The slow passage through a steady bifurcation: Delay and memory effects |
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Authors: | Paul Mandel Thomas Erneux |
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Affiliation: | (1) Service de Chimie-Physique II, Université Libre de Bruxelles, 1050 Bruxelles, Belgium;(2) Department of Engineering Sciences and Applied Mathematics, Northwestern University, 60201 Evanston, Illinois |
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Abstract: | We consider the following problem as a model for the slow passage through a steady bifurcation: dy/dt = (t) y – y3 +, where is a slowly increasing function oft given by=i + t (i,<0). Both and are small parameters. This problem is motivated by laser experiments as well as theoretical studies of laser problems. In addition, this equation is a typical amplitude equation for imperfect steady bifurcations with cubic nonlinearities. When=0, we have found that=0 is not the point where the bifurcation transition is observed. This transition appears at a value =j > 0. We call j the delay of the bifurcation transition. We study this delay as a function ofi, the initial position of, and, the imperfection parameter. To this end, we propose an asymptotic study of this equation as 0, small but fixed. Our main objective is to describe this delay in terms of the relative magnitude of and. Since time-dependent imperfections are always present in experiments, we analyze in the second part of the paper the effect of a small-amplitude but time-periodic imperfection given by (t) = cos(t). |
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Keywords: | Slowly varying bifurcation parameter effects of steady and time-periodic imperfections control of laser instabilities |
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