Topological classification of global dynamics of planar polynomial Hamiltonian systems with separable variables

In the paper, we completely characterize the local dynamics of polynomial Hamiltonian systems with separable variables and provide a method to determine its global dynamics on Poincaré disk. It is shown that there are three (four) topological classifications for finite (infinite, resp.) singular points of the Hamiltonian system with any degree n, and its global dynamics can be determined by the number of singular points and their separatrix skeleton. This provides an approach to characterize the topological classification of real algebraic curves $H_1\left(x\right)+H_2\left(y\right)=0$ in $R^2$, where $H_1\left(x\right)$ and $H_2\left(y\right)$ are real polynomials of degrees m and n, respectively.