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

TL;DR

This paper introduces a new stability criterion for non-degenerate equilibrium states in completely integrable systems, linking the sign of a scalar quantity to stability and nearby periodic orbits.

Contribution

It defines a scalar stability criterion for equilibrium states in integrable systems and illustrates its application with the Rikitake system.

Findings

Sign of the scalar quantity determines stability.

Existence of periodic orbits near stable equilibria.

Period of orbits approaches a specific value as they shrink.

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.