Upper bounds for the number of limit cycles through linear differential equations

Consider the differential equation x = y, y = h0(x)+ h1(x) y+ h2(x) y2 + y3 in the plane. We prove that if a certain solution of an associated linear ordinary differential equation does not change sign, there is an upper bound for the number of limit cycles of the system. The main ingredient of the proof is the Bendixson-Dulac criterion for ℓ-connected sets. Some concrete examples are developed.