Abstract:

In this article, the bifurcations of limit cycles at the degenerate critical point and at infinity for a class of quintic polynomial sy stem are investigated. In the system, the origin is degenerate critical point and the equator contains no real critical point. Firstly,algebraic recursive formulas for computing singular point quantities of the origin and infinity are derived resp ectively . The first five singular point quantities at the origin and first four singular point quantities at infinity for the system are given in order to get conditions of center and investigate bifurcations of limit cycles. At last, the authors construct a quintic system which allows the appearance of five limit cycles in the neighborhood of the origin and two limit cycles around infinity. As far as the authors know, this is the first time that the problem of limit cycles bifurcated from a degenerate singular point and from infinity under the sync hronous perturbed conditions is investigated.

Key words: Degenerate critical point, Infinity, Focal value, Singul ar , point quantity, Bifurcation of limit cycles.