This paper investigates the bifurcation of limit cycles in sixth-order polynomial differential systems of the form \({\left\{ \begin{array}{ll} \dot{\textrm{x}}=\textrm{U}_{1}(\textrm{x},\textrm{y})+\textrm{U}_{6}(\textrm{x},\textrm{y}), \\ \dot{\textrm{y}}=\textrm{V}_{1}(\textrm{x},\textrm{y})+\textrm{V}_{6}(\textrm{x},\textrm{y}), \end{array}\right. }\) where \(U_{i}\) and \(V_{i}\) are homogeneous polynomials of degree i. Using the second-order averaging method, we derive explicit conditions for the existence of limit cycles emanating from both the origin and a perturbed linear center. The analysis involves a transformation of the system into an Abel differential equation. We provide closed-form approximate expressions for the limit cycles in polar coordinates, detailing their global shape. The theoretical results are validated through two specific examples, confirming the effectiveness of the approach and contributing to the qualitative analysis of high-order polynomial systems.
