BIFURCATION OF LIMIT-CYCLES FROM QUADRATIC ISOCHRONES

For a one parameter family of plane quadratic vector fields X(.,ε) depending analytically on a small real parameter ε, we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous (or linearizable) center. Techniques are developed for treating the bifurcations of all orders, and these are applied to prove the following results. For the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three. For one class of nonlinear isochrones, at most one continuous family of limit cycles can emerge, whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge. Moreover, for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center. © 1991.