Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation over(x, ̇) = A (t) x3 + B (t) x2, where A (t) and B (t) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x = 0. © 2007 Elsevier Inc. All rights reserved.