Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations

By a suitable variable transformation we show that detuned lasers can be described by complex Lorenz equations and establish a close analogy between detuned lasers and baroclinic instability. The analogy enables us to get all analytical and exact results for the second threshold of the detuned single-mode lasers. In order to discriminate sub- from supercritical Hopf bifurcations, we use a combined approach of elimination procedure and normal form techniques to make a systematic calculation of the criterion for both of the systems. The influence of the parameter variations on the nature of the bifurcation is discussed in detail. For detuned lasers it is shown that, if we restrict ourselves to the case of b≤1 (b=${\gamma }_{\mathrm{?}}$/${\gamma }_{\mathrm{\perp }}$, i.e., the ratio of the longitudinal to the transversal relaxation constant of the atoms) and not-too-small k (=κ/${\gamma }_{\mathrm{\perp }}$, with κ denoting the cavity relaxation constant) and Δ, for given b and Δ (Δ and k or k and b), there exists a ${\mathit{k}}_{\mathit{c}}$ (${\mathit{b}}_{\mathit{c}}$ or ${\mathrm{\Delta }}_{\mathit{c}}$) such that the Hopf bifurcation is subcritical if k<${\mathit{k}}_{\mathit{c}}$ (b>${\mathit{b}}_{\mathit{c}}$ or Δ<${\mathrm{\Delta }}_{\mathit{c}}$) and supercritical if k>${\mathit{k}}_{\mathit{c}}$ (b<${\mathit{b}}_{\mathit{c}}$ or Δ>${\mathrm{\Delta }}_{\mathit{c}}$). Numerical investigations show that period-doubling bifurcations to chaos exist not only for the subcritical but also for the supercritical case.