On the Coexistence of Two Limit Cycles Enclosing a Node in a Class of Planar Differential Systems of Degree Six

Abstract

ABSTRACT In this paper, we look at a special type of six‐degree polynomial differential systems in the plane. We establish the coexistence of two limit cycles in these systems, both surrounding a single node. It is well known that, for systems of even degree, proving the coexistence of multiple limit cycles is highly challenging. To the best of our knowledge, this configuration is obtained here for the first time. We provide a rigorous proof of the result and present explicit examples illustrating the occurrence of two nested limit cycles surrounding the same singular point.