Gabriele Villari and  Fabio Zanolin
Phase-portrait of a Modified Generalized Lienard Type System
Dynamic Systems and Applications 34 (2024) 23-60

https://doi.org/10.46719/dsa2025.34.02

ABSTRACT.
This work deals with a new class of generalized Lienard type systems of the form

ẋ = y – λ(A – R(y))F(x), ẏ = – 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 R(y) ≈ |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.

AMS (MOS) Subject Classification. 34C25, 34C05, 34C23.

Key Words and Phrases. Generalized Lienard systems, limit cycles, multiplicity, stability, bifurcation.