On the limit cycles of quasihomogeneous polynomial systems

L. Rocío González-Ramírez; Osvaldo Osuna; Rubén Santaella-Forero

doi:10.15446/recolma.v49n2.60443

Published

2015-07-01

On the limit cycles of quasihomogeneous polynomial systems

DOI:

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

Keywords:

Integrating factors, Inverse integrating factors, Limit cycles, p - q-quasi-homogeneous systems (en)

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Authors

L. Rocío González-Ramírez Universidad Michoacana
Osvaldo Osuna Universidad Michoacana
Rubén Santaella-Forero Universidad Michoacana

Abstract (en)

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

On the limit cycles of quasihomogeneous polynomial systems

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

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.

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.

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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)

How to Cite

APA

González-Ramírez, L. R., Osuna, O. and Santaella-Forero, R. (2015). On the limit cycles of quasihomogeneous polynomial systems. Revista Colombiana de Matemáticas, 49(2), 261–268. https://doi.org/10.15446/recolma.v49n2.60443

ACM

[1]

González-Ramírez, L.R., Osuna, O. and Santaella-Forero, R. 2015. On the limit cycles of quasihomogeneous polynomial systems. Revista Colombiana de Matemáticas. 49, 2 (Jul. 2015), 261–268. DOI:https://doi.org/10.15446/recolma.v49n2.60443.

ACS

(1)

González-Ramírez, L. R.; Osuna, O.; Santaella-Forero, R. On the limit cycles of quasihomogeneous polynomial systems. rev.colomb.mat 2015, 49, 261-268.

ABNT

GONZÁLEZ-RAMÍREZ, L. R.; OSUNA, O.; SANTAELLA-FORERO, R. On the limit cycles of quasihomogeneous polynomial systems. Revista Colombiana de Matemáticas, [S. l.], v. 49, n. 2, p. 261–268, 2015. DOI: 10.15446/recolma.v49n2.60443. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/60443. Acesso em: 22 jan. 2025.

Chicago

González-Ramírez, L. Rocío, Osvaldo Osuna, and Rubén Santaella-Forero. 2015. “On the limit cycles of quasihomogeneous polynomial systems”. Revista Colombiana De Matemáticas 49 (2):261-68. https://doi.org/10.15446/recolma.v49n2.60443.

Harvard

González-Ramírez, L. R., Osuna, O. and Santaella-Forero, R. (2015) “On the limit cycles of quasihomogeneous polynomial systems”, Revista Colombiana de Matemáticas, 49(2), pp. 261–268. doi: 10.15446/recolma.v49n2.60443.

IEEE

[1]

L. R. González-Ramírez, O. Osuna, and R. Santaella-Forero, “On the limit cycles of quasihomogeneous polynomial systems”, rev.colomb.mat, vol. 49, no. 2, pp. 261–268, Jul. 2015.

MLA

González-Ramírez, L. R., O. Osuna, and R. Santaella-Forero. “On the limit cycles of quasihomogeneous polynomial systems”. Revista Colombiana de Matemáticas, vol. 49, no. 2, July 2015, pp. 261-8, doi:10.15446/recolma.v49n2.60443.

Turabian

González-Ramírez, L. Rocío, Osvaldo Osuna, and Rubén Santaella-Forero. “On the limit cycles of quasihomogeneous polynomial systems”. Revista Colombiana de Matemáticas 49, no. 2 (July 1, 2015): 261–268. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/60443.

Vancouver

1.

González-Ramírez LR, Osuna O, Santaella-Forero R. On the limit cycles of quasihomogeneous polynomial systems. rev.colomb.mat [Internet]. 2015 Jul. 1 [cited 2025 Jan. 22];49(2):261-8. Available from: https://revistas.unal.edu.co/index.php/recolma/article/view/60443