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

Abstract

We consider the Abel equation x ˙ = A ( t ) x 3 + B ( t ) x 2 , 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 .