Bifurcation at the equator for a class of quintic polynomial differential system (Q945419)

The authors study simultaneous bifurcations from the origin and infinity for the family of quintic polynomial differential systems of the form \[ x'=(\delta_1 x-\lambda y)(x^2+y^2)+P_4(x,y)+(-y+\delta_2 x)(x^2+y^2)^2, \] \[ y'=(\lambda x+\delta_1 y)(x^2+y^2)+Q_4(x,y)+(x+\delta_2 y)(x^2+y^2)^2, \] where \(\lambda\neq0\) and \(P_4\) and \(Q_4\) are homogeneous polynomials of degree 4, depending on five real parameters, chosen to simplify further computations. Notice that for these systems the infinity of their Poincaré compactification has no critical points. It is said that a family of differential systems presents the \(\{(n),m\}\) configuration of limit cycles if it is possible to bifurcate simultaneously \(n\) limit cycles from the origin and \(m\) limit cycles from infinity. The main result of this paper is that the above systems present \(\{(7),2\}\) and \(\{(2),6\}\) configurations. This result is proved by computing the Lyapunov quantities (focal values) at the origin and at infinity.