Nonlinear stability around an elliptic equilibrium point in a Hamiltonian system. (English) Zbl 0917.58015

Summary: Using a scheme given by Lochak, we derive a result of stability over exponentially long times with respect to the inverse of the distance to an elliptic equilibrium point which has a definite torsion. At the price of this assumption, our study is valid without arithmetical properties of the linearized system while the previous theorems of this kind rely on a Diophantine condition on the linear spectrum. Actually, under the latter condition and a definite torsion, a result of stability over superexponentially long times can be proved. Finally, the same kind of theorems are also valid for an elliptic lower-dimensional invariant torus.


37C75 Stability theory for smooth dynamical systems
70K20 Stability for nonlinear problems in mechanics
70K30 Nonlinear resonances for nonlinear problems in mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37G05 Normal forms for dynamical systems
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