SMALL AMPLITUDE LIMIT-CYCLES FOR CUBIC SYSTEMS

In this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systems x = y(ax2 + y2 - 1)-2cy2 + epsilonxy(-4 + y) y = -x(x2 + by2 - 1) + 2dx2 + epsilonxy(-4 + x). With a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.