Existence of Non-trivial Limit Cycles in Abel Equations with Symmetries

We study the periodic solutions of the generalized Abel equation x′ = a1A1(t)x 1+a2A2(t)x 2+a3A3(t)x n3 , where n1, n2, n3 > 1 are distinct integers, a1, a2, a3 ∈ R, and A1, A2, A3 are 2π-periodic analytic functions such that A1(t) sin t,A2(t) cos t, A3(t) sin t cos t are πperiodic positive even functions. When (n3−n1)(n3−n2) < 0 we prove that the equation has no nontrivial (different from zero) limit cycle for any value of the parameters a1, a2, a3. When (n3 − n1)(n3 − n2) > 0 we obtain under additional conditions the existence of non-trivial limit cycles. In particular, we obtain limit cycles not detected by Abelian integrals.