Phase portrait of a modified generalized Liénard type system

This work deals with a new class of generalized Li enard type systems of the form x' = y- k(A - R(y))F(x); y' = -g(x); where A is a positive constant and g(x) and F(x) are the typical nonlinearities occurring in the study of the van der Pol equation. For the function R(y) we assume that R(y) behaves like |y|^p . The study of the above model is motivated by recent works which already appeared in the literature for the case A = 0: In the present paper, we discuss the problem of existence, non-existence, uniqueness and multiplicity of the limit cycles, by moving the parameter A: We show that this case has a rich dynamics and, in particular, the presence of some new features which also put in evidence the existence of subtle relation between the distribution of the zeros of F(x) and the existence/non-existence of limit cycles.