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

Abstract: 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, $\overline{x}e$, we define a scalar quantity, $\mathcal{I}{X}(\overline{x}e)$, whose sign determines the stability of the equilibrium. Moreover, if $\mathcal{I}{X}(\overline{x}e)>0$, then around $\overline{x}_e$ there exist one-parameter families of periodic orbits shrinking to ${\overline{x}_e }$, whose periods approach $2\pi/\sqrt{\mathcal{I}{X}(\overline{x}_e)}$ as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system.