Existence and Nonexistence of Limit Cycles for a Certain Class of Planar Systems of Lienard Type
Year: 2024
Type: article
Abstract: The phase-portrait of a Liénard system ẋ = y-F (x), ẏ = -g(x) is investigated under a classical assumption on F (x), namely that there are α < 0 < β such that F (α) = F (0) = F (β) = 0 with F (x)x < 0 for x ∈ (α, β) \ {0} and F is monotone increasing outside (α, β).It is well known that such a system has at least a limit cycle provided that G(x) → +∞ or F (x)sign(x) → +∞ for x → ±∞.Clearly, such assumptions imply that G(x) ± F (x) → +∞ as x → ±∞.In [19] it has been proved that this is actually a necessary and sufficient condition for the intersection of the trajectories with the vertical isocline.In this paper we treat the case in which this assumption is not fulfilled and prove that there are cases where both existence and nonexistence may occur.
Source: Dynamic Systems and Applications
Authors: Gabriele Villari, Fabio Zanolin
Institutions: University of Florence, University of Udine
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4406569661