ON BIFURCATION OF LIMIT-CYCLES FROM CENTERS

For a one parameter family of plane vector fields X(., epsilon) depending analytically on a small real parameter epsilon, we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding a center. Aside from the intrinsic interest in the example we choose, it serves to illustrate some techniques which are developed for treating similar bifurcation problems when the first order methods are inconclusive. Actually, we are able to treat the bifurcations of all orders.