Quadratic Liénard Equations with Quadratic Damping

In this paper we study the generalized Liénard equations x+f(x) x+g(x)=0 with quadratic polynomialsfandg. We prove that these kind of equations can have at most one limit cycle, and we give the complete bifurcation diagram and classification of the phase portraits. The paper also contains a shorter proof for the result in A. Lins, W. de Melo, and C. C. Pugh, 1977,Lecture Notes in Math.597, 335–357 on the unicity of the limit cycle for (standard) Liénard equations with quadratic damping.