Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential

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In this paper we study the global dynamics of the Hamiltonian systems x˙=Hy(x,y), y˙=−Hx(x,y), where the Hamiltonian function H has the particular form H(x,y)=y2/2+P(x)/Q(x), P(x),Q(x)∈R[x] are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable μ-symplectic change of variables. Then, the global topological classification of the phase portraits of these systems having canonical forms in the Poincaré disk in the cases where degree(P)=0,1,2 and degree(Q)=0,1,2 are studied as a function of the parameters that define each polynomial. We use a blow-up technique for finite equilibrium points and the Poincaré compactification for the infinite equilibrium points. Finally, we show some applications.