Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

In this paper, we study the global dynamical behavior of the Hamiltonian system <(x)over dot> = H-y(x, y), <(y)over dot> = -H-x(x, y) with the rational potential Hamiltonian H(x, y) = y(2)/2 + P(x)/Q(y), where P(x) and Q(y) are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincare disk and provide their bifurcation diagrams.