Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior

Abstract

Dynamical properties of laser models based on the Maxwell Bloch equation are studied. Instances of stability and chaotic behavior are investigated. Special solutions of the system one of which reduces to the Lotka Volterra system under simplifying assumptions are derived. Reasons for the absence of oscillating solutions in the modified systems are studied.

Identifying weak foci and centers in the Maxwell–Bloch system

“…As known in [15], the authors give the qualitative properties of all equilibria for system (1.1). The reduced system (1.2) containing less parameters helps us in latter computation of center manifolds, normal forms and those determining quantities.…”

“…In 2010 Hacinliyan, Kusbeyzi and Aybar [15] indicated that the pair of equilibria which are C 2 -symmetric with respect to the Δ-axis are both of center-focus type and showed the rise of a stable limit cycle from a Hopf bifurcation, verifying that a pair of conjugate complex eigenvalues crosses the imaginary axis and varying the parameter g to obtain a negative first Lyapunov coefficient numerically. However, the work of [15] neither answers the order of weak foci nor identifies a center. Actually, it causes great complexity in computation to identify weak foci and centers for 3-dimensional systems [9,14,[19][20][21].…”

“…Arecchi [1] proposed the 3-dimensional quadratic differential system ⎧ ⎪ ⎨ ⎪ ⎩Ė = −k E + g P, P = −γ ⊥ P + g EΔ, Δ = −γ (Δ − δ 0 ) − 4g P E, (1.1) called the Maxwell-Bloch system, where E is the coupling of the fundamental cavity mode, P is the collective atomic polarization, Δ represents the population inversion, k, γ ⊥ , γ are the loss rates of field, polarization and population respectively, g is a coupling constant (g = 0) and δ 0 is the population inversion which would be established by the pump mechanism in the atomic medium, in the absence of coupling. As indicated in [1,15,17], the Maxwell-Bloch system describes Type I laser (He-Ne), Type II laser (Ruby and CO 2 ) and Type III laser (far infrared) when γ ⊥ ≈ γ k, when γ ⊥ γ ≈ k and when Δ 0 is large enough, respectively. Numerical simulations [1] show that Type I laser and Type II laser are both stable but Type III laser is unstable.…”

“…A few papers [15,23] were published to discuss this 3-dimensional system in mathematical aspect. In 1998 Puta [23] considered the integrability in the special case that k = γ ⊥ = γ = 0.…”

The diffusive Lotka–Volterra predator–prey system with delay

“…This model can also be used to describe chemical reactions and physical systems such as resonantly coupled lasers [11,15]. Many theoretical and experimental studies have considered the stability of the Lotka-Volterra predator-prey model.…”