Lengyel-Epstein Model Research Articles

A novel image inpainting method based on a modified Lengyel–Epstein model

With the increasing popularity of digital images, developing advanced algorithms that can accurately reconstruct damaged images while maintaining high visual quality is crucial. Traditional image restoration algorithms often struggle with complex structures and details, while recent deep learning methods, though effective, face significant challenges related to high data dependency and computational costs. To resolve these challenges, we propose a novel image inpainting model, which is based on a modified Lengyel–Epstein (LE) model. We discretize the modified LE model by using an explicit Euler algorithm. A series of restoration experiments are conducted on various image types, including binary images, grayscale images, index images, and color images. The experimental results demonstrate the effectiveness and robustness of the method, and even under complex conditions of noise interference and local damage, the proposed method can exhibit excellent repair performance. To quantify the fidelity of these restored images, we use the peak signal-to-noise ratio (PSNR), a widely accepted metric in image processing. The calculation results further demonstrate the applicability of our model across different image types. Moreover, by evaluating CPU time, our method can achieve ideal repair results within a remarkably brief duration. The proposed method validates significant potential for real-world applications in diverse domains of image restoration and enhancement.

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability.

Dynamic Exploration and Control of Bifurcation in a Fractional-Order Lengyel-Epstein Model Owing Time Delay

Delayed differential equation plays a vital role in revealing the dynamics of chemical reaction law. In this work, we propose a novel fractional-order Lengyel-Epstein model owing time delay. By regarding the delay as parameter and investigating the distribution of roots of the associated characteristic equation of the formulated fractional-order delayed Lengyel-Epstein model, we set up a new delay-dependent criterion on stability and bifurcation of the involved fractional-order delayed Lengyel-Epstein model. Making use of nonlinear delayed feedback controller, we can effectually control the stability domain and the time of bifurcation phenomenon of the formulated fractional-order delayed Lengyel-Epstein model. Taking advantage of hybrid controller, we are able to adjust the stability domain and the time of bifurcation phenomenon of the established fractional-order delayed Lengyel-Epstein model. The study shows that delay is a vital factor which affects the stability and bifurcation behavior of the addressed fractional-order delayed Lengyel-Epstein model. In order to illustrate the rationality of the acquired theoretical outcomes, we execute Matlab simulations to check this fact. The gained outcomes in this work are absolutely innovative and possess enormous theoretical significance in adjusting concentrations of different chemical substance.

Turing patterns on rotating spiral growing domains.

We investigate the emergence of Turing patterns in a system growing as a rotating spiral in two dimensions, utilizing the photosensitivity of the chlorine dioxide-iodine-malonic acid (CDIMA) reaction to control the growth process. We observe the formation of single and multiple (double and triple) stationary spiral patterns as well as transitional patterns. From numerical simulations of the Lengyel-Epstein model with an additional term to account for the effects of illumination on the reaction, we analyze the relationship between the final morphologies and the radial and angular growth velocities, identify conditions conducive to the formation of transitional structures, examine the importance of the size of the initial nucleation site in determining the spiral's multiplicity, and evaluate the stability and robustness of these Turing patterns. Our results indicate how inclusion of rotational degrees of freedom in the growth process may lead to the formation of a diverse new class of patterns in chemical and biological systems.

Symmetry-breaking rhythms in coupled, identical fast-slow oscillators.

Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.

Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model

Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model

Effect of obstructions on growing Turing patterns.

We study how Turing pattern formation on a growing domain is affected by discrete domain discontinuities. We use the Lengyel-Epstein reaction-diffusion model to numerically simulate Turing pattern formation on radially expanding circular domains containing a variety of obstruction geometries, including obstructions spanning the length of the domain, such as walls and slits, and local obstructions, such as small blocks. The pattern formation is significantly affected by the obstructions, leading to novel pattern morphologies. We show that obstructions can induce growth mode switching and disrupt local pattern formation and that these effects depend on the shape and placement of the objects as well as the domain growth rate. This work provides a customizable framework to perform numerical simulations on different types of obstructions and other heterogeneous domains, which may guide future numerical and experimental studies. These results may also provide new insights into biological pattern growth and formation, especially in non-idealized domains containing noise or discontinuities.

Effects of spatial periodic forcing on Turing patterns in two-layer coupled reaction diffusion system

Periodic forcing of pattern-forming systems is always a research hot spot in the field of pattern formation since it is one of the most effective methods of controlling patterns. In reality, most of the pattern-forming systems are the multilayered systems, in which each layer is a reaction-diffusion system coupled to adjacent layers. However, few researches on this issue have been conducted in the multilayered systems and their responses to the periodic forcing have not yet been well understood. In this work, the influences of the spatial periodic forcing on the Turing patterns in two linearly coupled layers described by the Brusselator (Bru) model and the Lengyel-Epstein (LE) model respectively have been investigated by introducing a spatial periodic forcing into the LE layer. It is found that the subcritical Turing mode in the LE layer can be excited as long as one of the external spatial forcing and the supercritical Turing mode (referred to as internal forcing mode) of the Bru layer is a longer wave mode. These three modes interact together and give rise to complex patterns with three different spatial scales. If both the spatial forcing mode and the internal forcing mode are the short wave modes, the subcritical Turing mode in the LE layer cannot be excited. But the superlattice pattern can also be generated when the spatial resonance is satisfied. When the eigenmode of the LE layer is supercritical, a simple and robust hexagon pattern with its characteristic wavelength appears and responds to the spatial forcing only when the forcing intensity is large enough. It is found that the wave number of forcing has a powerful influence on the spatial symmetry of patterns.

Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

<p style='text-indent:20px;'>Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.</p>

Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

In this paper, a Lengyel–Epstein model with two delays is proposed and considered. By choosing the different delay as a parameter, the stability and Hopf bifurcation of the system under different situations are investigated in detail by using the linear stability method. Furthermore, the sufficient conditions for the stability of the equilibrium and the Hopf conditions are obtained. In addition, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained with the normal form theory and the center manifold theorem to delay differential equations. Some numerical examples and simulation results are also conducted at the end of this paper to validate the developed theories.

Turing instability of the periodic solutions for reaction-diffusion systems with cross-diffusion and the patch model with cross-diffusion-like coupling

In this paper, we determine how the diffusion could destabilize an otherwise stable spatially homogeneous periodic solutions for the diffusive systems. This is known as Turing instability of the periodic solutions. In a general setting, we establish a formulae in terms of the diffusion rates (not necessarily limited to either larger diffusivity or smaller diffusivity) to determine Turing instability of the Hopf bifurcating periodic solutions for reaction-diffusion systems with cross-diffusions and the patch model of n-coupled reactors with cross-diffusion-like coupling. Then, we supply an example of the Lengyel-Epstein model which is used to characterize the CIMA chemical reactions to demonstrate how our results can be applied.

Numerical analysis on formation and transition of white-eye square superlattice patterns in dielectric barrier discharge system

The mechanism of formation and transformation of white-eye square patterns in dielectric barrier discharge system is investigated numerically, using the two-layer Lengyel–Epstein model with asymmetric and symmetric coupling. When the scale of the simulation system [Formula: see text] is two to three times of pattern wavelength [Formula: see text], it is found that an obvious intermediate state with square distribution appears by adjusting the ratio of diffusion coefficients [Formula: see text]/[Formula: see text]. When it is coupled with a suitable short-wavelength Turing mode in the range of [Formula: see text] to [Formula: see text], a new spatial resonance structure can be formed in the short-wavelength mode subsystem, and the pattern evolves from a simple square pattern to a white-eye square pattern. Although the two coupling methods achieve the same results, the duration time of the white-eye square pattern in the symmetric coupling method is significantly longer than that in the asymmetric coupling method. Because the quadratic coefficient of the amplitude equation in the reaction–diffusion system is not zero, the simple square pattern of the long wavelength mode subsystem gradually transits into a stable hexagon pattern gradually. As a result, the white-eye pattern transits from a square to a hexagon.

Numerical study and stability of the Lengyel\u2013Epstein chemical model with diffusion

In this paper, a nonlinear mathematical model with diffusion is taken into account to review the dynamics of Lengyel–Epstein chemical reaction model to describe the oscillating chemical reactions. For this purpose, the dimensionless Lengyel–Epstein model with diffusion and homogeneous boundary condition is considered. The steady states with and without diffusion of the Lengyel–Epstein model are studied. The basic reproductive number is computed and the global steady states for the system are calculated. Numerical results are offered for two systems using three well known techniques to validate the main outcomes. The consequences established from this qualitative study are supported by numerical simulations characterized by distinct programs, adopting forward Euler method, Crank–Nicolson method, and nonstandard finite difference method.

Super-lattice patterns in two-layered coupled non-symmetric reaction diffusion systems

The coupling mechanism is one of most important approaches to generating multiple-scaled spatial-temporal patterns. In this paper, the mode interaction between two different Turing modes and the pattern forming mechanisms in the non-symmetric reaction diffusion system are numerically investigated by using a two-layered coupled model. This model is comprised of two different reaction diffusion models: the Brusselator model and the Lengyel-Epstein model. It is shown that the system gives rise to superlattice patterns if these two Turing modes satisfy the spatial resonance condition, otherwise the system yields simple patterns or superposition patterns. A suitable wave number ratio and the same symmetry are two necessary conditions for the spatial resonance of Turing modes. The eigenvalues of these two Turing modes can only vary in a certain range in order to make the two sub-system patterns have the same symmetry. Only when the long wave mode becomes the unstable mode, can it modulate the other Turing mode and result in the formation of spatiotemporal patterns with multiple scale. As the wave number ratio increases, the higher-order harmonics of the unstable mode appear, and the sub-system with short wave mode undergoes a transition from the black-eye pattern to the white-eye pattern, and finally to a temporally oscillatory hexagon pattern. It is demonstrated that the resonance between the Turing mode and its higher-order harmonics located in the wave instability region is the dominant mechanism of the formation of this oscillatory hexagon pattern. Moreover, it is found that the coupling strength not only determines the amplitudes of these patterns, but also affects their spatial structures. Two different types of white-eye patterns and a new super-hexagon pattern are obtained as the coupling strength increases. These results can conduce to understanding the complex spatial-temporal behaviors in the coupled reaction diffusion systems.

RETRACTED ARTICLE: Fractional order Lengyel–Epstein chemical reaction model

The proposed model in this study is the fractional order differential equations system with multi-orders of the dimensionless Lengyel–Epstein model being the oscillating chemical reactions. It is founded the positive equilibrium point. Additionally, the stability of the positive equilibrium point obtained from this system is analyzed. The results founded from this qualitative analysis are corroborated by numerical simulations drawn by various programs using two different techniques.

Modulation of Turing Patterns in the CDIMA Reaction by Ultraviolet and Visible Light.

We have carried out the first systematic study of the effects of ultraviolet light, both alone and in combination with visible white light, on Turing patterns in the chlorine dioxide-iodine-malonic acid (CDIMA) reaction. The ultraviolet light used has a sharp peak at 368 nm and can perturb the system selectively. It primarily decomposes chlorine dioxide in a zeroth-order reaction, and when it is used to illuminate Turing patterns, shrunken spots are formed with an imperfect hexagonal arrangement. The ultraviolet light competes directly with the visible white light via the photoreaction with dissolved chlorine dioxide, which prevents the total suppression of patterns at intermediate intensities of white light. These results suggest that specific wavelengths of light in the ultraviolet spectrum selectively modify the chemistry behind the pattern formation and can be utilized to generate novel self-organized structures under forcing conditions. We propose a modified Lengyel-Epstein model to incorporate the effect of ultraviolet illumination and obtain good qualitative agreement between simulations and experiments. These results support the idea that chlorine dioxide photoreaction is a key step in modulating CDIMA patterns under ultraviolet illumination.

Extended Global Asymptotic Stability Conditions for a Generalized Reaction–Diffusion System

In this paper, we consider the general reaction–diffusion system proposed in Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017) as a generalization of the original Lengyel–Epstein model developed for the revolutionary Turing-type CIMA reaction. We establish sufficient conditions for the global existence of solutions. We also follow the footsteps of Lisena (Appl. Math. Comput. 249:67–75, 2014) and other similar studies to extend previous results regarding the local and global asymptotic stability of the system. In the local PDE sense, more relaxed conditions are achieved compared to Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017). Also, new extended results are achieved for the global existence, which when applied to the Lengyel–Epstein system, provide weaker conditions than those of Lisena (Appl. Math. Comput. 249:67–75, 2014). Numerical examples are used to affirm the findings and benchmark them against previous results.

On the dynamics of the Lengyel–Epstein model with forcing intensity

This paper is concerned with the Lengyel–Epstein model for interacting chemicals under Dirichlet–Neumann boundary data. This model describe the reaction between iodide, malonic and clorite acid (CIMA reaction). In particular the Lengyel–Epstein model that takes into account the effect of illumination of the reaction cell is investigated. It is shown that the solutions are bounded. The linear stability of the steady states is discussed. Conditions guaranteeing the nonlinear stability are also obtained.

Numerical simulation of the zebra pattern formation on a three-dimensional model

In this paper, we numerically investigate the zebra skin pattern formation on the surface of a zebra model in three-dimensional space. To model the pattern formation, we use the Lengyel–Epstein model which is a two component activator and inhibitor system. The concentration profiles of the Lengyel–Epstein model are obtained by solving the corresponding reaction–diffusion equation numerically using a finite difference method. We represent the zebra surface implicitly as the zero level set of a signed distance function and then solve the resulting system on a discrete narrow band domain containing the zebra skin. The values at boundary are dealt with an interpolation using the closet point method. We present the numerical method in detail and investigate the effect of the model parameters on the pattern formation on the surface of the zebra model.

Multifold Increases in Turing Pattern Wavelength in the Chlorine Dioxide-Iodine-Malonic Acid Reaction-Diffusion System.

Turing patterns in the chlorine dioxide-iodine-malonic acid reaction were modified through additions of sodium halide salt solutions. The range of wavelengths obtained is several times larger than in the previously reported literature. Pattern wavelength was observed to significantly increase with sodium bromide or sodium chloride. A transition to a uniform state was found at high halide concentrations. The observed experimental results are qualitatively well reproduced in numerical simulations with the Lengyel-Epstein model with an additional chemically realistic kinetic term to account for the added halide and an adjustment of the activator diffusion rate to allow for interhalogen formation.