Stability of equilibrium positions of periodic Hamiltonian systems under third and fourth order resonances

The problem of the stability of an equilibrium position of a nonautonomous $2 \pi$-periodic Hamiltonian system with $n$ degrees of freedom ($n \geqslant 2$), in a nonlinear setting, is studied in the presence of a single third and fourth order resonance. We give conditions of instability in the sense of Lyapunov and formal stability of the equilibrium position, depending on the coefficients of the Hamiltonian function.