Abstract
We consider the following problem as a model for the slow passage through a steady bifurcation: dy/dt = γ(t) y − y3 +δ, whereλ is a slowly increasing function oft given byλ=λ i + εt (λ i,<0). Both ε and δ 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. Whenδ=0, we have found thatλ=0 is not the point where the bifurcation transition is observed. This transition appears at a valueλ =λ j > 0. We call λj the delay of the bifurcation transition. We study this delay as a function ofλ i, the initial position ofλ, andδ, the imperfection parameter. To this end, we propose an asymptotic study of this equation asδ → 0,ε small but fixed. Our main objective is to describe this delay in terms of the relative magnitude ofδ andε. 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 δ(t) = δ cos(σt).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Goldbeter and L. A. Segel,Differentiation 17:127–135 (1980).
A. K. Kapila,SIAM J. Appl. Math. 41:29–42 (1981).
G. Ahlers, M. C. Cross, P. C. Hohenberg, and S. Safran,J. Fluid Mech. 110:297–334 (1981).
M. C. Cross, P. C. Hohenberg, and M. Lucke,J. Fluid Mech. 136:269–276 (1983).
T. Erneux and P. Mandel,Phys. Rev. A 28:869–909 (1983).
M. Matalon and T. Erneux,SIAM J. Appl. Math. 44:734–744 (1984).
E. Arimondo, C. Gabbanini, E. Menchi, D. Dangoisse, and P. Glorieux, inOptical Instabilities (Cambridge Studies in Modern Optics, Vol. 4, 1986, pp. 277–280).
D. Dangoisse, P. Glorieux, and T. Midavaine, inOptical Instabilities (Cambridge Studies in Modern Optics, Vol. 4, 1986, pp. 293–296).
B. Morris and F. Moss,Phys. Lett. 118:117–120 (1986).
N. R. Lebovitz and R. J. Schaar,Stud. Appl. Math. 54:229–260 (1975);56:1–50 (1977).
R. Haberman,SIAM J. Appl. Math. 37:69–105 (1979).
T. Erneux and P. Mandel,Phys. Rev. A 28:896–909 (1983).
T. Erneux and P. Mandel,Phys. Rev. A 30:1902–1909 (1984).
P. Mandel and T. Erneux,Phys. Rev. Lett. 53:1818–1820 (1984).
T. Erneux and P. Mandel,SIAM J. Appl. Math. 46:1–16 (1986).
P. Mandel, inFrontiers in Quantum Optics, E. R. Pike and S. Sarkar, eds., pp. 430–452 (Adam Hilger, Bristol, 1986).
R. Kapral and P. Mandel,Phys. Rev. A 32:1076–1081 (1985).
G. Broggi, A. Colombo, L. A. Lugiato, and P. Mandel,Phys. Rev. A 33:3635–3637.
D. K. Kondepudi, F. Moss, and P. V. E. McClintock, Observation of symmetry breaking, state selection and sensitivity in a noisy electronic system,Physica 21D:296–306 (1986).
R. Mannella, F. Moss, and P. V. E. McClintock, Postponed bifurcation of a ring laser model with a swept parameter and additive coloured noise, Preprint.
C. G. Lange,Stud. Appl. Math. 68:227–257 (1983).
S. M. Baer and T. Erneux, Singular Hopf bifurcation to relaxation oscillations,SIAM J. Appl. Math. 46:721–739 (1986).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mandel, P., Erneux, T. The slow passage through a steady bifurcation: Delay and memory effects. J Stat Phys 48, 1059–1070 (1987). https://doi.org/10.1007/BF01009533
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01009533