A Necessary and Sufficient Condition for General Rayleigh–Liénard System with an Elementary Center

Authors

• Xiuli Cen
• Hebai Chen
• Zhaosheng Feng, The University of Texas Rio Grande Valley
• Rui Zhang

Document Type

Article

Publication Date

4-27-2026

Abstract

The elementary center–focus problem is studied for a general separable planar analytic Rayleigh–Liénard system dxdt=y,dydt=−g(x)+f(x)ψ(y)y,

where g(x), f(x), and ψ(y) are analytic functions. By means of the Melnikov function method, a necessary and sufficient condition is obtained for the origin to be an elementary center: g′(0)>0 and either xf(ϕ−1(x))g(ϕ−1(x)) or ψ(y) is an odd function in a neighborhood of the origin, where ϕ(x)=√2∫x0g(s)ds⋅sgn(x).

Comments

https://rdcu.be/fluM2

Recommended Citation

Cen, Xiuli, Hebai Chen, Zhaosheng Feng, and Rui Zhang. "A Necessary and Sufficient Condition for General Rayleigh–Liénard System with an Elementary Center." Journal of Nonlinear Science 36, no. 3 (2026): 50. https://doi.org/10.1007/s00332-026-10257-9

Publication Title

Journal of Nonlinear Science

DOI

10.1007/s00332-026-10257-9