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

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Abstract

In this paper we consider the 3D real-valued Maxwell-Bloch equations with a parametric control given by \(\dot {x}=y+az+byz,\dot {y}=xz,\dot {z}=-xy\) (\(a,b\in \mathbb {R}\)). 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.

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Acknowledgments

We would like to thank the referees very much for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Politehnica University of Timişoara, Piaţa Victorei, Nr. 2, 300006, Timişoara, Romania

    Cristian Lăzureanu

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  1. Cristian Lăzureanu
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Correspondence to Cristian Lăzureanu.

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Lăzureanu, C. On a Hamilton-Poisson Approach of the Maxwell-Bloch Equations with a Control. Math Phys Anal Geom 20, 20 (2017). https://doi.org/10.1007/s11040-017-9251-3

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  • DOI: https://doi.org/10.1007/s11040-017-9251-3

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