On the limit cycles of quasihomogeneous polynomial systems

In this work, the nonexistence of limit cycles for classes of p - q-quasi-homogeneous polynomial planar systems of weighted degree l is established. Furthermore, we rule out the existence of limits cycles for certain perturbations of such planar systems. We present applications and examples in order to illustrate our results.

DOI: https://doi.org/10.15446/recolma.v49n2.60443

Sobre los ciclos límite de sistemas polinomiales cuasi-homogeneos

L. Rocío González-Ramírez^1,2, Osvaldo Osuna^1,*, Rubén Santaella-Forero¹

¹ Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México
e-mail: rgonzalez@ifm.umich.mx
e-mail: osvaldo@ifm.umich.mx
² Catedrática CONACYT, D.F., México
e-mail: rusanfo@matmor.unam.mx

---

Abstract

Key words and phrases. Integrating factors, Inverse integrating factors, Limit cycles, p − q-quasi-homogeneous systems.

---

2010 Mathematics Subject Classification. 34C05, 34C07, 34C25.

---

Resumen

En este trabajo, se establece la no existencia de ciclos límite para la clase de sistemas bidimensionales, polinomiales p − q-cuasi-homogeneos de grado ponderado l. Además, se descarta la existencia de ciclos límite para ciertas perturbaciones de tales sistemas. Finalmente, se presentan aplicaciones y ejemplos para ilustrar los resultados obtenidos.

Palabras y frases clave. Factores Integrantes, Factores integrantes inversos, ciclos límite, sistemas p − q-cuasi-homogeneos.

---

Texto completo disponible en PDF

---

References

[1] Giacomini H. Berrone L., On the vanishing set of inverse integrating factors, Qual. Theory Dyn. Syst. 1 (2000), 211-213.

[2] Murray J. D., Mathematical Biology I: An Introduction, Springer-Verlag, 2004.

[3] García B. Llibre J. Pérez del Río J., Planar quasi-homogeneous polynomial differential systems and their integrability, Journal of Differential Equations 255 (2013), no. 16, 3185-3204.

[4] Chavarriga J. García I., Lie symmetries of quasihomogeneous polynomial planar vector fields and certain perturbations, Acta Math. Sinica 21 (2005), no. 1, 185-192.

[5] Chavarriga J. Giacomini H. Giné J. Llibre J., Darboux integrability and the inverse integrating factor, J. Diff. Equations 194 (2003), 116-139.

[6] Chavarriga J. Giné J., Integrable systems via inverse integrating factor, Extracta Math. 13 (1998), 41-60.

[7] García I. Grau M., A survey on the inverse integrating factor, Qual. Theory Dyn. Syst. 9 (2010), no. 1-2, 115-166.

[8] Giacomini H. Llibre J. Viano M., On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity 9 (1996), no. 1, 501-516.

[9] Fulton W., Algebraic curves: An introduction to algebraic geometric, Addison-Wesley, 1989.

(Recibido en marzo de 2015. Aceptado en octubre de 2015)