Imperfect Bifurcation with a Slowly-Varying Control Parameter

Abstract

We consider a general class of imperfect bifurcation problems described by the following first order nonlinear differential equation:\[ y_i = ky^p + \lambda (t)y + \delta , \] where $k = 1$ or $-1$ and $p = 2$ or 3 are fixed quantities. The solution depends on the values of the “imperfection” parameter $\delta (0 < \delta \ll 1)$ and the time-dependent control parameter $\lambda (t) = \lambda _0 + \varepsilon t$ and $0 < \varepsilon \ll 1$). If $\delta = \varepsilon = 0$, this equation admits at $\lambda = 0$ a bifurcation from the basic state $y = 0$ to nonzero steady states. In the first part of the paper, we analyze the perturbation of the bifurcation solutions produced both by the small imperfection $(\delta \ne 0)$ and the slow variation of $\lambda (\varepsilon \ne 0)$. We show that $\lambda = 0$ does not correspond to the transition between the two branches of slowly-varying steady states. This transition appears at a larger value of $\lambda = \lambda _1 $. Provided that $\delta $ is sufficiently small compared to $\varepsilon ,\lambda _1 $ is an 0(1) quantity which only depends on $\lambda _0 $, i.e., the initial position of $\lambda (t)$. Our analysis is motivated by problems appearing in laser physics. In the second part of the paper, we show how the semiclassical equations for the simple laser and the laser with a saturable absorber can be reduced to this simple first-order nonlinear equation. We then discuss the practical interests of our results.