zbMATH — the first resource for mathematics

Bifurcation analysis of the Eckhaus instability. (English) Zbl 0721.35008
Summary: The bifurcation diagram for the Eckhaus instability is presented, based on the Ginzburg-Landau equation in a finite domain with either free-slip or periodic boundary conditions. The conductive state is shown to undergo a sequence of destabilizing bifurcations giving rise to branches of pure- mode states; all branches but the first are necessarily unstable at onset. Each pure-mode branch undergoes a sequence of secondary restabilizing bifurcations, the last of which is shown to correspond to the Eckhaus instability. The restabilizing bifurcations arise from mode interactions between the pure-mode branches, and can be related directly to the destabilizing bifurcations of the conductive state. The downwards shift of the Eckhaus parabola calculated by Kramer and Zimmerman for the case of finite geometry is stressed. Through a center manifold reduction, it is proved that for the Ginzburg-Landau equation all restabilizing bifurcations of the pure-mode states are subcritical, and hence that the Eckhaus instability is itself subcritical.

35B32 Bifurcations in context of PDEs
35Q40 PDEs in connection with quantum mechanics
Full Text: DOI
[1] Eckhaus, W., Studies in nonlinear stability theory, (1965), Springer Berlin · Zbl 0125.33101
[2] Newell, A.C.; Whitehead, J., J. fluid mech., 38, 279, (1969)
[3] Segel, L.A., J. fluid mech., 38, 203, (1969)
[4] Zippelius, A.; Siggia, E., Phys. fluids, 26, 2905, (1983)
[5] Cross, M.C.; Newell, A.C., Physica D, 10, 299, (1984)
[6] Pomeau, Y.; Manneville, P., J. phys. lett., 40, 609, (1979)
[7] Kramer, L.; Zimmerman, W., Physica D, 16, 221, (1985)
[8] Daniels, P.G., (), 25, 539, (1978)
[9] Cross, M.C.; Daniels, P.G.; Hohenberg, P.C.; Siggia, E.D., J. fluid mech., 127, 15, (1983)
[10] Zaleski, S., J. fluid mech., 149, 101, (1984)
[11] Pomeau, Y.; Zaleski, S., J. phys. Paris, 42, 515, (1981)
[12] Riecke, H.; Paap, H.-G., Phys. rev. A, 33, 547, (1986)
[13] P. Coullet and L. Gil, preprint
[14] Croquette, V.; Pocheau, A., ()
[15] Ahlers, G.; Cannella, D.S.; Dominguez-Lerma, M.A.; Heinrichs, R., Physica D, 23, 202, (1986)
[16] Lowe, M.; Gollub, J.P., Phys. rev. lett., 55, 2575, (1985)
[17] B. Matkowsky, private communication.
[18] Leray, J.; Schauder; Benjamin, T.B., (), 51, 373, (1934)
[19] Golubitsky, M.; Schaeffer, D.; Golubitsky, M.; Schaeffer, D.; Stewart, I., ()
[20] Arnol’d, V.I., Geometrical methods in the theory of ordinary differential equations, (1983), Springer Berlin · Zbl 0569.58018
[21] Palis, J.; de Melo, W., Geometric theory of dynamical systems, (1982), Springer Berlin
[22] Bauer, L.; Keller, H.B.; Reiss, E.L.; Bauer, L.; Keller, H.B.; Reiss, E.L., SIAM rev., SIAM rev., 101, (1975)
[23] Keener, J.P.; Keener, J.P.; Schaeffer, D.; Schaeffer, D.; Shearer, M.; Shearer, M.; Riley, D.S.; Winters, K.H.; Riley, D.S.; Winters, K.H., (), SIAM J. math. anal., SIAM J. math. anal., J. fluid mech., J. fluid mech., 325, (1989)
[24] Seydel, R.; Seydel, R., Zamp, Zamp, 713, (1975)
[25] Seydel, R., From equilibrium to chaos: practical bifurcation and stability analysis, (1988), Elsevier New York · Zbl 0652.34059
[26] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001
[27] Tsiveriotis, K.; Brown, R.A., Phys. rev. lett., 63, 2048, (1989)
[28] L.S. Tuckerman and D. Barkley, Phys. Rev. Lett., submitted for publication.
[29] Tuckerman, L.S.; Barkley, D.; Barkley, D.; Tuckerman, L.S., Phys. rev. lett., Physica D, 37, 288, (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.