The period function in a class of quadratic Kolmogoroff systems

  • Marco Uribe S. Universidad de la Santísima Concepción.
  • Myrna Wallace C. Universidad de la Santísima Concepción.
Palabras clave: Invariants, period functions, invariantes, funciones periódicas.

Resumen

In this paper we consider the family of quadratic Kolmogoroff systems with a center in the real quadrant:where 1 < a < ∞:This system has three invariant lines (the coordinate axes and the line x + y - 1 = 0) and a family of periodic solutions nested around a center and filling out the triangle determined by the three invariant lines. Using integrability of this system we reduce the abelian integral representing the period function and its derivative. The main result is that the corresponding period function is monotone increasing for values of the parameter near a = 3.

Biografía del autor

Marco Uribe S., Universidad de la Santísima Concepción.
Facultad de Ingeniería, Area de Ciencias Básicas.
Myrna Wallace C., Universidad de la Santísima Concepción.
Facultad de Ciencias Físicas y Matemáticas, Departamento de Matemática.

Citas

[1] Chicone C. and Jacob M. Bifurcation of critical periods for plane vector fields. Trans. Amer. Math. Soc. 312, N 12, (1989).

[2] Chicone C. The monotonicity of period function for planar hamiltonian vector fields. Journal of Differential Equation 69, (1987).

[3] Gasull A., Guillamon A. and Mañosa V. An expression for the first Liapunov and period constants with application. Preprint, Universidad Autonoma de Barcelona, España.

[4] Gasull A., Guillamon A. and Mañosa V. Center and Isochronicity conditions for systems with homogeneus nonlinearities. Proceeding of the 2a catalan days on applied mathematics. (1995).

[5] Van Horssen W and Reyn J. Bifurcation of limit cycles in a particular class of quadratic systems. Differential and Integral Equation Vol. 8 , N 4, (1995).
Publicado
2017-06-14
Cómo citar
Uribe S., M., & Wallace C., M. (2017). The period function in a class of quadratic Kolmogoroff systems. Proyecciones. Journal of Mathematics, 19(2), 197-205. https://doi.org/10.22199/S0716-09172000000200006
Sección
Artículos