A stability criterion for non-degenerate equilibrium states of completely integrable systems

We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton-Poisson realization of a completely integrable system generated by a smooth n-dimensional vector field, X, and a non-degenerate regular (in the Poisson sense) equilibrium state, x‾e, we define a scalar quantity, IX(x‾e), whose sign determines the stability of the equilibrium. Moreover, if IX(x‾e)>0, then around x‾e, there exist one-parameter families of periodic orbits shrinking to {x‾e}, whose periods approach 2π/IX(x‾e) as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system.