Stability and bifurcations of limit cycles of the equator in a class of cubic polynomial systems

In this paper, we study the appearance of limit cycles from the equator in a class of cubic polynomial vector fields with no singular points at infinity and the stability of the equator of the systems. We start by deducing the recursion formula for quantities at infinity in these systems, then specialize to a particular case of these cubic systems for which we study the bifurcation of limit cycles from the equator. We compute the quantities at infinity with computer algebraic system Mathematica 2.2 and reach with relative ease an expression of the first six quantities at infinity of the system, and give a cubic system, which allows the appearance of six limit cycles in the neighborhood of the equator. As far as we know, this is the first time that an example of cubic system with six limit cycles bifurcating from the equator is given. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of quantities at infinity is linear and then avoids complex integrating operations. Therefore, the calculation can be readily done with using computer symbol operation system such as Mathematica.