Localized structures in coupled Ginzburg–Landau equations

Physica D: Nonlinear Phenomena, 2001

We introduce a general system of two coupled cubic complex Ginzburg-Landau (GL) equations that admits exact solitarypulse (SP) solutions with a stable zero background. Besides representing a class of systems of the GL type, it also describes a dual-core nonlinear optical fiber with gain in one core and losses in the other. By means of systematic simulations, we study generic transformations of SPs in this system, which turn out to be: cascading multiplication of pulses through a subcritical Hopf bifurcation, which eventually leads to a spatio-temporal chaos; splitting of SP into stable traveling pulses; and a symmetry-breaking bifurcation transforming a standing SP into a traveling one. In some parameter region, the Hopf bifurcation is found to be supercritical, which gives rise to stable breathers. Travelling breathers are also possible in the system considered. In a certain parameter region, stable standing SPs, moving permanent-shape ones, and traveling breathers all coexist. In that case, we study collisions between various types of the pulses, which, generally, prove to be strongly inelastic.

SIAM Journal on Applied Dynamical Systems, 2004

We study in this article the bifurcation and stability of the solutions of the Ginzburg-Landau equation, using a notion of bifurcation called attractor bifurcation. We obtain in particular a full classification of the bifurcated attractor and the global attractor as λ crosses the first critical value of the linear problem. Bifurcations from the rest of the eigenvalues of the linear problem are obtained as well.

Physics Letters A, 2005

Stationary to pulsating soliton bifurcation analysis of the complex Ginzburg-Landau equation (CGLE) is presented. The analysis is based on a reduction from an infinite-dimensional dynamical dissipative system to a finite-dimensional model. Stationary solitons, with constant amplitude and width, are associated with fixed points in the model. For the first time, pulsating solitons are shown to be stable limit cycles in the finite-dimensional dynamical system. The boundaries between the two types of solutions are obtained approximately from the reduced model. These boundaries are reasonably close to those predicted by direct numerical simulations of the CGLE.

International Journal of Bifurcation and Chaos, 1999

We study the spatiotemporal dynamics, in one and two spatial dimensions, of two complex fields which are the two components of a vector field satisfying a vector form of the complex Ginzburg-Landau equation. We find synchronization and generalized synchronization of the spatiotemporally chaotic dynamics. The two kinds of synchronization can coexist simultaneously in different regions of the space, and they are mediated by localized structures. A quantitative characterization of the degree of synchronization is given in terms of mutual information measures.

Discrete and Continuous Dynamical Systems, 1999

It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial 2π-periodic vortex solutions that have 2n simple zeros ("vortices") per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutions of the CGL equation. 1991 Mathematics Subject Classification. 35K55, 35Q35, 58F14.

The phenomenon of time-periodic evolution of spatial chaos is investigated in the frames of one-and two-dimensional complex Ginzburg-Landau equations. It is found that there exists a region of the parameters in which disordered spatial distribution of the field behaves periodically in time; the boundaries of this region are determined. The transition to the regime of spatiotemporal chaos is investigated and the possibility of describing spatial disorder by a system of ordinary differential equations is analyzed. The effect of the size of the system on the shape and period of oscillations is investigated. It is found that in the two-dimensional case the regime of time-periodic spatial disorder arises only in a narrow strip, the critical width of which is estimated. The phenomenon investigated in this paper indicates that a family of limit cycles with finite basins exists in the functional phase space of the complex Ginzburg-Landau equation in finite regions of the parameters.

Physica Scripta, 1996

ABSTRACT The two-dimensional Ginzburg-Landau (GL) equation in the weakly dissipative regime (real parts of the coefficients are assumed to be small in comparison with the imaginary ones) is considered in a square cell with reflecting (Neumann) boundary conditions. Following the lines of the analysis developed earlier for the analogous 1D equation, we demonstrate that, near the threshold of the modulational instability, the GL equation can be consistently approximated by a five-dimensional dynamical system which possesses a three-dimensional attracting invariant manifold. On the manifold, the dynamics are governed by a modified Lorenz model containing an additional cubic term. By means of numerical simulations of this approximation, a diagram of dynamical regimes is constructed, in a relevant parameter space. A region of chaos is found. Unlike the previously studied case of the 1D GL equation, in the present case a blow-up is possible, depending on initial conditions.

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

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.

2000

We characterize the synchronization of two nonidentical spatially extended fields ruled by onedimensional Complex Ginzburg-Landau equations, in the two regimes of phase and amplitude turbulence. If two fields display the same dynamical regime, the coupling induces a transition to a completely synchronized state. When, instead, the two fields are in different dynamical regimes, the transition to complete synchronization is mediated by defect synchronization. In the former case, the synchronized manifold is dynamically equivalent to that of the unsynchronized systems, while in the latter case the synchronized state substantially differs from the unsynchronized one, and it is mainly dictated by the synchronization process of the space-time defects.

Physica D: Nonlinear Phenomena, 2005

Numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one-dimensional complex Ginzburg-Landau equation (CGL) on the unit, spatially periodic domain. These cycles connect different spatially and temporally inhomogeneous time-periodic solutions as t → ±∞. A careful analysis of the connections is made using a projection onto 5 complex Fourier modes. It is shown first that the time-periodic solutions can be treated as (relative) equilibria after consideration of the symmetries of the CGL. Second, the cycles are shown to be robust since the individual heteroclinic connections exist in invariant subspaces. Thirdly, after constructing appropriate Poincaré maps around the cycle, a criteria for temporal stability is established, which is shown numerically to hold in specific parameter regions where the cycles are found to be of Shil'nikov type. This criterion is also applied to a much higher-mode Fourier truncation where similar results are found. In regions where instability of the cycles occurs, either Shilni'kov-Hopf or blow-out bifurcations are observed, with numerical evidence of competing attractors. Implications for observed spatio-temporal intermittency in situations modelled by the CGL are discussed.

Physica D: Nonlinear Phenomena, 1997

We study a globally coupled version of the complex Ginzburg-Landau equation (GC-CGLE) which consists of a large number N of identical two-dimensional oscillators coupled through their mean amplitude. Depending on parameter values, different dynamical regimes are attained. We focus particularly on an interesting regime where the individual oscillators follow erratic motion but in a sufficiently coherent way so that the average motion does not vanish when N becomes large and is also chaotic. A simple description of this state is proposed by considering the motion of a single forced two-dimensional system which has both a limit cycle and a fixed point as stable attractors. Determining which of these two deterministic attractors is selected by a weak noise and how this depends on the parameter of the reduced system allows us to determine self-consistently the average amplitude and dominant frequency of the collective behaviour of the full system. Finally, we show that adding a vmall noise to the GC-CGLE transforms the chaotic collective behaviour into a purely periodic one.

Abstract and Applied Analysis, 2012

Physica D-nonlinear Phenomena, 2003

Coupled Ginzburg-Landau equations appear in a variety of contexts involving instabilities in oscillatory media. When the relevant unstable mode is of vectorial character (a common situation in nonlinear optics), the pair of coupled equations has special symmetries and can be written as a vector complex Ginzburg-Landau (CGL) equation. Dynamical properties of localized structures of topological character in this vector-field case are considered. Creation and annihilation processes of different kinds of vector defects are described, and some of them interpreted in theoretical terms. A transition between different regimes of spatiotemporal dynamics is described.

Physica A: Statistical Mechanics and its Applications, 1996

After a brief introduction to the complex Ginzburg-Landau equation, some of its important features in two space dimensions are reviewed. A comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented. The nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed.

Physical Review E, 2005

Generalized chaotic synchronization regime is observed in the unidirectionally coupled one-dimensional Ginzburg-Landau equations. The mechanism resulting in the generalized synchronization regime arising in the coupled spatially extended chaotic systems demonstrating spatiotemporal chaotic oscillations has been described. The cause of the generalized synchronization occurrence is studied with the help of the modified Ginzburg-Landau equation with additional dissipation.