Reachability of maximal number of critical periods without independence

Abstract

The reachability of the maximal number of critical periods bifurcated from a weak center is judged with independence of period quantities. Since the independence is a sufficient condition, in this paper we give a method to judge the reachability without the independence. Our main idea is to consider how the number of zeros for derivative of the period function changes as a curve in the parameter space scans near a specific point and choose an appropriate curve to find the largest number of zeros. We apply this method to two polynomial differential systems, having zero-dimensional isochronicity variety and nonzero-dimensional one separately, to show that the maximal number is reached and obtain greater numbers than those obtained by classical independence separately. With our method, the two examples give the possibility that more critical periods arise from an isochronous center than a finite order weak center, an opposite fact to many known results.