Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with n-Degrees of Freedom in the Case of Single Resonance

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

“…The Lie normal form of the Hamiltonian function H in (2) was characterized in [7] according to the generators of the module M ω . In fact, we have: • If M ω = k 1 Z + · · · + k s Z = {0}, we have that H m = H m (r, k 1 · ϕ, .…”

“…For single third and four order resonances we can cite [2] and [10] for the autonomous case, while the periodic case was considered in [20]. In [7] it was considered the case of a single resonance in the general case where the matrix of the linearized system is diagonalizable. Here, the authors improve and generalize several results existing in the literature.…”

“…The main purpose of this paper is to show new results about the stability or instability of equilibrium points of the system (1) under the presence of multiple resonance in the autonomous as in the periodic case. In order to enunciate our main results we need to remember the concept of Lie-stability which was used in [7] because of the use of the Lie normal form. This was first introduced by Khazin in [11] because of the use of the Birkhoff normal form.…”

The linearization of periodic Hamiltonian systems with one degree of freedom under the Diophantine condition

“…In the case of resonance, i.e., ω is a rational number, the above result cannot be applied directly. For this case, there also are many results, see [2], [3], [7], [8], [22], [23], [24] and the references therein. For example, Mansilla [7] obtained some sufficient conditions for stability and instability of the trivial solution by using Moser's twist theorem and Liapunov theorem, respectively.…”

Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates

“…More recent papers on Lie stable and unstable systems are due to dos Santos and coworkers [33,34,35] where the authors establish several criteria dealing with Lie stable equilibria in cases of resonances. They also treat instability using suitable Chetaev functions [24].…”