Brusselator Equations Research Articles

Model order reduction for the cross-diffusive Brusselator Equation

The cross-diffusive Brusselator equation is a reaction-diffusive system that models complex chemical and biological process with both self-diffusion and cross-diffusion effects. These equations exhibit rich spatiotemporal dynamics, including Turing patterns and instability-driven pattern formations. Despite its significance, the computational cost of solving high-dimensional discretized versions of the cross-diffusive Brusselator equation can be prohibitive, particularly in parameter-dependent or long-time simulations. This study presents a model order reduction (MOR) framework tailored to the Brusselator equation, leveraging Proper Orthogonal Decomposition (POD) combined with Galerkin projection along with the Discrete Empirical Interpolation Method (DEIM) and the Dynamic Mode Decomposition Method (DMD) to efficiently approximate nonlinear dynamics. The reduced models are constructed to preserve key features of the original system, including stability and accuracy, while achieving substantial computational savings. Numerical experiments validate the proposed approach, demonstrating its effectiveness in capturing the essential dynamics of the Brusselator equation under various parameter settings. These findings provide a robust pathway for efficient simulation and analysis of reaction-diffusion systems in scientific and engineering applications.

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We present a method for computing actions of the exponential-like φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions for a Kronecker sum K of d arbitrary matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}. It is based on the approximation of the integral representation of the φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call phiks, evaluates the required actions by means of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode products involving exponentials of the small sized matrices Aμ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mu $$\\end{document}, without forming the large sized matrix K itself. phiks, which profits from the highly efficient level 3 BLAS, is designed to compute different φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on the same vector or a linear combination of actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of phiks in the exponential integration context. In fact, our newly designed method has been tested in popular exponential Runge–Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document}-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection–diffusion–reaction, Allen–Cahn, and Brusselator equations show the superiority of the proposed μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-mode approach.

The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations

PurposeThis study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology.Design/methodology/approachFirst, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations.FindingsThis study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space.Originality/valueThis study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties.

High-Order Analysis of Canard Explosion in the Brusselator Equations

The aim of this paper is to obtain a high-order approximation of the canard explosion in the Brusselator equations. This classical chemical system has been extensively studied but, until now, only first-order approximation to the canard explosion has been provided. Here, with the help of the nonlinear time transformation method, we are able to obtain an approximation to any desired order. Our results strongly agree with those obtained by numerical continuation.

Random attractors of the stochastic extended Brusselator system with a multiplicative noise

In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.

A spectral method for nonlocal diffusion operators on the sphere

We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics, the computation of their eigenvalues to high relative accuracy using quadrature and asymptotic formulas, and a fast spherical harmonic transform. These techniques also lead to an efficient implementation of high-order exponential integrators for time-dependent models. We apply our method to the nonlocal Poisson, Allen–Cahn and Brusselator equations.

Numerical Study of a Mathematical Model of an Autocatalytic Reaction with Diffusion in a Tubular Reactor

The article analyzes analytically and numerically the model of the autocatalytic reaction with diffusion in the degenerate case on a finite connected directed graph G with the Showalter -- Sidorov condition. The mathematical model of the autocatalytic reaction with diffusion is based on the system of distributed Brusselator equations. The system of degenerate equations of a distributed Brusselator whose functions satisfy the conditions of continuity and flow balance belongs to a wide class of semilinear Sobolev type equations. To investigate the existence of a solution of this system of equations, the phase space method which was developed by G.A.Sviridyuk and his students to study the solvability of Sobolev type equations will be used. We will show the simplicity of the phase space and the existence of a unique local solution of the given Showalter -- Sidorov problem. The theoretical results of this article served as the basis for developing an algorithm for numerical study of the model in a Maple environment. The algorithm of numerical investigation is based on the Galerkin method, which allows us to take into account the phenomenon of degeneracy of the equation. The article gives several examples illustrating the results of the computational experiment obtained on the three-ribbed and five-ribbed graphs.

Edge of chaos in reaction diffusion CNN model

Abstract In this paper, we study the dynamics of a reaction-diffusion Cellular Nonlinear Network (RD-CNN) nodel in which the reaction term is represented by Brusselator cell. We investigate the RD-CNN dynamics by means of describing function method. Comparison with classical results for Brusselator equation is provided. Then we introduce a new RD-CNN model with memristor coupling, for which the edge of chaos regime in the parameter space is determined. Numerical simulations are presented for obtaining dynamic patterns in the RD-CNN model with memristor coupling.

Third-Order Approximate Solution of Chemical Reaction-Diffusion Brusselator System Using Optimal Homotopy Asymptotic Method

The objective of this paper is to investigate the effectiveness and performance of optimal homotopy asymptotic method in solving a system of nonlinear partial differential equations. Since mathematical modeling of certain chemical reaction-diffusion experiments leads to Brusselator equations, it is worth demanding a new technique to solve such a system. We construct a new efficient recurrent relation to solve nonlinear Brusselator system of equations. It is observed that the method is easy to implement and quite valuable for handling nonlinear system of partial differential equations and yielding excellent results at minimum computational cost. Analytical solutions of Brusselator system are presented to demonstrate the viability and practical usefulness of the method. The results reveal that the method is explicit, effective, and easy to use.

The Prediction of Wave Competitions in Inhomogeneous Brusselator Systems* *Supported by National Natural Science Foundation of China under Grant Nos. 11105051, 11104071, 11247272, Fundamental Research Funds for Central Universities, Beijing Higher Education Elite Young Teacher Project, and Youth Scholars Program of Beijing Normal University

The competition of waves has remained a hot topic in physics over the past few decades, especially the area of pattern control. Because of improved understanding of various dynamic behaviors, many practical applications have sprung up recently. The prediction of wave competitions is also very important and quite useful in these fields. This paper considers the behaviors of wave competitions in simple, inhomogeneous media which is modeled by Brusselator equations. We present a simple rule to judge the results of wave competitions utilizing the dispersion relation curves and the waves coming from different wave sources. Moreover, this rule can also be used to predict the results of wave propagation. It provides methods of obtaining the desired waves with given frequencies in inhomogeneous media. All our results are concluded and verified by computer simulations.

Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x = 0 and x = - 1 . We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p , any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε → 0 ) of the form ∑ Y n ( x / ε ′ ) ε ′ n , where ε ′ p + 1 = ε , for x = ε ′ X with X large enough, but independent of ε . In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.

Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems

In this paper numerical results are reviewed [D.L. Ropp, J.N. Shadid, C.C. Ober, Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comput. Phys. 194 (2) (2004) 544–574] that demonstrate that common second-order operator-splitting methods can exhibit instabilities when integrating the Brusselator equations out to moderate times of about seven periods of oscillation. These instabilities are manifested as high wave number spatial errors. In this paper, we further analyze this problem, and present a theorem for stability of operator-splitting methods applied to linear reaction-diffusion equations with indefinite reaction terms which controls both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. In the absence of L-stability, an additional time step condition that suppresses the high wave number modes appears to guarantee convergence at the asymptotic order for the operator-splitting method. Numerical results for a model problem confirm this theory, and results for the Brusselator problem agree as well.

Multiple bifurcations generated by mode interactions in a reaction–diffusion problem

We study multiple bifurcations in a system of reaction–diffusion equations defined on a unit square with Robin boundary conditions. First we investigate linear stabilities of the system at the uniform steady state solution. Then we discuss how multiple bifurcations can be generated by mode interactions of the system, and how these multiple bifurcations can be preserved in the associated discrete system. A continuation-unsymmetric Lanczos algorithm is described to trace discrete solution curves. Numerical experiments on the Brusselator equations are reported.

Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking

Recently it has been observed, that perturbations of symmetric ODE’s can lead to highly nontrivial dynamics. In this paper we want to establish a similar result for certain nonlinear partial differential systems. Our results are applied to equations which are motivated from chemical reactions. In fact we show that the theory applies to the Brusselator on a sphere. To be more precise, we consider solutions of a semi-linear parabolic equation on the 2-sphere. When this equation has an axisymmetric equilibrium $u_\alpha$, the group orbit of $u_\alpha$ (under rotations) gives a whole (invariant) manifold $M$ of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) $L\subset {{\textbf {O}(3)}}$-equivariant perturbation, $M$ persists as an invariant manifold $\widetilde M$. However, the flow on $\widetilde M$ is in general no longer trivial. Indeed, we find slow dynamics on $\widetilde M$ and, in the case $L=\mathbb {T}$ (the tetrahedral subgroup of ${{\textbf {O}(3)}}$), we observe heteroclinic cycles. In the application to chemical systems we would expect intermittent behaviour. However, for the Brusselator equations this phenomenon is not stable. In order to see it in a physically relevant situation we need to introduce further terms to get a higher codimension bifurcation.

Numerical Continuation at Double Bifurcation Points of a Reaction–Diffusion Problem

We study double bifurcations of a reaction–diffusion problem, and numerical methods for the continuation of bifurcating solution branches. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce imperfection of bifurcations, we consider a preservation of multiplicities of the bifurcation points in the discrete problems. A continuation-Arnoldi algorithm is exploited to trace the solution branches, and to detect secondary bifurcations. Numerical results on the Brusselator equations confirm our analysis.

Hopf bifurcation near a double singular point

The nonlinear equation f( x, λ, α) = 0, f: X × R 2→ X, where X is a Banach space and f satisfies a Z 2-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric. We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.

Parallel Implementations of Iterated Runge-Kutta Methods

We investigate different parallel algorithms for the iter ated Runge-Kutta method on distributed memory mul tiprocessors for the solution of systems of ordinary differential equations (ODEs). The iterated Runge-Kutta method is an iteration scheme for the numerical solu tion of initial-value problems of nonstiff ODEs; embed ded approximation formulae are used to control the step size. The parallel algorithms realize potential par allelism across the method and the system in a group- SPMD (single-program, multiple-data) programming style, using an appropriate set of communication prim itives that can be implemented on all common topolo gies. A theoretical performance analysis with run-time formulae and a run-time simulation show the value of the algorithms. The implementation on the Intel iPSC/ 860 confirms the predicted run times. The speedup values depend strongly on the particular system of ODEs to be solved. The parallel iterated Runge-Kutta method is applied to a typical discretization problem, the discretized Brusselator equation. Application-spe cific modifications of the general parallel ODE solver are developed, which result in a considerable reduc tion in the parallel execution time.

Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Several results concerning asymptotical mean square stability of equilibria of specific linear stochastic systems are presented and proven. These discrete time systems can be interpreted as numerical solution of stochastic differential equations driven by Wiener noise. Effects of the presented mean square calculus are shown by the Kubo oscillator perturbed by white noise and a simplified system of noisy Brusselator equations

Normal form for Hopf bifurcation of partial differential equations on the square

We derive and analyse a normal form governing dynamics of Hopf bifurcations of partial differential evolution equations on a square domain. We assume that the differential operator for the linearized problem decomposes into two one-dimensional self-adjoint operators and a local 'reaction' operator; this gives a basis of i.e. of the form u(x1,x2)=f1(x1)f2(x2) The normal form reduces to that investigated by Swift (1988) for bifurcation of modes with odd parity but is new for modes with even parity where the centre eigenspace carries a reducible action of D4*Si. We consider the Brusselator equations as an example and discover that a separable linearization introduces a degeneracy which causes the three new third order terms in the normal form to be related in an unexpected but simple way.

DETERMINING THE DELAY TIME OF THE TWO-DIMENSIONAL RECONSTRUCTION FOR ORDINARY DIFFERENTIAL EQUATIONS

The second order Jacobian J has been calculated and the delay time τ for the two-dimensional reconstruction of ordinary differential equations has been determined according to the first maximum of the absolute value of J. This method reveals more features about reconstruction than others, and is used in the Rössler and the forced Brusselator equation.