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In this paper, we study the Hamiltonian differential systems with ahomogeneous nonlinearity part on $\mathbb{C}^2$. Firstly, we give a series oftopological properties of the polynomial Hamiltonian function, especially theinformation on the singular points and the non-trivial cycles vanishing atinfinity. Secondly, by using those topological properties, we successfullyprovide a complete set of the necessary and sufficient conditions ofisochronous centers for such systems with any degree. Our method avoids tediouscomputation of the coefficients of normalization occurring in the usual toolsto deal with the isochronicity problem.

The topology and isochronicity on complex Hamiltonian systems with homogeneous nonlinearities