Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems

This paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems. The basic theme throughout the paper is that the number, positions, and multiplicities of the limit cycles that bifurcate under perturbations are related to the number, positions, and multiplicities of the zeros of the Melnikov function for the system. The general theory is illustrated by a number of examples from the literature, some of which are extended to include new results.