On the Hamiltonian dynamics and geometry of the Rabinovich system

Răzvan M. Tudoran; Anania Gîrban

doi:10.3934/dcdsb.2011.15.789

Article Contents

2011, Volume 15, Issue 3: 789-823. Doi: 10.3934/dcdsb.2011.15.789

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On the Hamiltonian dynamics and geometry of the Rabinovich system

Răzvan M. Tudoran^1, and
Anania Gîrban^2,

1.
The West University of Timişoara, Faculty of Mathematics and C.S., Department of Mathematics, B-dul. Vasile Pârvan, No. 4, 300223 - Timişoara
2.
"Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei nr. 2, 300006 - Timişoara

Received: March 31, 2010

Revised: August 31, 2010

Published: September 2011

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Abstract

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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Mathematics Subject Classification: Primary: 70H05; Secondary: 70H14, 70K44.

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