Bifurcation at the equator for a class of quintic polynomial differential system

The center problem and bifurcation of limit cycles for degenerate singular point and infinity are far from being solved in general. In this paper, we studied center conditions and bifurcation of limit cycles at the degenerate singular point and infinity (the equator) in a class of quintic polynomial differential system with two small parameters and eight normal parameters. The method is based on two homeomorphic transformations of the infinity and degenerate singular point into linear singular point, which allows us to compute the generalized Lyapunov constants (the singular point quantities) for the origin and infinity. The center conditions for the degenerate singular point and infinity are derived respectively. The limit cycle configurations of {(7), 2} and {(2), 6} are obtained under simultaneous perturbation at the origin and infinity. As far as we know, it is the first time that the bifurcation of limit cycles from a degenerate singular point and infinity simultaneously is studied in planar differential system.