Stability of equilibrium solutions of autonomous and periodic Hamiltonian systems in the case of multiple resonances

•We study Hamiltonians with n-degrees of freedom.•We have multiple resonances.•We give new criteria of (un)stability.

In this paper, we give necessary conditions for Lie-stability of the equilibrium solutions of autonomous and periodic Hamiltonian systems with n degrees of freedom which possess multiple resonances of arbitrary order. Necessary conditions for instability in the sense of Lyapunov in some cases of multiple resonances are provided. These conditions depend on the order of the resonance and the coefficients of the Hamiltonian function.