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JGM

We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.

keywords:
characteristic multipliers
,
stability
,
periodic orbits
,
Dissipative dynamics
,
control.

DCDS-B

In this paper, we describe some relevant dynamical and geometrical properties of the Rabinovich system from the Poisson geometry and the dynamics point of view. Starting with a Lie-Poisson realization of the Rabinovich system, we determine and then completely analyze the Lyapunov stability of all the equilibrium states of the system, study the existence of periodic orbits, and then using a new geometrical approach, we provide the complete dynamical behavior of the Rabinovich system, in terms of the geometric semialgebraic properties of a two-dimensional geometric figure, associated with the problem. Moreover, in tight connection with the dynamical behavior, by using this approach, we also recover all the dynamical objects of the system (e.g. equilibrium states, periodic orbits, homoclinic and heteroclinic connections). Next, we integrate the Rabinovich system by Jacoby elliptic functions, and give some Lax formulations of the system. The last part of the article discusses some numerics associated with the Poisson geometrical structure of the Rabinovich system.

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