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Transitions to chaos in the Ginzburg-Landau equation. (English) Zbl 0558.58030
Order in chaos, Proc. int. Conf., Los Alamos/N.M. 1982, Physica 7D, 135-150 (1983).
Summary: The amplitude evolution of instability waves in many dissipative systems is described close to criticality, by the Ginzburg-Landau partial differential equation. A numerical study of the long-time behaviour of amplitude-modulated waves governed by this equation allows the identification of two distinct routes of the Ruelle-Takens-Newhouse type as the modulation wavenumber is decreased. The first route involves a sequence of bifurcations from a limit cycle to a two-torus to a three- torus and to a turbulent régime, the last stage being preceded by frequency locking. The turbulent régime is itself followed by a new two-torus. In the second route, this two-torus exhibits a single subharmonic bifurcation which immediately results in transition to chaos. A description of the various possible dynamical states is tentatively given in the plane of the two control parameters \(c_ d\) and \(c_ n\).

MSC:
58J99 Partial differential equations on manifolds; differential operators
37G99 Local and nonlocal bifurcation theory for dynamical systems
35Q56 Ginzburg-Landau equations
References:
[1] Lorenz, E. N.: J. atmos. Sci.. 20, 130 (1963)
[2] Curry, J. H.: Commun. math. Phys.. 60, 193 (1978)
[3] Curry, J. H.: Phys. rev. Lett.. 43, 1013 (1979)
[4] Mclaughlin, J.; Martin, P. C.: Phys. rev.. 12, 186 (1975)
[5] Newell, A. C.: Lect. appl. Math.. 15, 157 (1974)
[6] Gibbon, J. D.; Mcguinness, M. J.: Proc. R. Soc. lond.. 377, 185 (1981)
[7] Stewartson, K.; Stuart, J. T.: J. fluid mech.. 48, 529 (1971)
[8] Hocking, L. M.; Stewartson, K.; Stuart, J. T.: J. fluid mech.. 51, 705 (1972)
[9] Newell, A. C.; Whitehead, J. A.: J. fluid mech.. 38, 279 (1969)
[10] Kogelman, S.; Diprima, R. C.: Phys. fluids. 13, 1 (1970)
[11] Hasimoto, H.; Ono, H.: J. phys. Soc. Japan. 33, 805 (1972)
[12] Zakharov, V. E.; Shabat, A. B.: Soc. phys. JETP. 34, 62 (1972)
[13] Yuen, H. C.; Lake, B. M.: Ann. rev. Fluid mech.. 12, 303 (1980)
[14] Hafizi, B.: Phys. fluids. 24, 1791 (1981)
[15] Ma, Y. C.; Ablowitz, M. J.: Stud. appl. Math.. 65, 113 (1981)
[16] Howard, L. N.: Dynamics and modelling of reactive systems. (1980)
[17] Kopell, N.; Howard, L. N.: Stud. appl. Math.. 64, 1 (1981)
[18] Kuramoto, Y.; Tsuzuki, T.: Prog. theor. Phys.. 52, 1399 (1974)
[19] Kuramoto, Y.; Tsuzuki, T.: Prog. theor. Phys.. 54, 687 (1975)
[20] Kuramoto, Y.: Suppl. prog. Theor. phys.. 64, 346 (1978)
[21] Stuart, J. T.; Diprima, R. C.: Proc. R. Soc. lond.. 362, 27 (1978)
[22] Moon, H. T.: Transition to chaos in the Ginzburg-Landau equation. Ph.d. dissertation (1982)
[23] Fornberg, B.; Whitham, G. B.: Phil. trans. R. soc. Lond.. 289, 373 (1978)
[24] Moon, H. T.; Huerre, P.; Redekopp, L. G.: Phys. rev. Lett.. 49, 458 (1982)
[25] Gollub, J. P.; Benson, S. V.: J. fluid mech.. 100, 449 (1980)
[26] Ruelle, D.; Takens, F.: Commun. math. Phys.. 20, 167 (1971) · Zbl 0227.76084
[27] Newhouse, S.; Ruelle, D.; Takens, F.: Commun. math. Phys. 64, 35 (1978)
[28] S.J. Shenker, ”Scaling Behavior in a Map of the Circle onto itself: Empirical results”, University of Chicago, preprint.
[29] D.A. Rand, S. Ostlund, J. Sethna and E.D. Siggia, ”A universal transition from quasi-periodicity to chaos in dissipative systems”, preprint. · Zbl 0538.58025
[30] D.A. Rand, ”Dynamics and symmetry. Predictions for modulated waves in rotating fluids”, to appear in Arch. Rational Mech. Anal. · Zbl 0495.76031
[31] Gorman, M.; Swinney, H. L.; Rand, D. A.: Phys. rev. Lett.. 46, 992 (1981)
[32] Feigenbaum, M. J.: J. stat. Phys.. 19, 25 (1978)
[33] Feigenbaum, M. J.: Commun. math. Phys.. 77, 25 (1980)
[34] V. Franceschini, ”Bifurcations of tori and phase locking in a dissipative system of differential equations”, submitted to Physica D. · Zbl 1194.34075
[35] Nayfeh, A. H.: Perturbation methods. (1973) · Zbl 0265.35002
[36] Kuramoto, Y.; Tsuzuki, T.: Prog. theor. Phys.. 55, 356 (1976)
[37] Benney, D. J.: J. math. And phys.. 45, 150 (1966)
[38] Kuramoto, Y.: Prog. theor. Phys.. 63, 1885 (1980)
[39] Sivashinsky, G. I.; Michelson, D. M.: Prog. theor. Phys.. 63, 2112 (1980)
[40] J.P. Crutchfield and N.H. Packard, private communication.
[41] D. Bennett, A.R. Bishop and S.E. Trullinger, ”Coherence and chaos in the driven, damped sine-Gordon chain”, to appear in Zeitschrift für Physik.
[42] Farmer, J. D.: Physica. 4D, 366 (1982)
[43] Farmer, J. D.; Weidman, P. D.; Hart, J.: A phase space analysis of baroclinic flow. Phys. lett. 91A, 22 (1982)
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