Secondary structures in a one-dimensional complex Ginzburg–Landau equation with homogeneous boundary conditions

Nana, Laurent; Ezersky, Alexander; Mutabazi, Innocent

doi:10.1098/rspa.2009.0002

Cited by 14 publications

(11 citation statements)

References 39 publications

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“…[27,42] in the study of the amplitude of defects in spiral pattern observed in Newtonian Couette-Taylor flow between counter-rotating cylinders and in the Taylor-Dean system [24]. They confirm the fact that the spatio-temporal defects are characterized by vanishing amplitude A = 0 [20,29,32].…”

Section: B Velocity Field Of the Ribbon Patternssupporting

confidence: 69%

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Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow

et al. 2016

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Transition to defect-mediated turbulence in the ribbon patterns observed in a viscoelastic Taylor-Couette flow is investigated when the rotation rate of the inner cylinder is increased while the outer cylinder is fixed. In four polymer solutions with different values of the elasticity number, the defects appear just above the onset of the ribbon pattern and trigger the appearance of disordered oscillations when the rotation rate is increased. The flow structure around the defects is determined and the statistical properties of these defects are analyzed in the framework of the complex Ginzburg-Landau equation.

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Section: B Velocity Field Of the Ribbon Patternssupporting

confidence: 69%

“…Theoretical analysis of the defects dynamics has been developed by Refs. [20,[30][31][32] in the framework of the complex Ginzburg-Landau equation (CGLE) describing the space-time dynamics of the complex order parameter A of the field pattern:…”

Section: Introductionmentioning

confidence: 99%

Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow

et al. 2016

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“…We note the similarity of the flowfield with the knotted wTWI solution in figure 22, from which we infer that our solution is a streamwise and spanwise localized, or knotted, wTWI. The knot features of this solution could also be analysed in terms of the defects observed in Taylor-Couette and Taylor-Dean systems as discussed in Bot & Mutabazi (2000), Nana et al (2009), Ezersky et al (2010 and Abcha et al (2013). Nana et al (2009) and Ezersky et al (2010) use a complex Ginsburg-Landau equation model to defects of spiral solutions in Taylor-Couette flow, and it is plausible that the production of knots in our localized solution could be a result of the same mechanism, though we do not pursue evidence for this here.…”

Section: Romentioning

confidence: 99%

Secondary instability and tertiary states in rotating plane Couette flow

et al. 2014

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Recent experimental studies have shown rich transition behaviour in rotating plane Couette flow (RPCF). In this paper we study the transition in supercritical RPCF theoretically by determination of equilibrium and periodic orbit tertiary states via Floquet analysis on secondary Taylor vortex solutions. Two new tertiary states are discovered which we name oscillatory wavy vortex flow (oWVF) and skewed vortex flow (SVF). We present the bifurcation routes and stability properties of these new tertiary states, and in addition, we describe a bifurcation procedure whereby a set of defected wavy twist vortices are approached. Further to this, transition scenarios at flow parameters relevant to experimental works are investigated by computation of the set of stable attractors which exist on a large domain. The physically observed flow states are shown to share features with states in our set of attractors.

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“…In the case of the finite domain, we use as initial condition a pulse solution. Wave patterns are described by CGL equation in which the amplitude of the wave pattern vanishes at the lateral boundaries of the domain in order to retrieve numerically some coherent structures observed experimentally, in the case of absolute or convective instabilities [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamical Regimes and Control of Turbulence through the Complex Ginzburg-Landau Equation

Tafo¹,

Nana²,

Tabi³

et al. 2020

Research Advances in Chaos Theory

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The dynamical behavior of pulse and traveling hole in a one-dimensional system depending on the boundary conditions, obeying the complex Ginzburg-Landau (CGL) equation, is studied numerically using parameters near a subcritical bifurcation. In a spatially extended system, the criterion of Benjamin-Feir-Newell (BFN) instability near the weakly inverted bifurcation is established, and many types of regimes such as laminar regime, spatiotemporal regime, defect turbulence regimes, and so on are observed. In finite system by using the homogeneous boundary conditions, two types of regimes are detected mainly the convective and the absolute instability. The convectively unstable regime appears below the threshold of the parameter control, and beyond, the absolute regime is observed. Controlling such regimes remains a great challenge; many methods such as the nonlinear diffusion parameter control are used. The unstable traveling hole in the onedimensional cubic-quintic CGL equation may be effectively stabilized in the chaotic regime. In order to stabilize defect turbulence regimes, we use the global time-delay auto-synchronization control; we also use another method of control which consists in modifying the nonlinear diffusion term. Finally, we control the unstable regimes by adding the nonlinear gradient term to the system. We then notice that the chaotic system becomes stable under strong nonlinearity.

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Secondary structures in a one-dimensional complex Ginzburg–Landau equation with homogeneous boundary conditions

Cited by 14 publications

References 39 publications

Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow

Defect-mediated turbulence in ribbons of viscoelastic Taylor-Couette flow

Secondary instability and tertiary states in rotating plane Couette flow

Nonlinear Dynamical Regimes and Control of Turbulence through the Complex Ginzburg-Landau Equation

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