Identifying weak foci and centers in the Maxwell–Bloch system

Abstract

In this paper we identify weak foci and centers in the Maxwell-Bloch system, a three dimensional quadratic system whose three equilibria are all possible to be of centerfocus type. Applying irreducible decomposition and the isolation of real roots in computation of algebraic varieties of Lyapunov quantities on an approximated center manifold, we prove that at most 6 limit cycles arise from Hopf bifurcations and give conditions for exact number of limit cycles near each weak focus. Further, applying algorithms of computational commutative algebra we find Darboux polynomials and give some center manifolds in closed form globally, on which we identify equilibria to be centers or singular centers by integrability and time-reversibility on a center manifold. We prove that those centers are of at most second order.

First integrals of the Maxwell–Bloch system

“…As indicated in [1,7,8], it can be used to model Type I laser (He-Ne), Type II laser (Ruby and CO2) and Type III laser (far infrared) in the case of γ ⊥ ≈ γ κ, γ ⊥ γ ≈ κ and 0 large enough, respectively. This system has been analyzed as a dynamical system by many researchers, see for instance [5,7,13,18] and the references therein. In this paper, we try to understand its complexity and chaotic properties from the view of integrability and non-integrability.…”

New Symmetric Periodic Solutions for the Maxwell-Bloch Differential System

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Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

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