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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Tudoran, Razvan M.¹ 

[1] West Univ Timisoara, Dept Math, Fac Math & Comp Sci, Blvd Vasile Parvan 4, Timisoara 300223, Romania

关键词: Completely integrable systems;    Stability;    Equilibrium states;    Periodic orbits;    Rikitake system;   

DOI  :  10.1016/j.jde.2017.07.032

来源: Elsevier

【 摘 要 】

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, Ye, we define a scalar quantity, I-x((x) over bar (e)), whose sign determines the stability of the equilibrium. Moreover, if I-x ((x) over bar (e)) > 0, then around (x) over bar (e), there exist one -parameter families of periodic orbits shrinking to {(x) over bar (e)}, whose periods approach 2 pi / root I-x((x) over bar (e)) as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free