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.

34 pages

References in corpus (2)

• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
• Bifurcations of limit cycles of perturbed completely integrable systems