Stationary Spatial Structures Research Articles

Suppression of modulation instability in VCSEL by external optical injection.

The dynamics of a broad area semiconductor vertical-cavity surface-emitting laser (VCSEL) is analytically and numerically investigated considering the linewidth enhancement factor resulting in the modulation instability of a spatially homogeneous generation mode. We find the aperture width depends on the parameters of the laser system, above which the modulation instability leads to chaotic filamentalization of the optical field and complicates the practical application of such lasers. We show that using the external optical injection of small amplitude, it is possible both to transform chaotic filaments into regular stationary spatial optical structures (stripes and hexagons) and to stabilize the broad-area VCSELs.

Multistability and Stochastic Phenomena in the Distributed Brusselator Model

Abstract An influence of random disturbances on the pattern formation in reaction–diffusion systems is studied. As a basic model, we consider the distributed Brusselator with one spatial variable. A coexistence of the stationary nonhomogeneous spatial structures in the zone of Turing instability is demonstrated. A numerical parametric analysis of shapes, sizes of deterministic pattern–attractors, and their bifurcations is presented. Investigating the corporate influence of the multistability and stochasticity, we study phenomena of noise-induced transformation and generation of patterns.

Bifurcations of spatiotemporal structures in a medium of FitzHugh–Nagumo neurons with diffusive coupling

We study the boundaries of existence of traveling waves and stationary spatial structures in an active medium model by varying the control parameters. The medium is represented by a ring of diffusively coupled FitzHugh–Nagumo neurons, which, when uncoupled, can demonstrate excitable, self-sustained oscillatory or bistable dynamics depending on control parameter values. The dynamical regimes realized in the medium are compared with those ones observed in an individual FitzHugh–Nagumo neuron. Possible bifurcations of traveling waves are analyzed when the dynamics of the medium elements changes. We also explore the influence of the relaxation level of FitzHugh–Nagumo neurons on the medium dynamics.

Flow-induced arrest of spatiotemporal chaos and transition to a stationary pattern in the Gray-Scott model.

We examine the prototypical Gray-Scott model, which mimics cubic autocatalytic reaction with linear decay of the autocatalyst, to model the kinetics of a reaction-diffusion system subjected to advective streamline flow. For a proper choice of boundary conditions and parameter space, the system admits wave-induced spatiotemporal chaos in the absence of flow. We show that flow above a critical value leads to an arrest of the spatiotemporal chaos due to a change in the instability from absolute to convective type. Furthermore, stationary spatial structures are borne out of a second successive bifurcation for yet another critical flow value. The theoretical formulations are corroborated by extensive numerical simulation of the full reaction-diffusion-advection system in one dimension.

RECONSTRUCTION OF NON-STATIONARY COMPLEX SPATIAL STRUCTURES BY A NOVEL FILTER-BASED MULTI SCALE MPS ALGORITHM

Geologic modeling of complex structures needs realistic algorithms that can extract and reproduce position-dependent statistical relations in these phenomena. For this purpose, methods based on Multi Point Statistics (MPS) should be developed because spatial patterns such as curvilinearity cannot be recognized via two-point statistical concepts. MPS extracts high order spatial relations from a Training Image (TI) as a visually and statistically explicit model and some regional images in non-stationary cases. In this work a novel pattern to pattern filter-based algorithm is first presented for modeling of stationary spatial structures. Then for nonstationary images, in an automatic preprocessing step, one scale and azimuth for each node in final realization map is inferred from regional/auxiliary images. These values are used in patterns similarity measurements and pasting steps in a multi-scale simulation process. Performance of proposed algorithms is investigated by applying the methods on several binary 2D training images. In comparison, lower running time and better visual quality in all cases is clear. In the final non-stationary images, spatial statistics of training images and physical features of regional images are reproduced and traditional methods related concepts such as search tree and realization zoning and their unintended effects are eliminated.

Spatial structure of surface soil water content in a natural forested headwater catchment with a subtropical monsoon climate

Summary The surface or near-surface soil moisture at the interface between atmospheric and terrestrial environments has significant implications for the complex interactions between hydrological processes and ecosystems. Although a number of previous studies have reported the spatial patterns and structures of surface moisture in pasture environments based on large sample sizes, information regarding the detailed spatial patterns and structure of surface moisture in a natural forest environment is still sparse, particularly for steep headwater catchments due to the labor intensity involved and the difficulty of making field measurements. In this study, we measured the detailed surface soil water content at depths of 0–12 cm and 0–20 cm at 470 measurement points using the TDR (time domain reflectometry) method. Thirteen surveys were conducted within 1 year in a steep natural forested headwater catchment with a subtropical monsoon climate in Taiwan. The sample and model variograms were used to evaluate the spatial structure of the surface soil moisture. We also compared the spatial structures of surface moisture in this study with those in a similar study conducted in a flat pasture catchment ( Western et al., 1998 ). For the results at our site, exponential model variograms with the nugget effect well fitted sample variograms, indicating an obvious regional dependence of surface moisture and stationary spatial structures in all surveys. The geostatistical structural characteristics of the nugget varied between 22.70 and 47.40 (% V/V) 2 ; the sill varied between 54.99 and 98.51 (% V/V) 2 ; and the range varied between 10.20 and 58.65 m. The nugget, sill, and range all increased with an increase in soil moisture. The notable anisotropy under dry conditions was attributed to the year-round surface runoff pathway. The sample variograms that represented the spatial structure of soil moisture and reliably estimated the nugget, sill, and range required a sample size of at least 235 at our 0.15-ha site (i.e., a measurement resolution of at least 6.4 m 2 per sample). The pattern of soil moisture was relatively stationary in two gullies and on the side of a hillslope, but it varied on the valley-head hillslope depending on whether conditions were dry or wet. In contrast to previous results for a flat pasture catchment ( Western et al., 1998 ), our results produced a right-skewed histogram of soil moisture under dry or moderately wet conditions, variograms that tended to increase with an increase in soil moisture, anisotropy of soil moisture structure under dry, but not wet, conditions, a need for a 55 times higher measurement resolution associated with the large sample size, and a larger nugget effect in the steep natural forested headwater catchment. This study identified the spatial structure and patterns of surface soil moisture as the primary step in understanding hydrologic processes in a natural forested headwater catchment.

Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources

We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. They can be related to the emergence of biological species due to the intra-specific competition and random mutations. Various types of travelling waves are observed.

Pattern formation induced by nonequilibrium global alternation of dynamics.

We recently proposed a mechanism for pattern formation based on the alternation of two dynamics, neither of which exhibits patterns. Here we analyze the mechanism in detail, showing by means of numerical simulations and theoretical calculations how the nonequilibrium process of switching between dynamics, either randomly or periodically, may induce both stationary and oscillatory spatial structures. Our theoretical analysis by means of mode amplitude equations shows that all features of the model can be understood in terms of the nonlinear interactions of a small number of Fourier modes.

Transition from Self-Replicating Behavior to Stationary Patterns Induced by Concentration-Dependent Diffusivities

In this Letter, we report the observation of a transition from self-replicating behavior to stationary spatial structures induced by concentration-dependent diffusivities in the excitable Gray-Scott medium. Notably, the transition occurs even though there is no change in the relative diffusivities between the activator and the inhibitor. In contrast to the well-known Turing patterns, the obtained time-independent spatial structure has no intrinsic wavelength and the asymptotic state depends exclusively on the initial perturbations. This study illustrates that variable diffusivities can also have profound effects on pattern formation and selection in excitable media.

Second harmonic generation in a two-dimensional diatomic lattice

Resonant interactions of modes in two-dimensional (2D) diatomic lattices are studied within the framework of the multiscale expansion. Equations governing three-wave processes and second harmonic generation are derived for a lattice having a finite number of cells. A particular example of the second harmonic generation in a two-dimensional lattice with nearest-neighbor interactions is considered in detail. It is shown that the 2D case reveals new features like, for example, possibility of simultaneous resonant interaction of one mode with two other ones, existence of solitary wave pulses localized in space and moving along the lattice, and creation of stationary spatial structures subject to proper boundary conditions.

Controlled optical structures in a nonlinear system involving the suppression of low spatial frequencies in the feedback loop

A nonlinear optical system with spatially distributed feedback was studied both theoretically and experimentally. The phase-to-intensity transformation in this system was performed by a spatial filter capable of suppressing low spatial frequencies. Hard excitation of stationary spatial structures observed in this system was explained by an analysis of the phase space structure of the amplitude equations. The developed theoretical approach, which uses a step-function approximation of the steady-state solution, allows one to determine the main quantitative characteristics of the generated structures. The basic properties of the response of the system to external perturbations with various symmetries were investigated experimentally. The obtained experimental data qualitatively agree with the results of the theoretical analysis.

Dissipative effects in a nonlinear wave system with an unstable linear spectrum

We consider a system describing the interaction of two resonantly coupled waves, which includes linear coupling terms, different group velocities, cubic nonlinear terms and linear losses (either symmetric or nonsymmetric). The system’s dynamics are driven by an instability induced by the linear coupling. The collapse-free situation is considered. First, modulational instability of uniform continuous-wave solutions is investigated analytically. Then, numerical simulations demonstrate that, in most cases, development of an instability in the model with symmetric friction leads, after an initial ‘latent’ stage, followed by a ‘turbulent’ transient, to establishment of quasi-periodic or chaotic stationary spatial structures (the spatially chaotic structure is only quasi-stationary, demonstrating persistent residual ‘breathings’). However, when the dissipative constant gets close to its limiting value, beyond which the system’s dynamics are trivial, the duration of the transient turbulent stage diverges, so that persistent spatio-temporal chaos is observed. Qualitative arguments interpreting these features are presented. In the case when the dissipation is present only in one mode, stationary states do not exist. A specific nonstationary regime, with the undamped mode indefinitely growing and the damped one gradually vanishing but still playing an essential role, is predicted analytically and found numerically in this case. The simulations produce a nontrivial value of the scaling index controling the growth of the undamped mode. A general inference is that, in this system with an unstable linear spectrum, which generates spatio-temporal chaos in the absence of dissipation, adding dissipative terms leads, in most cases, to suppression of the temporal chaos, and generation instead of spatially chaotic quasi-stationary structures. But in some cases spatio-temporal chaos survives in the presence of dissipation.

Modes selection in polymer mixtures undergoing phase separation by photochemical reactions.

Phase separation kinetics and morphology of binary polymer mixtures (A/B) in the presence of photochemical reactions were investigated by using phase-contrast optical microscopy combined with digital image analysis. The polymers were chemically designed in such a way that two types of chemical reactions, intermolecular photodimerization and intramolecular photoisomerization, of polymer segments can be induced and controled by irradiation with ultraviolet light. Unlike the conventional case, the phase separation in the presence of these reactions is spontaneously frozen due to the suppression of the long-wavelength instabilities, resulting in stationary spatial structures with intrinsic periodicities. These characteristic length scales are determined by the competition between the two antagonistic interactions: phase separation as a relatively short-range activation and the photochemical reaction as a long-range inhibition. Furthermore, it was found that the spatial symmetry breaking of concentration fluctuations can emerge from the elastic stress associated with the nonhomogeneous kinetics of the reactions. Experimental data obtained with three types of reactions: A-A only cross-link, A-A and B-B simultaneous cross-links and the reversible A<-->B photoisomerization are described. These results do not only indicate that combination of chemical reactions and phase separation could provide a novel method to control the morphology of multiphase polymer materials, but also suggest that photoreactive polymers can be used as a chemical system to study the mode-selection process in polymers far from thermodynamic equilibrium. (c) 1999 American Institute of Physics.

WAVE-SPLITTING IN THE BISTABLE GRAY-SCOTT MODEL

The Gray-Scott model describes a chemical reaction in which an activator species grows autocatalytically on a continuously fed substrate. For certain feed rates and activator life times the model allows for the coexistence of two different homogeneous steady states, the red state in which the conditions correspond to those of the pure substrate, and the blue state where the activator concentration is relatively high and limited by substrate depletion. The blue state may again undergo a Hopf bifurcation leading to self-sustained oscillations and, for a spatially extended system, this state may also exhibit a subcritical Turing bifurcation capable of producing global as well as localized, stationary spatial structures. Interactions between these instabilities can lead to a variety of different phenomena including Turing-Hopf mixed modes and spatio-temporal chaos. The paper presents the results of a computer simulation study of some of these far-from-equilibrium phenomena. Special emphasis is given to the propagation, collision and splitting of the traveling pulses which can develop in response, for instance, to strong local perturbations of the red state in a region of parameter space where the blue state is Turing and Hopf unstable.

Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems

The development of one-dimensional Turing patterns characteristic of the chlorite-iodide-malonic acid/starch reaction as well as analogous Brussellator/immobilizer and Schnackenberg/immobilizer model systems is investigated by means of a weakly nonlinear stability analysis applied to the appropriately scaled governing equations. Then the theoretical predictions deduced from these pattern formation studies are compared with experimental evidence relevant to the Turing diffusive instabilities under examination in order to explain more fully the transition to such stationary symmetry-breaking spatial structures when the temperature or pool species concentrations vary.

Necessary condition of the Turing instability

A reaction-diffusion system in any number of spatial variables consisting of an arbitrary number of chemical species cannot exhibit Turing instability if none of the reaction steps expresses cross inhibition. A corollary of this result underlines the importance of nonlinearity in the formation of stationary spatial structures, a kind of self-organization on a chemical basis.

Two types of performance of an isotropic cellular automaton: stationary (Turing) patterns and spiral waves

An isotropic cellular automaton containing only the crudest caricature of activator-inhibitor systems, is shown to be capable of producing stationary spatial structures (Turing patterns) in qualitative agreement with a variety of experiments in chemistry. Due to the isotropy of our automaton, we can obtain spots arranged in regular hexagons, a pattern that cannot be obtained with a classical square lattice. Labyrinthine stripes, like those observed in magnetic domains in thin films, are also obtained. Spiral waves are obtained upon increasing the radius of action of the activator relative to that of the inhibitor, as predicted by theory.

Period-doubling bifurcations and modulational instability in the nonlinear ring cavity: an analytical study

A new theoretical approach for the study of the influence of diffraction on Ikeda instability in the Kerr-type nonlinear ring cavity is proposed. The simplicity of this model allows for a complete analytical study of transverse modulational instability in the period-2 regime of the cavity. In particular the emergence of stationary spatial dissipative structures in the profile of the period-doubled transmitted beam is predicted.

Spatial and temporal instabilities in aCO2laser

We show experimentally the appearance of time-dependent and stationary complex spatial structures in a ${\mathrm{CO}}_{2}$ laser. A comparison of the data with a theoretical model supports the notion that the observed phenomena result from the nonlinear interaction of transverse cavity modes with the active medium. The simultaneous appearance of spatial and temporal instabilities hints to the interesting prospect that one may be able to study turbulence in a laser system.

Cooperative frequency locking and stationary spatial structures in lasers

We investigate the spontaneous emergence of transverse patterns in lasers by using both the standard two-level model and the so-called cubic approximation, which is generally valid in the threshold regions. The stationary intensity configurations fall into two distinct classes. The first includes solutions of the single-mode type with the frequency and spatial structure of one of the transverse resonances. The solutions of the second group involve the simultaneous oscillation of several cavity modes, operating in such a way as to produce a stationary intensity profile. The stationary character of these multimode configurations emerges from the fact that the transverse modes of the resonator lock onto a common frequency during the nonlinear transient. We call this phenomenon cooperative frequency locking.