The topology and isochronicity on complex Hamiltonian systems with homogeneous nonlinearities

In this paper, we study the Hamiltonian differential systems with homogeneous nonlinearity parts on $\mathbb{C}^2$. Firstly, we present a series of topological properties of polynomial Hamiltonian functions, with a particular focus on the characteristics of critical points and non-trivial cycles that vanish at infinity. Secondly, we use these topological properties to derive a complete set of necessary and sufficient conditions for isochronous centers in this class of systems of any degree. Our method avoids tedious computation of the coefficients of normalization occurring in the usual tools to deal with the isochronicity problem.