## Abstract and Applied Analysis

### The Number of Limit Cycles of a Polynomial System on the Plane

#### Abstract

We perturb the vector field $\stackrel{˙}{x}=-yC\left(x,y\right)$, $\stackrel{˙}{y}=xC\left(x,y\right)$ with a polynomial perturbation of degree $n$, where $C\left(x,y\right)=\left(\mathrm{1}-{y}^{\mathrm{2}}{\right)}^{m}$, and study the number of limit cycles bifurcating from the period annulus surrounding the origin.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 482850, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511960

Digital Object Identifier
doi:10.1155/2013/482850

Mathematical Reviews number (MathSciNet)
MR3073507

Zentralblatt MATH identifier
1300.34068

#### Citation

Liu, Chao; Han, Maoan. The Number of Limit Cycles of a Polynomial System on the Plane. Abstr. Appl. Anal. 2013 (2013), Article ID 482850, 7 pages. doi:10.1155/2013/482850. https://projecteuclid.org/euclid.aaa/1393511960

#### References

• V. I. Arnold, “Some unsolved problems in the theory of differential equations and mathematical physics,” Russian Mathematical Surveys, vol. 44, pp. 157–171, 1989.
• J. Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 1, pp. 47–106, 2003.
• H. Giacomini, J. Llibre, and M. Viano, “On the nonexistence, existence and uniqueness of limit cycles,” Nonlinearity, vol. 9, no. 2, pp. 501–516, 1996.
• J. Llibre, J. S. Pérez del Río, and J. A. Rodríguez, “Averaging analysis of a perturbated quadratic center,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 46, no. 1, pp. 45–51, 2001.
• G. Xiang and M. Han, “Global bifurcation of limit cycles in a family of polynomial systems,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 633–644, 2004.
• A. Buică and J. Llibre, “Limit cycles of a perturbed cubic polynomial differential center,” Chaos, Solitons and Fractals, vol. 32, no. 3, pp. 1059–1069, 2007.
• B. Coll, J. Llibre, and R. Prohens, “Limit cycles bifurcating from a perturbed quartic center,” Chaos, Solitons & Fractals, vol. 44, no. 4-5, pp. 317–334, 2011.
• A. Atabaigi, N. Nyamoradi, and H. R. Z. Zangeneh, “The number of limit cycles of a quintic polynomial system,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 677–684, 2009.
• A. Gasull, J. T. Lázaro, and J. Torregrosa, “Upper bounds for the number of zeroes for some Abelian integrals,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 13, pp. 5169–5179, 2012.
• A. Gasull, R. Prohens, and J. Torregrosa, “Bifurcation of limit cycles from a polynomial non-global center,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 945–960, 2008.
• G. Xiang and M. Han, “Global bifurcation of limit cycles in a family of multiparameter system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 9, pp. 3325–3335, 2004.
• A. Gasull, C. Li, and J. Torregrosa, “Limit cycles appearing from the perturbation of a system with a multiple line of critical points,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 75, no. 1, pp. 278–285, 2012.