematicae, Tome 26 (1999), p. 357-362 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

We prove that every quadratic plane differential system having an isochronous center commutes with a polynomial differential system.

Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:219245 
@article{bwmeta1.element.bwnjournal-article-zmv26i3p357bwm,
     author = {M. Sabatini},
     title = {Quadratic Isochronous centers commute},
     journal = {Applicationes Mathematicae},
     volume = {26},
     year = {1999},
     pages = {357-362},
     zbl = {1005.34022},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p357bwm}
}

Sabatini, M. Quadratic Isochronous centers commute. Applicationes Mathematicae, Tome 26 (1999) pp. 357-362. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p357bwm/

[000] [AFG] A. Algaba, E. Freire and E. Gamero, Isochronicity via normal form, preprint. | Zbl 1021.34022

[001] [C] R. Conti, Centers of polynomial systems in R2, preprint, Firenze, 1990.

[002] [CDL] C. J. Christopher, J. Devlin and N. G. Lloyd, On the classification of Liénard systems with amplitude-independent periods, preprint.

[003] [CGG1] J. Chavarriga, J. Giné and I. García, Isochronous centers of cubic systems with degenerate infinity, Differential Equations Dynam. Systems 7 (1999), to appear. | Zbl 0982.34025

[004] [CGG2] J. Chavarriga, J. Giné and I. García, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomials, Bull. Sci. Math. 123 (1999), 77-96. | Zbl 0921.34032

[005] [CGG3] J. Chavarriga, J. Giné and I. García, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials, preprint, Univ. de Lleida. | Zbl 0978.34028

[006] [CJ] C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), 433-486. | Zbl 0678.58027

[007] [D] J. Devlin, Coexisting isochronous and nonisochronous centers, Bull. Lond. Math. Soc. 28 (1996), 495-500. | Zbl 0853.34032

[008] [GGM1] A. Gasull, A. Guillamon and V. Mañosa, An explicit expression of the first Lyapunov and period constants with applications, J. Math. Anal. Appl. 211 (1997), 190-212. | Zbl 0882.34040

[009] [GGM2] A. Gasull, A. Guillamon and V. Mañosa, Centre and isochronicity conditions for systems with homogeneous nonlinearities, in: Proc. 2nd Catalan Days on Appl. Math., Collect. Études, Presses Univ. Perpignan, Perpignan, 1995, 105-116. | Zbl 0909.34030

[010] [L] W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations 3 (1964), 21-36. | Zbl 0139.04301

[011] [MRT] P. Mardešić, C. Rousseau and B. Toni, Linearization of isochronous centers, J. Differential Equations 121 (1995), 67-108. | Zbl 0830.34023

[012] [MS] L. Mazzi and M. Sabatini, Commutators and linearizations of isochronous centers, preprint UTM 482, Univ. of Trento, 1996.

[013] [NS] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, NJ, 1960. | Zbl 0089.29502

[014] [O] Z. Opial, Sur les périodes des solutions de l'équation différentielle x'' + g(x) = 0, Ann. Polon. Math. 10 (1961), 49-72. | Zbl 0096.29604

[015] [P] I. I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Differential Equations 5 (1969), 796-802.

[016] [S1] M. Sabatini, On the period function of Liénard systems, J. Differential Equations 152 (1999), 467-487. | Zbl 0922.34028

[017] [S2] M. Sabatini, Quadratic isochronous centers commute, preprint UTM 461, Univ. of Trento, 1995.

[018] [S3] M. Sabatini, Qualitative analysis of commuting flows on two-dimensional manifolds, in: EQUADIFF 95-International Conf. on Differential Equations (Lisboã, 1995), L. Magalhaes, C. Rocha and L. Sanchez (eds.), World Sci., Singapore, 1998, 494-497.

[019] [S4] M. Sabatini, Characterizing isochronous centers by Lie brackets, Differential Equations Dynam. Systems 5 (1997), 91-99.

[020] [S5] M. Sabatini, Dynamics of commuting systems on two-dimensional manifolds, Ann. Mat. Pura Appl. (4) 173 (1997), 213-232. | Zbl 0941.34018

[021] [SC] G. Sansone e R. Conti, Equazioni differenziali non lineari, Cremonese, Roma, 1956. | Zbl 0075.26803

[022] [U] M. Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961), 569-578. | Zbl 0100.29901

[023] [V] M. Villarini, Regularity properties of the period function near a center of a planar vector field, Nonlinear Anal. 19 (1992), 787-803. | Zbl 0769.34033