Front propagation sustained by additive noise

Patchy landscapes in arid environments: Nonlinear analysis of the interaction-redistribution model

We consider a generic interaction-redistribution model of vegetation dynamics to investigate the formation of patchy vegetation in semi-arid and arid landscapes. First, we perform a weakly nonlinear analysis in the neighborhood of the symmetry-breaking instability. Following this analysis, we construct the bifurcation diagram of the biomass density. The weakly nonlinear analysis allows us to establish the condition under which the transition from super- to subcritical symmetry-breaking instability takes place. Second, we generate a random distribution of localized patches of vegetation numerically. This behavior occurs in regimes where a bare state coexists with a uniform biomass density. Field observations allow to estimate the total biomass density and the range of facilitative and competitive interactions.

Chaotic motion of localized structures

Mobility properties of spatially localized structures arising from chaotic but deterministic forcing of the bistable Swift-Hohenberg equation are studied and compared with the corresponding results when the chaotic forcing is replaced by white noise. Short structures are shown to possess greater mobility, resulting in larger root-mean-square speeds but shorter displacements than longer structures. Averaged over realizations, the displacement of the structure is ballistic at short times but diffusive at larger times. Similar results hold in two spatial dimensions. The effects of chaotic forcing on the stability of these structures is also quantified. Shorter structures are found to be more fragile than longer ones, and their stability region can be displaced outside the pinning region for constant forcing. Outside the stability region the deterministic fluctuations lead either to the destruction of the structure or to its gradual growth.

Bifurcations of front motion in passive and active Allen-Cahn-type equations

The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occurrence of different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyze fronts for more involved AC-type models. In particular, we consider a cubic-quintic variational AC model and two different nonvariational generalizations. We determine and compare the bifurcation diagrams of front solutions in the four considered models.

Front propagation steered by a high-wavenumber modulation: Theory and experiments

Homogeneously driven dynamical systems exhibit multistability. Depending on the initial conditions, fronts present a rich dynamical behavior between equilibria. Qualitatively, this phenomenology is persistent under spatially modulated forcing. However, the understanding of equilibria and front dynamics organization is not fully established. Here, we investigate these phenomena in the high-wavenumber limit. Based on a model that describes the reorientation transition of a liquid crystal light valve with spatially modulated optical forcing and the homogenization method, equilibria and fronts as a function of forcing parameters are studied. The forcing induces patterns coexisting with the uniform state in regions where the system without forcing is monostable. The front dynamics is characterized theoretically and numerically. Experimental results verify these phenomena and the law describing bistability, showing quite good agreement.

Phase Stochastic Resonance in a Forced Nanoelectromechanical Membrane

Stochastic resonance is a general phenomenon usually observed in one-dimensional, amplitude modulated, bistable systems. We show experimentally the emergence of phase stochastic resonance in the bidimensional response of a forced nanoelectromechanical membrane by evidencing the enhancement of a weak phase modulated signal thanks to the addition of phase noise. Based on a general forced Duffing oscillator model, we demonstrate experimentally and theoretically that phase noise acts multiplicatively, inducing important physical consequences. These results may open interesting prospects for phase noise metrology or coherent signal transmission applications in nanomechanical oscillators. Moreover, our approach, due to its general character, may apply to various systems.

Characterization of spatiotemporal chaos in a Kerr optical frequency comb and in all fiber cavities

Complex spatiotemporal dynamics have been a subject of recent experimental investigations in optical frequency comb microresonators and in driven fiber cavities with Kerr-type media. We show that this complex behavior has a spatiotemporal chaotic nature. We determine numerically the Lyapunov spectra, allowing us to characterize different dynamical behavior occurring in these simple devices. The Yorke-Kaplan dimension is used as an order parameter to characterize the bifurcation diagram. We identify a wide regime of parameters where the system exhibits a coexistence between the spatiotemporal chaos, the oscillatory localized structure, and the homogeneous steady state. The destabilization of an oscillatory localized state through radiation of counter-propagating fronts between the homogeneous and the spatiotemporal chaotic states is analyzed. To characterize better the spatiotemporal chaos, we estimate the front speed as a function of the pump intensity.

Recurrent noise-induced phase singularities in drifting patterns

We show that the key ingredients for creating recurrent traveling spatial phase defects in drifting patterns are a noise-sustained structure regime together with the vicinity of a phase transition, that is, a spatial region where the control parameter lies close to the threshold for pattern formation. They both generate specific favorable initial conditions for local spatial gradients, phase, and/or amplitude. Predictions from the stochastic convective Ginzburg-Landau equation with real coefficients agree quite well with experiments carried out on a Kerr medium submitted to shifted optical feedback that evidence noise-induced traveling phase slips and vortex phase-singularities.

Internal noise and system size effects induce nondiffusive kink dynamics

We investigate the effects of inherent fluctuations and system size in the dynamics of domain between uniform symmetric states. In the case of monotonous kinks, this dynamics is characterized by exhibiting nonsymmetric random walks, being attracted to the system borders. For nonmonotonous interface, the dynamics is replaced by a hopping dynamic. Based on bistable universal models, we characterize the origin of these unexpected dynamics through use of the stochastic kinematic laws for the interface position and the survival probability. Numerical simulations show a quite good agreement with the theoretical predictions.

Spatially modulated kinks in shallow granular layers

We report on the experimental observation of spatially modulated kinks in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow. We show the appearance of these solutions as the layer undergoes a parametric instability. Due to the inherent fluctuations of the granular layer, the kink profile exhibits an effective wavelength, a precursor, which modulates spatially the homogeneous states and drastically modifies the kink dynamics. We characterize the average and fluctuating properties of this solution. Finally, we show that the temporal evolution of these kinks is dominated by a hopping dynamics, related directly to the underlying spatial structure.

Symmetry-induced pinning-depinning transition of a subharmonic wave pattern

The stationary to drifting transition of a subharmonic wave pattern is studied in the presence of inhomogeneities and drift forces as the pattern wavelength is comparable with the system size. We consider a pinning-depinning transition of stationary subharmonic waves in a tilted quasi-one-dimensional fluidized shallow granular bed driven by a periodic air flow in a small cell. The transition is mediated by the competition of the inherent periodicity of the subharmonic pattern, the asymmetry of the system, and the finite size of the cell. Measurements of the mean phase velocity of the subharmonic pattern are in good agreement with those inferred from an amplitude equation, which takes into account asymmetry and finite-size effects of the system, emphasizing the main ingredients and mechanism of the transition.

Continuous description of lattice discreteness effects in front propagation

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls-Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.

Analytical studies of fronts, colonies, and patterns: Combination of the Allee effect and nonlocal competition interactions

We present an analytic study of traveling fronts, localized colonies, and extended patterns arising from a reaction-diffusion equation which incorporates simultaneously two features: the well-known Allee effect and spatially nonlocal competition interactions. The former is an essential ingredient of most systems in population dynamics and involves extinction at low densities, growth at higher densities, and saturation at still higher densities. The latter feature is also highly relevant, particularly to biological systems, and goes beyond the unrealistic assumption of zero-range interactions. We show via exact analytic methods that the combination of the two features yields a rich diversity of phenomena and permits an understanding of a variety of issues including spontaneous appearance of colonies.

Front dynamics and pinning-depinning phenomenon in spatially periodic media

Front propagation in one- and two-dimensional spatially modulated media is studied both experimentally and theoretically. The pinning-depinning phenomenon, long ago predicted by Pomeau [Physica D 23, 3 (1986)], is obtained and verified experimentally in a nematic liquid-crystal cell under various configurations of optical forcing. The front dynamics is characterized with respect to the different forcing parameters and the observations are compared with numerical simulations of a full model for the tilt angle of the liquid crystals under optical feedback. A spatially forced dissipative ϕ4 model is derived near the points of nascent bistability. From this model we derive analytical results that account qualitatively for the observed front dynamics and pinning range. Localized structures of different sizes and shapes are found to exist inside the pinning range and experimentally proved to be stable states of the spatially forced system.

Subharmonic wave transition in a quasi-one-dimensional noisy fluidized shallow granular bed

We present an experimental and theoretical study of the pattern formation process of standing subharmonic waves in a fluidized quasi-one-dimensional shallow granular bed. The fluidization process is driven by means of a time-periodic air flow, analogous to a tapping type of forcing. Measurements of the amplitude of the critical mode close to the transition are in quite good agreement with those inferred from a universal stochastic amplitude equation. This allows us to determine both the bifurcation point of the deterministic system and the corresponding noise intensity. We also show that the probability density distribution is well described by a generalized Rayleigh distribution, which is the stationary solution of the corresponding Fokker-Planck equation of the universal stochastic amplitude equation that describes our system.

Rectified oscillatory motion of the self-ordered front under zero-mean ac force: role of symmetry of the rate function

The rectified oscillatory motion of the "bistable" fronts (BFs) joining two states of the different stability in a spatially extended system with two stable equilibria is studied by use of the macroscopic kinetic equation of the reaction-diffusion type. The adiabatic approximation is used: We assume that the period of the ac force acting on the front in the system significantly exceeds the characteristic relaxation time of the system. By using the arguments based on the symmetry properties of the rate function in the governing equation of the ac driven front, we show that a close corelation (one-to-one correspondence) between the rate functions of the different symmetry, the symmetrical and asymmetrical ones, and the response functions performing the "input-output" conversion between the oscillatory forcing (input) function and the speed (output) function, which describes the temporal oscillations of the moment velocity of the ac driven BF, exists. Making use of the symmetry analysis we are able to show that the average characteristics of the ratchetlike transport of the ac driven BFs derivable by the symmetrical and asymmetrical rate functions radically differ. In particular, we find that depending on the symmetry of the rate function used, either symmetrical or asymmetrical one, the complete ensemble of the forward and backward running fronts propagating at the different initial velocities in the ac driven system remains either permanently at rest on average or it travels at some fixed nonzero velocity. We confirm our predictions being derived with the rate function of the general form by the direct calculations carried out by use of the cubic polynomial rate function and its piecewise linear emulations satisfying the different symmetry properties.

Local theory of the slanted homoclinic snaking bifurcation diagram

Localized states in out of equilibrium one-dimensional systems are described by the homoclinic snaking associated with the infinite sequence of multibump localized solutions of the corresponding time reversible dynamical system. We show that when the pattern undergoes a saddle-node bifurcation the homoclinic snaking bifurcation diagram becomes slanted and a finite set of localized states continue to exist outside the region of bistability. This generic behavior offers a local theory resolution of the discrepancy between models and experiments.

Universal shape law of stochastic supercritical bifurcations: theory and experiments

A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional (1D) systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement.