# On a Hamilton-Poisson Approach of the Maxwell-Bloch Equations with a Control

On a Hamilton-Poisson Approach of the Maxwell-Bloch Equations with a Control In this paper we consider the 3D real-valued Maxwell-Bloch equations with a parametric control given by x ̇ = y + az + byz , y ̇ = xz , Ż = − xy $\dot {x}=y+az+byz,\dot {y}=xz,\dot {z}=-xy$ ( a , b ∈ ℝ $a,b\in \mathbb {R}$ ). We give two Lie-Poisson structures of this system that are related with well-known Lie algebras. Moreover, we construct infinitely many Hamilton-Poisson realizations of this system. We also analyze the stability of the equilibrium points, as well as the existence of periodic orbits. In addition, we emphasize some connections between the energy-Casimir mapping of the considered system and the above-mentioned dynamical elements. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Mathematical Physics, Analysis and Geometry" Springer Journals

# On a Hamilton-Poisson Approach of the Maxwell-Bloch Equations with a Control

, Volume 20 (3) – Aug 8, 2017
22 pages

/lp/springer_journal/on-a-hamilton-poisson-approach-of-the-maxwell-bloch-equations-with-a-ApPoX3CCOw
Publisher
Springer Netherlands
Subject
Physics; Theoretical, Mathematical and Computational Physics; Analysis; Geometry; Group Theory and Generalizations; Applications of Mathematics
ISSN
1385-0172
eISSN
1572-9656
D.O.I.
10.1007/s11040-017-9251-3
Publisher site
See Article on Publisher Site

### Abstract

In this paper we consider the 3D real-valued Maxwell-Bloch equations with a parametric control given by x ̇ = y + az + byz , y ̇ = xz , Ż = − xy $\dot {x}=y+az+byz,\dot {y}=xz,\dot {z}=-xy$ ( a , b ∈ ℝ $a,b\in \mathbb {R}$ ). We give two Lie-Poisson structures of this system that are related with well-known Lie algebras. Moreover, we construct infinitely many Hamilton-Poisson realizations of this system. We also analyze the stability of the equilibrium points, as well as the existence of periodic orbits. In addition, we emphasize some connections between the energy-Casimir mapping of the considered system and the above-mentioned dynamical elements.

### Journal

"Mathematical Physics, Analysis and Geometry"Springer Journals

Published: Aug 8, 2017

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