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A quintic polynomial differential system with eleven limit cycles at the infinity. (English) Zbl 1152.34329
Summary: In this article, a recursion formula for computing the singular point quantities of the infinity in a class of quintic polynomial systems is given. The first eleven singular point quantities are computed with the computer algebra system Mathematica. The conditions for the infinity to be a center are derived as well. Finally, a system that allows the appearance of eleven limit cycles in a small enough neighborhood of the infinity is constructed.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34-04 Machine computation, programs (ordinary differential equations)
Mathematica
Full Text:
##### References:
 [1] Li, J.: Hilbert’s 16th problem and bifurcation of planar polynomial vector fields. International journal of bifurcation and chaos 13, 47-106 (2003) · Zbl 1063.34026 [2] Ye, Y.: Qualitative theory of polynomial differential systems. (1995) · Zbl 0854.34003 [3] Liu, Y.: Theory of center-focus for a class of higher-degree critical points and infinite points. Science in China (series A) 44, 37-48 (2001) · Zbl 1012.34027 [4] Blows, T. R.; Rousseau, C.: Bifurcation at infinity in polynomial vector fields. Journal of differential equations 104, 215-242 (1993) · Zbl 0778.34024 [5] Liu, Y.; Chen, H.: Stability and bifurcation of limit cycles of the equator in a class of cubic polynomial systems. Computers & mathematics with applications 44, 997-1005 (2002) · Zbl 1084.34523 [6] Huang, Wen-Tao; Liu, Yi-Rong: A cubic polynomial system with six limit cycles at infinity. Journal of central south university 34, No. 4, 690-693 (2004) [7] Zhang, Q.; Liu, Y.: A cubic polynomial system with seven limit cycles at infinity. Applied mathematics and computation 177, No. 1, 319-329 (2006) · Zbl 1096.65130 [8] W. Huang, Y. Liu, A cubic system with seven limit cycles at the infinity. Ph.D. Thesis, Central South University, 2004(3) pp. 23--30 (in Chinese) [9] Liu, Y.; Meichun, Z.: Stability and bifurcation of limit cycles of the equator in a class of fifth polynomial systems. Chinese journal of contemporary mathematics 23, No. 1, 75-78 (2002) · Zbl 1007.34040 [10] Huang, W.; Liu, Y.: Bifurcation of limit cycles from infinity for a class of quintic polynomial system. Bulletin des sciences mathématiques 128, 291-302 (2004) · Zbl 1070.34064 [11] Liu, Y.; Li, J.: Theory of values of singular point in complex autonomous differential system. Science in China (series A) 3, 245-255 (1989)