Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems.

Abstract The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems. [ABSTRACT FROM AUTHOR]