Nonlinear stability of elliptic equilibria in Hamiltonian systems with resonances of order four with interactions

In this paper, we advance the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows, which is one of the oldest problems in mathematical physics. More precisely, in this work we study the nonlinear stability in the Lyapunov sense of one equilibrium solution in autonomous Hamiltonian systems with $ n $-degrees of freedom, assuming the existence of two vectors of resonance, both of order four, with interaction in one frequency. We provide conditions to obtain a type of formal stability, called Lie stability. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems with three and four degrees of freedom. In addition, we give sufficient conditions for the instability in the sense of Lyapunov. We apply some of our results in the spatial satellite problem at one of its equilibrium points, which is a novelty in this problem.