Transitions to chaos in the Ginzburg-Landau equation

Moon, Hie Tae; Huerre, Patrick; Redekopp, L. G.

doi:10.1016/0167-2789(83)90124-0

Cited by 90 publications

(48 citation statements)

References 29 publications

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“…The equations are scaled so that these terms are +1 and -1 respectively. Interesting results for the case that plane wave solutions are unstable, were obtained by Schopf and Kramer [19], Sirovich et al [13,14,18,21,22], Keefe [5], Kuramoto and Koga [7], Moon et al [10], Nozaki and Bekki [15], and others. We set r = u + iv and seek normal mode solutions of the form 6) where u and v are constants.…”

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confidence: 98%

Stability of plane wave solutions of complex Ginzburg-Landau equations

Matkowsky¹,

Volpert²

1993

Quart. Appl. Math.

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Abstract. We consider the stability of plane wave solutions of both single and coupled complex Ginzburg-Landau equations and determine stability domains in the space of coefficients of the equations. [6,11] and the references therein).Actually, CGL equations describe the evolution of the amplitudes of unstable modes for any process exhibiting Hopf bifurcation, for which the continuous band of unstable wave numbers is taken into account. Therefore, the equations have become a self-significant object of study.Coupled CGL equations govern the amplitudes, on slow spatial and temporal scales, of traveling waves propagating in opposite directions on the faster temporal and spatial scales associated with the original problem from which the GinzburgLandau equations were derived. The case when one of the amplitudes equals zero corresponds to a traveling wave solution of the original problem, while the case when the moduli of the two amplitudes are equal corresponds to a standing wave.The simplest solutions of CGL equations are plane wave solutions and the present paper is devoted to an analysis of their stability. Stability of traveling wave solutions of a single CGL equation, which arises when the most unstable wave number is

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confidence: 98%

Stability of plane wave solutions of complex Ginzburg-Landau equations

Matkowsky¹,

Volpert²

1993

Quart. Appl. Math.

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“…Eq. (1.1) with periodic boundary conditions is studied by Doering et al [1] and Ghidaglia and H&on [2]: in these papers (sharp) estimates on the dimension of the (chaotic) attractor are derived; numerical studies of this situation appeared in Moon et al [3] and Keefe [4] (and other papers). The stability of periodic wave solutions (~ = R e itkx+'~t)) is investigated in Stuart and DiPrima [5]; slowly varying waves are studied by Bernhoff [6].…”

Section: Introductionmentioning

confidence: 99%

Slow time-periodic solutions of the Ginzburg-Landau equation

Doelman

1989

Physica D: Nonlinear Phenomena

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In this paper we study the behaviour of solutions of the form if(z, t) = q~(z) e-i~wt (e << 1) of the rescaled Ginzburg-Landau equation, ~k, = [1-(1 + iB)l~kl2]~k + (1 + iA)~kzz, for A = ca, B = eb, w plays the role of free parameter. This leads to a perturbation analysis on a complex Duffing equation (similar to the analysis of Holmes [Physica D 23 (1986) 84]). We show that the spatial quasiperiodic solutions (of the unperturbed, e = 0, case) disappear due to the perturbation and prove the existence of degenerated periodic solutions which oscillate through the origin. We also establish the existence of several types of heteroclinic orbits connecting counterrotating periodic patterns.

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“…More recently the GinzburgLandau equation has become an increasingly popular subject of study, in particular since the possibility of chaotic behaviour has been discovered (for example refs. [1, 11,13,14]). On the other hand, in the non-chaotic case, Kramer and Zimmerman [12], for the case of real coefficients (A = B = 0), have studied stationary quasi-periodic and homoclinic solutions.…”

Section: Introductionmentioning

confidence: 99%