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We show two examples of noise--induced synchronization. We study a 1-d map and the Lorenz systems, both in the chaotic region. For each system we give numerical evidence that the addition of a (common) random noise, of large enough intensity, to different trajectories which start from different initial conditions, leads eventually to the perfect synchronization of the trajectories. The largest Lyapunov exponent becomes negative due to the presence of the noise terms., Comment: 5 pages, uses aipproc.cls and aipproc.sty (included). Five double figures are provided as ten separate gif files. Version with (large) postscript figures included available from http://www.imedea.uib.es/PhysDept/publicationsDB/date.html

We study the effect of spatial frequency-forcing on standing-wave solutions of coupled complex Ginzburg-Landau equations. The model considered describes several situations of nonlinear counterpropagating waves and also of the dynamics of polarized light waves. We show that forcing introduces spatial modulations on standing waves which remain frequency locked with a forcing-independent frequency. For forcing above a threshold the modulated standing waves unlock, bifurcating into a temporally periodic state. Below the threshold the system presents a kind of excitability., Comment: 4 pages, including 4 postscript figures. To appear in Physical Review Letters (1996). This paper and related material can be found at http://formentor.uib.es/Nonlinear/

We numerically study in the one-dimensional case the validity of the functional calculated by Graham and coworkers as a Lyapunov potential for the Complex Ginzburg-Landau equation. In non-chaotic regions of parameter space the functional decreases monotonically in time towards the plane wave attractors, as expected for a Lyapunov functional, provided that no phase singularities are encountered. In the phase turbulence region the potential relaxes towards a value characteristic of the phase turbulent attractor, and the dynamics there approximately preserves a constant value. There are however very small but systematic deviations from the theoretical predictions, that increase when going deeper in the phase turbulence region. In more disordered chaotic regimes characterized by the presence of phase singularities the functional is ill-defined and then not a correct Lyapunov potential., Comment: 20 pages,LaTeX, Postcript version with figures included available at http://formentor.uib.es/~montagne/textos/nepg

A classification of dynamical systems in terms of their variational properties is reviewed. Within this classification, front propagation is discussed in a non-gradient relaxational potential flow. The model is motivated by transient pattern phenomena in nematics. A front propagating into an unstable homogenous state leaves behind an unstable periodic pattern, which decays via a second front and a second periodic state. An interface between unstable periodic states is shown to be a source of propagating fronts in opposite directions., Comment: 13 pages of text including 6 Figures in a tar-compressed and uuencoded file (using uufiles package). To appear in "Instabilities and Nonequilibrium Structures V", Eds. E. Tirapegui and W. Zeller, Kluwer, Dordrecht, 1995. Latex version with figures available at http://formentor.uib.es/~montagne/textos/chile93

We study damage spreading in models of two-dimensional systems undergoing first order phase transitions. We consider several models from the same non-conserved order parameter universality class, and find unexpected differences between them. An exact solution of the Ohta-Jasnow-Kawasaki model yields the damage growth law $D \sim t^{\phi}$, where $\phi = t^{d/4}$ in $d$ dimensions. In contrast, time-dependent Ginzburg-Landau simulations and Ising simulations in $d= 2$ using heat-bath dynamics show power-law growth, but with an exponent of approximately $0.36$, independent of the system sizes studied. In marked contrast, Metropolis dynamics shows damage growing via $\phi \sim 1$, although the damage difference grows as $t^{0.4}$. PACS: 64.60.-i, 05.50.+q, Comment: 4 pags of revtex3 + 3 postscript files appended as a compressed and uuencoded file. UIB9403200

The dynamics of transient patterns formed by front propagation in extended nonequilibrium systems is considered. Under certain circumstances, the state left behind a front propagating into an unstable homogeneous state can be an unstable periodic pattern. It is found by a numerical solution of a model of the Fr\'eedericksz transition in nematic liquid crystals that the mechanism of decay of such periodic unstable states is the propagation of a second front which replaces the unstable pattern by a another unstable periodic state with larger wavelength. The speed of this second front and the periodicity of the new state are analytically calculated with a generalization of the marginal stability formalism suited to the study of front propagation into periodic unstable states. PACS: 47.20.Ky, 03.40.Kf, 47.54.+r, Comment: 12 pages

We study the effect of fluctuations in the vicinity of an Eckhaus instability. The classical stability limit, which is defined in the absence of fluctuations, is smeared out into a region in which fluctuations and nonlinearities dominate the decay of unstable states. The width of this region is shown to grow as $ D^{1/2} $, where $D$ is the intensity of the fluctuations. We find an effective stability boundary that depends on $D$. A numerical solution of the stochastic Swift-Hohenberg equation in one dimension is used to test this prediction and to study pattern selection when the initial unstable state lies within the fluctuation dominated region. The asymptotically selected state differs from the predictions of previous analyses. Finally, the nonlinear relaxation for $D > 0$ is shown to exhibit a scaling form., Comment: 13 pages REVTeX + 2 postscript figures

We introduce and analyze a general one-dimensional model for the description of transient patterns which occur in the evolution between two spatially homogeneous states. This phenomenon occurs, for example, during the Freedericksz transition in nematic liquid crystals.The dynamics leads to the emergence of finite domains which are locally periodic and independent of each other. This picture is substantiated by a finite-size scaling law for the structure factor. The mechanism of evolution towards the final homogeneous state is by local roll destruction and associated reduction of local wavenumber. The scaling law breaks down for systems of size comparable to the size of the locally periodic domains. For systems of this size or smaller, an apparent nonlinear selection of a global wavelength holds, giving rise to long lived periodic configurations which do not occur for large systems. We also make explicit the unsuitability of a description of transient pattern dynamics in terms of a few Fourier mode amplitudes, even for small systems with a few linearly unstable modes., Comment: 18 pages (REVTEX) + 10 postscript figures appended

We present a theoretical and numerical investigation of the effect of a time-varying external driving force on interface growth. First, we derive a relation between the roughening exponents which comes from a generalized Galilean invariance, showing how the critical dimension of the model is tunable with the external field. We further conjecture results for the exponents in two dimensions, and find consistency with data obtained through simulations of two models we expect to be in the same universality class. Finally, we discuss how our results can be investigated experimentally., Comment: 11 pages (LATEX file with a .PS file appended for two figures)