Multiparameter bifurcation for some particular reaction–diffusion systems

1989 ◽  
Vol 112 (1-2) ◽  
pp. 135-143 ◽  
Author(s):  
J. Esquinas ◽  
J. López-Gómez
Keyword(s):  
Linear Operator ◽  
Linear Part ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Operator Polynomial ◽  
Reaction Diffusion Systems ◽  
Abstract Equation ◽  
Diffusion Systems ◽  
Multiparameter Bifurcation

SynopsisIn some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

ABRAHAMS–TSUNETO REACTION–DIFFUSION SYSTEM IN SUPERCONDUCTIVITY AND SYMBOLIC COMPUTATION

2001 ◽  
Vol 12 (10) ◽  
pp. 1417-1423 ◽  
Author(s):  
YI-TIAN GAO ◽  
BO TIAN ◽  
GUANG-MEI WEI
Keyword(s):  
Symbolic Computation ◽  
Reaction Diffusion ◽  
Analytic Solutions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Ginzburg Landau ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Exact Analytic Solutions ◽  
Self Consistent

The reaction–diffusion systems represent various problems in the real world. For the Abrahams–Tsuneto reaction–diffusion system arising in superconductivity, we perform computerized symbolic computation and find its new exact analytic solutions, which are solitonic. We see the possibility that by way of the shock waves, the self–consistent superconducting interaction drives the Ginzburg–Landau order parameter, which might be observable.

scholarly journals A reaction–diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modelling

2021 ◽  
pp. 1-18
Author(s):  
ROMAN M. CHERNIHA ◽  
VASYL V. DAVYDOVYCH
Keyword(s):  
Exact Solutions ◽  
Reaction Diffusion ◽  
Interesting Case ◽  
Lie Symmetry ◽  
Point Of View ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Diffusion Systems

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

A geometrical approach to wave-type solutions of excitable reaction-diffusion systems

1991 ◽  
Vol 433 (1887) ◽  
pp. 151-164 ◽  
Keyword(s):  
Numerical Solution ◽  
Wave Front ◽  
Eikonal Equation ◽  
Reaction Diffusion ◽  
Stationary Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Wave Boundary

We formulate the eikonal equation approximation for travelling waves in excitable reaction-diffusion systems, which have been proposed as models for a large number of biomedical situations. This formulation is particularly suited, in a natural way, to numerical solution by finite difference methods. We show how this solution is independent of the parametric variable used for expressing the eikonal equation, and how a reduction of dimensionality implies a major saving over the time taken to solve the original reaction-diffusion system. Neumann boundary conditions on reactants in the original system translate into a geometric constraint on the wave boundary itself. We show how this leads to geometrically stable stationary wave boundaries in appropriately shaped non-convex domains. This analytical prediction is verified by numerical solution of the eikonal equation on a domain which supports geometrically stable stationary wave boundary configurations. We show how the concepts of geometrical stability and wave-front stability relate to a problem where a bi-stable reaction-diffusion system has a stable stationary wave-front configuration.

scholarly journals Coloured Noise from Stochastic Inflows in Reaction–Diffusion Systems

2020 ◽  
Vol 82 (4) ◽  
Author(s):  
Michael F. Adamer ◽  
Heather A. Harrington ◽  
Eamonn A. Gaffney ◽  
Thomas E. Woolley
Keyword(s):  
Steady State ◽  
Power Spectrum ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Coloured Noise ◽  
Reaction Systems ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Parameter Values

Abstract In this paper, we present a framework for investigating coloured noise in reaction–diffusion systems. We start by considering a deterministic reaction–diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction–diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction–diffusion system.

scholarly journals An Algebraic Method on the Eigenvalues and Stability of Delayed Reaction-Diffusion Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jian Ma ◽  
Baodong Zheng
Keyword(s):  
Asymptotic Stability ◽  
Linear Operator ◽  
Reaction Diffusion ◽  
Matrix Pencil ◽  
Algebraic Method ◽  
Delayed Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Matrix ◽  
Pure Imaginary Eigenvalues

The eigenvalues and stability of the delayed reaction-diffusion systems are considered using the algebraic methods. Firstly, new algebraic criteria to determine the pure imaginary eigenvalues are derived by applying the matrix pencil and the linear operator methods. Secondly, a practical checkable criteria for the asymptotic stability are introduced.

Mathematical and computational studies of fractional reaction-diffusion system modelling predator-prey interactions

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Kolade M. Owolabi ◽  
Edson Pindza
Keyword(s):  
Stability Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
System Modelling ◽  
Computational Studies ◽  
Mathematical Basis ◽  
Transform Method ◽  
Reaction Diffusion Systems ◽  
Fractional Reaction

Abstract This paper provides the essential mathematical basis for computational studies of space fractional reaction-diffusion systems, from biological and numerical analysis perspectives. We adopt linear stability analysis to derive conditions on the choice of parameters that lead to biologically meaningful equilibria. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. For the solution of the full reaction-diffusion system modelled by the fractional partial differential equations, we introduced the Fourier transform method to discretize in space and advance the resulting system of ordinary differential equation in time with the fourth-order exponential time differencing scheme. Numerical results.

scholarly journals Local controllability of reaction-diffusion systems around nonnegative stationary states

2020 ◽  
Vol 26 ◽  
pp. 55
Author(s):  
Kévin Le Balc’h
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Controllability ◽  
Laplacian Operator ◽  
Stationary States ◽  
Inverse Mapping ◽  
Smooth Bounded Domain ◽  
Reaction Diffusion Systems ◽  
Nonlinear Reaction

We consider a n × n nonlinear reaction-diffusion system posed on a smooth bounded domain Ω of ℝN. This system models reversible chemical reactions. We act on the system through m controls (1 ≤ m < n), localized in some arbitrary nonempty open subset ω of the domain Ω. We prove the local exact controllability to nonnegative (constant) stationary states in any time T > 0. A specificity of this control system is the existence of some invariant quantities in the nonlinear dynamics that prevents controllability from happening in the whole space L∞(Ω)n. The proof relies on several ingredients. First, an adequate affine change of variables transforms the system into a cascade system with second order coupling terms. Secondly, we establish a new null-controllability result for the linearized system thanks to a spectral inequality for finite sums of eigenfunctions of the Neumann Laplacian operator, due to David Jerison, Gilles Lebeau and Luc Robbiano and precise observability inequalities for a family of finite dimensional systems. Thirdly, the source term method, introduced by Yuning Liu, Takéo Takahashi and Marius Tucsnak, is revisited in a L∞-context. Finally, an appropriate inverse mapping theorem in suitable spaces enables to go back to the nonlinear reaction-diffusion system.

Replicating Spots in Reaction-Diffusion Systems

1997 ◽  
Vol 07 (05) ◽  
pp. 1149-1158 ◽  
Author(s):  
Kyoung J. Lee ◽  
Harry L. Swinney
Keyword(s):  
Cell Division ◽  
Numerical Simulations ◽  
Continuous Process ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Diffusion System ◽  
One Dimension ◽  
Reaction Diffusion Systems ◽  
Gel Layer ◽  
Diffusion Systems

We review the phenomenon of replicating spots in reaction-diffusion systems and discuss the mechanism of replication. This phenomenon was discovered in recent experiments on a ferrocyanide-iodate-sulfite reaction-diffusion system. Patterns form in a thin gel layer that is in contact with a continuously fed stirred reservoir. Patterns of spots are observed to undergo a continuous process of growth and multiplication through cell division and death through overcrowding. A similar phenomenon is also found in numerical simulations in one dimension on a four-species model of the ferrocyanide-iodate-sulfite reaction and in simulations in two dimensions of simpler two-species reaction-diffusion models: Gray–Scott model by J. Pearson and FitzHugh–Nagumo model by A. Hagberg and E. Meron.

scholarly journals A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

2015 ◽  
Vol 61 (1) ◽  
pp. 59-78 ◽  
Author(s):  
Monica-Dana Burlică ◽  
Daniela Roşu
Keyword(s):  
Fixed Point ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Nonlocal Initial Conditions ◽  
Diffusion Systems ◽  
Compactness Arguments ◽  
Fixed Point Techniques

Abstract We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C0-solution.

scholarly journals Reaction-diffusion systems with supercritical nonlinearities revisited

2021 ◽  
Author(s):  
Anna Kostianko ◽  
Chunyou Sun ◽  
Sergey Zelik
Keyword(s):  
Linear Part ◽  
Reaction Diffusion ◽  
Strong Solutions ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Standard Case ◽  
Reaction Diffusion Systems ◽  
Supercritical Case ◽  
Diffusion Systems ◽  
Monotonicity Assumption

AbstractWe give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the main attention to the supercritical case, where the nonlinearity is not subordinated to the linear part of the equation trying to put as small as possible amount of extra restrictions on this nonlinearity. The properties of such systems in the supercritical case may be very different in comparison with the standard case of subordinated nonlinearities. We examine the global existence and uniqueness of weak and strong solutions, various types of smoothing properties, asymptotic compactness and the existence of global and exponential attractors.

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