Periodic solutions of a quartic differential equation and Groebner bases

We consider first-order ordinary differential equations with quartic nonlinearities. The aim is to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. It is shown that this number is ten when the coefficients are certain cubic polynomials. Equations with the maximum number of such periodic solutions are also constructed. The paper is heavily dependent on computing Groebner bases.