Periodic orbits for perturbed non-autonomous differential equations

We consider non-autonomous differential equations, on the cylinder (t,r)∈S1×Rd, given by dr/dt=f(t,r,ε) and having an open continuum of periodic solutions when ε=0. From the study of the variational equations of low order we obtain successive functions such that the simple zeroes of the first one that is not identically zero control the periodic orbits that persist for the unperturbed equation. We apply these results to several families of differential equations with d=1,2,3. They include some autonomous polynomial differential equations and some Abel type non-autonomous differential equations.