Year: 2026
Type: preprint
Abstract: We establish criteria for at most one or at most two limit cycles of generalized Liénard systems in a single potential well. Two independent uniqueness tests use modified quotients of the damping and restoring forces; both allow a degenerate restoring force, and one extends to continuous potential profiles with corners. Opposite convexity of the two branches of the damping quotient in the potential coordinate gives an upper bound of two. Further criteria apply to weak damping with an analytic normalized profile on a bounded potential interval under an explicit smallness condition, to small profile perturbations vanishing near the equilibrium on a bounded range of half-widths, and to parameter families satisfying curvature, monotonicity and boundary conditions. Applications give another proof of the classical quartic bound. For Selkov systems, we obtain a two-cycle bound in a specified negative-parameter region and prove existence and uniqueness in a positive-parameter region. We also prove Conjecture~2.1 in [J. Math. Anal. Appl., 2025]. Examples attain the two-cycle bound.
Source: arXiv (Cornell University)
Authors: Cao, Chen, Chen, Hebai, Tang, Yilei
Institutions: Shanghai Jiao Tong University, Central South University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223347416