Global phase portraits of generalized polynomial Liénard system with a unique equilibrium and a periodic annulus

Abstract

This paper aims to study the sufficient and necessary conditions for a generalized polynomial Liénard system with a unique equilibrium and one periodic annulus. In such a case, there are two types of equilibrium: a centre or a neighbourhood of the equilibrium consisting of a hyperbolic sector and an elliptical sector. The sufficient and necessary conditions for both cases are established and global phase portraits in the Poincaré disc are presented under certain conditions.

Keywords:

• Liénard system
• periodic annulus
• Poincaré–Bendixson theorem
• global phase portrait
• centre

Mathematics Subject Classifications (2020):

• 34C07
• 34C23
• 34C37
• 34K18

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China (Nos. 12322109, 12171485) and Science and Technology Innovation Program of Hunan Province (No. 2023RC3040); Division of Mathematical Sciences [2312952].