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

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.