Abstract The limit cycles of the van der Pol oscillator X ̈ + AX − 2BX 3 +ε(z 3 +z 2X 2 +z 1X 4) X ̇ = 0 , for B > 0, are studied in first-order approximation, using the Jacobian elliptic functions with the method of harmonic balance. The transitory motion, and in consequence the limit cycles and their stability are also studied in an approximate quantitative way with a generalized method of the slowly varying amplitude and phase. The bifurcations of these non-linear oscillators are studied using the methods of differentiable dynamics to obtain the qualitative behaviour. Quantitative values for the radius, frequency and energy of the limit cycles are given. The presence and stability of zero, one, or two limit cycles depend on the parameters z i . The presence of bifurcations depends on z i and A.