scholarly journals An analytical connection between temporal and spatio-temporal growth rates in linear stability analysis

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130171 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Peter D. M. Spelt
Keyword(s):  
Stability Analysis ◽  
Landau Equation ◽  
Temporal Stability ◽  
Exact Formula ◽  
Complex Frequency ◽  
Ginzburg Landau ◽  
First Case ◽  
Wake Instability ◽  
Spatio Temporal ◽  
Parameter Values

We derive an exact formula for the complex frequency in spatio-temporal stability analysis that is valid for arbitrary complex wavenumbers. The usefulness of the formula lies in the fact that it depends only on purely temporal quantities, which are easily calculated. We apply the formula in two model dispersion relations: the linearized complex Ginzburg–Landau equation, and a model of wake instability. In the first case, a quadratic truncation of the exact formula applies; in the second, the same quadratic truncation yields an estimate of the parameter values at which the transition to absolute instability occurs; the error in the estimate decreases upon increasing the order of the truncation. We outline ways in which the formula can be used to characterize stability results obtained from purely numerical calculations, and point to a further application in global stability analyses.

scholarly journals Controlling spatio-temporal chaos in the scenario of the one-dimensional complex Ginzburg–Landau equation

2006 ◽  
Vol 364 (1846) ◽  
pp. 2383-2395 ◽  
Author(s):  
S Boccaletti ◽  
J Bragard
Keyword(s):  
Plane Wave ◽  
Landau Equation ◽  
Control Technique ◽  
Ginzburg Landau ◽  
Plane Wave Solution ◽  
One Dimensional ◽  
Ginzburg Landau Equation ◽  
System Extension ◽  
Spatio Temporal ◽  
The One

We discuss some issues related with the process of controlling space–time chaotic states in the one-dimensional complex Ginzburg–Landau equation. We address the problem of gathering control over turbulent regimes with the use of only a limited number of controllers, each one of them implementing, in parallel, a local control technique for restoring an unstable plane-wave solution. We show that the system extension does not influence the density of controllers needed in order to achieve control.

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

2009 ◽  
Vol 465 (2107) ◽  
pp. 2251-2265 ◽  
Author(s):  
Laurent Nana ◽  
Alexander B. Ezersky ◽  
Innocent Mutabazi
Keyword(s):  
Boundary Conditions ◽  
Landau Equation ◽  
State Diagram ◽  
Stable Region ◽  
Nonlinear Frequency ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Dean Flow ◽  
Ginzburg Landau Equation ◽  
Spatio Temporal

Experiments in extended systems, such as the counter-rotating Couette–Taylor flow or the Taylor–Dean flow system, have shown that patterns with vanishing amplitude may exhibit periodic spatio-temporal defects for some range of control parameters. These observations could not be interpreted by the complex Ginzburg–Landau equation (CGLE) with periodic boundary conditions. We have investigated the one-dimensional CGLE with homogeneous boundary conditions. We found that, in the ‘Benjamin–Feir stable’ region, the basic wave train bifurcates to state with periodic spatio-temporal defects. The numerical results match the observations quite well. We have built a new state diagram in the parameter plane spanned by the criticality (or equivalently the linear group velocity) and the nonlinear frequency detuning.

Dromion-like structures and stability analysis in the variable coefficients complex Ginzburg–Landau equation

2015 ◽  
Vol 360 ◽  
pp. 341-348 ◽  
Author(s):  
Pring Wong ◽  
Li-Hui Pang ◽  
Long-Gang Huang ◽  
Yan-Qing Li ◽  
Ming Lei ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Landau Equation ◽  
Variable Coefficients ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

Spatio-temporal stability analysis of linear arrays of 2D density stratified wakes and jets

2018 ◽  
Vol 30 (11) ◽  
pp. 114103 ◽  
Author(s):  
Jacob Sebastian ◽  
Benjamin Emerson ◽  
J. O’Connor ◽  
Tim Lieuwen
Keyword(s):  
Stability Analysis ◽  
Temporal Stability ◽  
Linear Arrays ◽  
Spatio Temporal

scholarly journals Amplitude Equations and Chemical Reaction–Diffusion Systems

1997 ◽  
Vol 07 (07) ◽  
pp. 1539-1554 ◽  
Author(s):  
M. Ipsen ◽  
F. Hynne ◽  
P. G. Sørensen
Keyword(s):  
Diffusion Equation ◽  
Chemical Reaction ◽  
Bifurcation Point ◽  
Landau Equation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Amplitude Equations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatio Temporal

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.

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