Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by H. Zoladek in the article (Eleven small limit cycles in a cubic vector field, Nonlinearity 8, 843--860, 1995) can indeed have eleven limit cycles under perturbations at least up to $7$th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to $39$th-order perturbations, and no more than eleven limit cycles are found.