Stability of odd periodic solutions in a resonant oscillator

We obtain an odd 2 π-periodic solution φ in a driven differential equation 

ẍ + g(x) = εp(t), where g and p are odd smooth functions with g^′(0) = n² for some n∈ N and g^″′(0) ≠ 0. The periodic solution φ is obtained by continuation of the equilibrium x≡ 0 of the unperturbed problem (ε= 0) for small ε. In order to prove this result, we establish an extension of a Loud’s version of the implicit function theorem at rank 0. Moreover, we present sufficient conditions for the existence of one or three odd 2 π-periodic continuations and also we give conditions for their linear stability.