Different Types of Instabilities and Complex Dynamics in Fractional Reaction-Diffusion Systems

2009 ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Fractional Derivatives ◽  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Main Attention ◽  
Van Der Pol ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Different Types ◽  
Pattern Formations

In this article we analyze a influence of possible instabilities on pattern formations in the reaction-diffusion systems with fractional derivatives. The results of qualitative analysis are confirmed by numerical simulations. The main attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer-van der Pol kinetics and to Brusselator model.

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Vasyl Gafiychuk ◽  
Bohdan Datsko
Keyword(s):  
Complex Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Linearized System ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Activator Inhibitor ◽  
Fractional Reaction

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

scholarly journals The space spectral interpolation collocation method for reaction-diffusion systems

2021 ◽  
pp. 22-22
Author(s):  
Xiao-Li Zhang ◽  
Wei Zhang ◽  
Yu-Lan Wang ◽  
Ting-Ting Ban
Keyword(s):  
Collocation Method ◽  
Complex Dynamics ◽  
Solution Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spectral Interpolation ◽  
Fractal Calculus ◽  
Pattern Formations ◽  
Good Agreement

A space spectral interpolation collocation method is proposed to study nonlinear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.

scholarly journals Collision patterns on mollusc shells

1997 ◽  
Vol 1 (1) ◽  
pp. 57-76 ◽  
Author(s):  
P. J. Plath ◽  
J. K. Plath ◽  
J. Schwietering
Keyword(s):  
Reaction Diffusion ◽  
Growth Front ◽  
Cell System ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Different Types ◽  
Classical Waves ◽  
Cellular Behaviour ◽  
Mollusc Shells

On mollusc shells one can find famous patterns. Some of them show a great resemblance to the soliton patterns in one-dimensional systems. Other look like Sierpinsky triangles or exhibit very irregular patterns. Meinhardt has shown that those patterns can be well described by reaction–diffusion systems [1]. However, such a description neglects the discrete character of the cell system at the growth front of the mollusc shell.We have therefore developed a one-dimensional cellular vector automaton model which takes into account the cellular behaviour of the system [2]. The state of the mathematical cell is defined by a vector with two components. We looked for the most simple transformation rules in order to develop quite different types of waves: classical waves, chemical waves and different types of solitons. Our attention was focussed on the properties of the system created through the collision of two waves.

Wawelet transform application for classification of solutions of reaction-diffusion systems

2017 ◽  
Vol 2017 (45) ◽  
pp. 50-55
Author(s):  
Z.I. Vasjunyk ◽  
◽  
Y.I. Maksymiv ◽  
V.V. Meleshko ◽  
◽  
...  
Keyword(s):  
Wavelet Transforms ◽  
Type System ◽  
Reaction Diffusion ◽  
Oscillatory Solutions ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Classification Of Solutions ◽  
Different Types ◽  
Wavelet Transformations

Girer–Meynxardt type system of reaction-diffusion with classical derivatives and Bryusselyator system with fractional time derivatives are investigated. On the basis of computer simulations it is shown that qualitatively different types of oscillatory solutions may arise due to instability in these systems. Wavelet transformations are applied to analyze and classify the solutions of such systems, A comparative estimation of wavelet transforms of oscillatory and chaotic solutions is given and it is shown that such method of classification of solutions is effective.

Moving Finite Element Simulations for Reaction-Diffusion Systems

2012 ◽  
Vol 4 (03) ◽  
pp. 365-381 ◽  
Author(s):  
Guanghui Hu ◽  
Zhonghua Qiao ◽  
Tao Tang
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Large Solution ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Monitor Functions ◽  
Moving Finite Element

AbstractThis work is concerned with the numerical simulations for two reaction-diffusion systems, i.e., the Brusselator model and the Gray-Scott model. The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients. High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions. Unlike [33], this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model. Several ways for verifying the quality of the numerical solutions are also proposed, which may be of important use for comparisons.

scholarly journals Analysis of stochastic sensitivity of Turing patterns in distributed reaction-diffusion systems

2020 ◽  
Vol 55 ◽  
pp. 155-163
Author(s):  
A.P. Kolinichenko ◽  
L.B. Ryashko
Keyword(s):  
Diffusion Coefficients ◽  
Stochastic Dynamics ◽  
Random Noise ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Reaction Diffusion Systems ◽  
Brusselator Model ◽  
Stochastic Sensitivity ◽  
Diffusion Systems ◽  
The Mean

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed.

Multistability and Stochastic Phenomena in the Distributed Brusselator Model

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Alexander Kolinichenko ◽  
Lev Ryashko
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatial Structures ◽  
Reaction Diffusion Systems ◽  
Corporate Influence ◽  
Basic Model ◽  
Brusselator Model ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Stochastic Phenomena

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.

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