Cubic Lienard equations with linear damping

Abstract

The authors present an extensive qualitative study of the phase portraits of Lienard equations x=A(x)+B(x)x, with A cubic and B linear. They encounter limit cycles surrounding one singularity and limit cycles surrounding two or three singularities. They prove that the equations can have at most one limit cycle of the first kind which can never be surrounded by one of the second kind. They adduce strong evidence that there is at most one limit cycle in each case. After an affine coordinate change in phase space these equations reduce to the vector fields y delta / delta x+(+or-x3+ mu 2x+ mu 1+y( nu +bx)) delta / delta y, with b>0, whereby they recover the families of vector fields obtained previously after (principal) rescaling. The results obtained can be used to complete some proofs missing in earlier work. They present a complete study for the case A quadratic and B linear, and show that at most one limit cycle is present.