Brusselator System Research Articles

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

Strong Solutions of Brusselator System

The study involves a mathematical analysis of the Brusselator system on a convex bounded three-dimensional open domain, considering Neumann boundary conditions. We establish the global existence and uniqueness of the strong solution for this system. Achieving high regularity for the strong solution requires stringent conditions on the initial data. The study demonstrates the continuous dependence of the solution on the initial conditions.

Investigation of transient extreme events in a mutually coupled star network of theoretical Brusselator system

In this article, we present evidence of a distinct class of extreme events that occur during the transient chaotic state within network modeling using the Brusselator with a mutually coupled star network. We analyze the phenomenon of transient extreme events in the network by focusing on the lifetimes of chaotic states. These events are identified through the finite-time Lyapunov exponent and quantified using threshold and statistical methods, including the probability distribution function (PDF), generalized extreme value (GEV) distribution, and return period plots. We also evaluate the transitions of these extreme events by examining the average synchronization error and the system’s energy function. Our findings, validated across networks of various sizes, demonstrate consistent patterns and behaviors, contributing to a deeper understanding of transient extreme events in complex networks.

Proof of Llibre–Valls’s conjecture for the Brusselator system

The aim of this paper is to prove Llibre–Valls’s conjecture for the Brusselator system. By transforming the Brusselator system into Liénard equation and applying the limit cycle theory of Liénard equation, we give a positive answer to this conjecture. Numerical simulations are also given to illustrate our theoretical results.

A novel conformable fractional approach to the Brusselator system with numerical simulation

A novel conformable fractional approach to the Brusselator system with numerical simulation

Physics-informed neural networks for the reaction-diffusion Brusselator model

In this work, we are interesting in solving the 1D and 2D nonlinear stiff reaction-diffusion Brusselator system using a machine learning technique called Physics-Informed Neural Networks (PINNs). PINN has been successful in a variety of science and engineering disciplines due to its ability of encoding physical laws, given by the PDE, into the neural network loss function in a way where the network must not only conform to the measurements, initial and boundary conditions, but also satisfy the governing equations. The utilization of PINN for Brusselator system is still in its infancy, with many questions to resolve. Performance of the framework is tested by solving some one and two dimensional problems with comparable numerical or analytical results. Validation of the results is investigated in terms of absolute error. The results showed that our PINN has well performed by producing a good accuracy on the given problems.

Solving the reaction-diffusion Brusselator system using Generalized Finite Difference Method

<abstract><p>In this paper, we investigate the numerical solution of the Brusselator system using a meshless method. A numerical scheme is derived employing the formulas of the Generalized Finite Difference Method, and the convergence of the approximate solution to the exact solution is examined. In order to demonstrate the applicability and accuracy of the method, several examples are proposed.</p></abstract>

Existence and uniqueness of solution for the nonlinear Brusselator system with Robin boundary conditions

Abstract The system of Brusselator-type reaction-diffusion equations (RDs) on open bounded convex domains 𝒟 ⊂ ℝ d {\mathcal{D}\subset\mathbb{R}^{d}} ( d ≤ 3 ) {(d\leq 3)} with Robin boundary conditions (Rbcs) has been mathematically analyzed. The Faedo–Galerkin approach is used to demonstrate the global existence and uniqueness of a weak solution to the system. The weak solution’s higher regularity findings are constructed under more regular conditions on the initial data. In addition, continuous dependence on the initial conditions has been proved.

Spatial Turing Patterns of Periodic Solutions for the Brusselator System with Cross-Diffusion-Like Coupling

In this paper, we consider the Brusselator system with cross-diffusion-like coupling and study Turing bifurcation of its stable Hopf bifurcating periodic solutions. According to the Brusselator system’s diffusivity, a formula is established to determine the Turing bifurcation of periodic solutions. Additionally, for the Brusselator system with cross-diffusion-like coupling, a proof method for its periodic solutions to generate Turing bifurcation is given. Finally, the theoretical results are further verified by numerical simulations.

Oscillating reaction in porous media under saddle flow

Pattern formation due to oscillating reactions represents variable natural and engineering systems, but previous studies employed only simple flow conditions such as uniform flow and Poiseuille flow. We studied the oscillating reaction in porous media, where dispersion enhanced the spreading of diffusing components by merging and splitting flow channels. We considered the saddle flow, where the stretching rate is constant everywhere. We generated patterns with the Brusselator system and classified them by instability conditions and Péclet number (Pe), which was defined by the stretching rate. The results showed that each pattern formation was controlled by the stagnation point and stable and unstable manifolds of the flow field due to the heterogeneous flow fields and the resulting heterogeneous dispersion fields. The characteristics of the patterns, such as the position of stationary waves parallel to the unstable manifold and the size of local stationary patterns around the stagnation point, were also controlled by Pe.

Vieta–Lucas Polynomials for the Brusselator System with the Rabotnov Fractional-Exponential Kernel Fractional Derivative

In this study, we provide an efficient simulation to investigate the behavior of the solution to the Brusselator system (a biodynamic system) with the Rabotnov fractional-exponential (RFE) kernel fractional derivative. A system of fractional differential equations can be used to represent this model. The fractional-order derivative of a polynomial function tp is approximated in terms of the RFE kernel. In this work, we employ shifted Vieta–Lucas polynomials in the spectral collocation technique. This process transforms the mathematical model into a set of algebraic equations. By assessing the residual error function, we can confirm that the provided approach is accurate and efficient. The outcomes demonstrate the effectiveness and simplicity of the technique for accurately simulating such models.

Exact distributed kinetic Monte Carlo simulations for on-lattice chemical kinetics: lessons learnt from medium- and large-scale benchmarks.

Kinetic Monte Carlo (KMC) simulations have been instrumental in multiscale catalysis studies, enabling the elucidation of the complex dynamics of heterogeneous catalysts and the prediction of macroscopic performance metrics, such as activity and selectivity. However, the accessible length- and time-scales have been a limiting factor in such simulations. For instance, handling lattices containing millions of sites with 'traditional' sequential KMC implementations is prohibitive owing to large memory requirements and long simulation times. We have recently established an approach for exact, distributed, lattice-based simulations of catalytic kinetics which couples the Time-Warp algorithm with the Graph-Theoretical KMC framework, enabling the handling of complex adsorbate lateral interactions and reaction events within large lattices. In this work, we develop a lattice-based variant of the Brusselator system, a prototype chemical oscillator pioneered by Prigogine and Lefever in the late 60s, to benchmark and demonstrate our approach. This system can form spiral wave patterns, which would be computationally intractable with sequential KMC, while our distributed KMC approach can simulate such patterns 15 and 36 times faster with 625 and 1600 processors, respectively. The medium- and large-scale benchmarks thus conducted, demonstrate the robustness of the approach, and reveal computational bottlenecks that could be targeted in further development efforts. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Connecting chaotic regions in the Coupled Brusselator System

A family of vector fields describing two Brusselators linearly coupled by diffusion is considered. This model is a well-known example of how identical oscillatory systems can be coupled with a simple mechanism to create chaotic behavior. In this paper we discuss the relevance and possible relation of two chaotic regions. One of them is located using numerical techniques. The another one was first predicted by theoretical results and later studied via numerical and continuation techniques. As a conclusion, under the constrains of our exploration, both regions are not connected and, moreover, the former one has a big size, whereas the later one is quite small and hence, it might not be detected without the support of theoretical results. Our analysis includes a detailed analysis of singularities and local bifurcations that permits to provide a global parametric study of the system.

General conditions for Turing and wave instabilities in reaction -diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction–diffusion system with n components and diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to infty , gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHugh–Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

Instabilities in hyperbolic reaction–diffusion system with cross diffusion and species-dependent inertia

The hyperbolic reaction–diffusion (HRD) equation may overcome the physical shortcomings of the parabolic reaction–diffusion (PRD) equation, where the initially localized disturbance propagates infinitely fast through space. Instead, species often exhibit inertia, resulting in delayed effect in their spatial movement. Incorporating such response time for the onset of species flow due to a concentration gradient leads to an HRD equation with inertia. In this paper, we develop the general theory for pattern-forming instabilities in a two-species HRD system, which becomes a PRD system in the limiting case, with cross-diffusion and species-dependent inertia to explore how they play a role in the pattern forming instabilities. In particular, we determine various criteria for diffusion-induced instabilities (like Turing, wave, wave–Turing) and Hopf-induced instabilities (like pure Hopf, Hopf–wave, Hopf–Turing, and Hopf–wave–Turing) arise due to the cross-diffusion and inertial time. The theoretical results are demonstrated with an example where the Brusselator system represents the local interaction.

Improvement of Split-Step Forward Milstein Schemes for SODEs Arising in Mathematical Physics

In the present investigation, new explicit approaches by the Milstein method and increment function of the Jacobian derivative of the drift coefficient are designed. Several numerical tests such as Cox–Ingersoll–Ross process, stochastic Brusselator, and Davis-Skodje system are presented to illustrate the accuracy and the efficiency of our schemes. Furthermore, we show that the strong convergence rate of our procedures is approximately one.

Analytical modeling of the approximate solution behavior of multi‐dimensional reaction–diffusion Brusselator system

This article presents an effective and simple approximation scheme to analytically approximate solution behavior of a multi‐dimensional reaction–diffusion Brusselator system. This system has applications in cooperative processes of chemical kinetics and biochemical reaction processes. The proposed residual power series method is based on multiple power series expansion and residual error function formulation. The convergence properties and error bounds of the proposed scheme are theoretically established and numerically verified. Two benchmark problems in two‐ and three‐dimensional space are considered to validate the efficiency and accuracy of the proposed scheme. The obtained results verified reliability, accuracy, and simplicity of the proposed scheme against the available results in literature.

Discrete Temimi-Ansari method for solving a class of stochastic nonlinear differential equations

<abstract> <p>In this paper, a numerical method to solve a class of stochastic nonlinear differential equations is introduced. The proposed method is based on the Temimi-Ansari method. The special states of the four systems are studied to show that the proposed method is efficient and applicable. These systems are stochastic Langevin's equation, Ginzburg-Landau equation, Davis-Skodje, and Brusselator systems. The results clarify the accuracy and efficacy of the presented new method with no need for any restrictive assumptions for nonlinear terms.</p> </abstract>

Stability, Chaos, and Bifurcation Analysis of a Discrete Chemical System

The local dynamics, chaos, and bifurcations of a discrete Brusselator system are investigated. It is shown that a discrete Brusselator system has an interior fixed point P(1, r) if r > 0. Then, by linear stability theory, local dynamical characteristics are explored at interior fixed point P(1, r). Furthermore, for the discrete Brusselator system, the existence of periodic points is investigated. The existence of bifurcations around an interior fixed point is also investigated and proved that the discrete Brusselator model undergoes hopf and flip bifurcations if (r, h) ∈ ℋℬ|P(1, r) = {(r, h), h = 2 − r} and , respectively. The next feedback control method is utilized to stabilize the chaos that exists in the discrete Brusselator system. Finally, obtained results are verified numerically.

Connecting Chaos in the Coupled Brusselator System

Connecting Chaos in the Coupled Brusselator System