Imperfect bifurcation with a slowly-varying control parameter
The authors consider a general class of imperfect bifurcation problems described by the following first order nonlinear differential equation: lambda/sub T/ = Ky/sup RHO/ + 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 << 1) and the time-dependent control parameter lambda(t) = lambda/sub 0/ + epsilont (lambda/sub 0/ < 0 and 0 < epsilon << 1). If delta = epsilon = 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, they analyze the perturbation of the bifurcation solutions produced both by the small imperfection (delta not equal to 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/sub 1/. Provided that delta is sufficiently small compared to epsilon, lambda/sub 1/ is an 0(1) quantity which only depends on lambda/sub 0/ i.e., the initial position of lambda(t). The analysis is motivated by problems appearing in laser physics. In the second part of the paper, they 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.
- Research Organization:
- Dept. of Engineering Sciences and Applied Mathematics, The Technological Inst., Northwestern Univ., Evanston, IL 60201
- DOE Contract Number:
- AC02-78ER04650
- OSTI ID:
- 5679882
- Journal Information:
- SIAM J. Appl. Math.; (United States), Journal Name: SIAM J. Appl. Math.; (United States) Vol. 46:1; ISSN SMJMA
- Country of Publication:
- United States
- Language:
- English
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