Isochronous oscillations for cubic systems


  • Víctor Guiñez Universidad de Santiago.
  • Jaime Figueroa Universidad Técnica Federico Santa María.
  • Eduardo Sáez Universidad Técnica Federico Santa María.



Planar vector fields

? = -y + P?(x,y)

? = x + q?(x, y).

with a center non necessarily at the origin are considered, where p? and q? are homogeneous polynomials of degree 3. We are concerned with the behavior of the periods of the periodic solutions near the center, and in determining when the center is isochronous, i.e., when all periodic solutions have the same period. It is proved that, modulus a linear change of coordinates, there are only four systems which have an isochronous center. Each of them has an isochronous center at the origin, and its other centers are also isochronous but not necessarily with the same period. Also the maximal weakness of the non isochronous centers, is obtained.

Author Biographies

Víctor Guiñez, Universidad de Santiago.

Departamento de Matemáticas y Cs. de la Computación.

Jaime Figueroa, Universidad Técnica Federico Santa María.

Departamento de Matemáticas.

Eduardo Sáez, Universidad Técnica Federico Santa María.

Departamento de Matemáticas.


[1] C. Chicone and M. Jacobs, Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312, No. 2, pp. 433-486, (1989).
[2] W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations 3, pp. (21- 36), (1964).
[3] N. A. Lukashevich, isochronicity of center for certain systems of differential equations, Differential Equations 1, pp. 220- 226, (1965).
[4] K. E. Malkin, Criteria for center of a differential equation, Volg. Matem. Sbornik 2, pp. 87- 91, (1964).
[5] I. I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Differential Equations 5, pp. 796-802, (1969).
[6] C. Rousseau and D. Schlomiuk, Cubic vector fields symmetric with respect toa center, preprint, Université de Montréal, (1992).
[7] C. K. Siegel and J. K. Moser, Lectures on celestial mechanics, SpringerVerlag, New York, (1971 ).
[8] K. S. Sibirsky, On the number of limit cycles in the neighborhood of a singular point, Differential Equations 1, pp. 36- 47, (1965).
[9] M. Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10, pp. 569-578, (1961).



How to Cite

V. Guiñez, J. Figueroa, and E. Sáez, “Isochronous oscillations for cubic systems”, Proyecciones (Antofagasta, On line), vol. 14, no. 2, pp. 115-132, Apr. 2018.




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