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**Bifurcation of limit cycles at the equator for a class of polynomial differential system.**
*(English)*
Zbl 1167.37341

Summary: Center conditions and bifurcation of limit cycles from the equator for a class of polynomial system of degree seven are studied. The method is based on converting a real system into a complex system. The recursion formula for the computation of singular point quantities of complex system at the infinity, and the relation of singular point quantities of complex system at the infinity with the focal values of its concomitant system at the infinity are given. Using the computer algebra system Mathematica, the first 14 singular point quantities of complex system at the infinity are deduced. At the same time, the conditions for the infinity of a real system to be a center and 14 order fine focus are derived respectively. A system of degree seven that bifurcates 13 limit cycles from the infinity is constructed for the first time.

### MSC:

37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |

34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |

### Keywords:

polynomial system of degree seven; infinity; singular point quantity; bifurcation of limit cycle### Software:

Mathematica
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\textit{Q. Zhang} et al., Nonlinear Anal., Real World Appl. 10, No. 2, 1042--1047 (2009; Zbl 1167.37341)

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### References:

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