Limit Cycles for a Class of Abel Equations

Abstract

The number of solutions of the Abel differential equation ${{dx(t)} /{dt}} = A(t)x(t)^3 + B(t)x(t)^2 + C(t)x(t)$ satisfying the condition $x(0) = x(1)$ is studied, under the hypothesis that either $A(t)$ or $B(t)$ does not change sign for $t \in [0,1]$. The main result obtained is that there are either infinitely many or at most three such solutions. This result is also applied to control the maximum number of limit cycles for some planar polynomial vector fields with homogeneous nonlinearities.