Global phase portraits of uniform isochronous centers system of degree six with polynomial commutator (Q6565521)

This paper provides an in-depth study of the global phase portraits of uniform isochronous centers in systems of degree six, characterized by a polynomial commutator. The authors focus on systems of the form \( \dot{x} = -y + xf(x, y) \) and \( \dot{y} = x + yf(x, y) \), where \( f(x, y) = x\sigma(y) \), exploring the geometric implications of invariant straight lines arising from the zeros of a specific polynomial. The results are effectively visualized on the Poincaré disk, offering a comprehensive representation of the global dynamics.