Dynamics of polynomial generalized Liénard system near the origin and infinity

We classify all topological phase portraits of the polynomial generalized Liénard system, determined by three arbitrary polynomials, at the origin and the infinity. This yields a complete characterization of monodromy at the origin and the infinity. Moreover, we obtain a necessary and sufficient condition for the local center via Cherkas’ method. When the origin is the only equilibrium and it is a center, there are 5 global phase portraits, including two types of global center. Further, we prove that the global center is not isochronous.