PATTERN FORMATION IN FRACTIONAL REACTION–DIFFUSION SYSTEMS WITH MULTIPLE HOMOGENEOUS STATES

2012 ◽  
Vol 22 (04) ◽  
pp. 1250087 ◽  
Author(s):  
BOHDAN DATSKO ◽  
YURY LUCHKO ◽  
VASYL GAFIYCHUK
Keyword(s):  
Reaction Diffusion ◽  
Self Organization ◽  
Cubic Nonlinearity ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Standard Reaction ◽  
Simulation Results ◽  
Spatio Temporal ◽  
Derivatives Of ◽  
Fractional Reaction

This paper is devoted to the investigation of self-organization phenomena in time-fractional reaction–diffusion systems with multiple homogeneous states. It is shown that the fractional reaction–diffusion systems possess some new properties compared to the systems with derivatives of integer orders. In particular, some complex spatio-temporal solutions that cannot be found in the standard reaction–diffusion systems are identified. The simulation results are presented for the case of a incommensurate time-fractional reaction–diffusion system with a cubic nonlinearity.

Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point

2018 ◽  
Vol 21 (1) ◽  
pp. 237-253 ◽  
Author(s):  
Bohdan Datsko ◽  
Vasyl Gafiychuk
Keyword(s):  
Nonlinear Dynamics ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Homogeneous State ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal ◽  
Complex Solutions ◽  
The Given ◽  
Fractional Reaction

Abstract In this article, we study complex spatio-temporal solutions in nonlinear time-fractional reaction-diffusion systems. The main attention is paid to nonlinear dynamics near a bifurcation point. Despite the fact that the homogeneous state is stable at the parameters lower than bifurcation ones, a variety of complex solutions can also form in the subcritical domain. As an example, we consider a generalized fractional FitzHugh-Nagumo model. Depending on the given standard bifurcation parameters and the order of fractional derivative, the new types of steady auto-wave solutions in such systems have revealed. By computer simulation, it is shown that fractional reaction-diffusion possess much more complex nonlinear dynamics than their integer counterparts even at a subcritical bifurcation.

scholarly journals Self-organization of nanoparticles and molecules in periodic Liesegang-type structures

2021 ◽  
Vol 7 (16) ◽  
pp. eabe3801
Author(s):  
Amanda J. Ackroyd ◽  
Gábor Holló ◽  
Haridas Mundoor ◽  
Honghu Zhang ◽  
Oleg Gang ◽  
...  
Keyword(s):  
Constant Velocity ◽  
Periodic Structures ◽  
Reaction Diffusion ◽  
Self Organization ◽  
Reaction Diffusion Systems ◽  
Structural Hierarchy ◽  
Preparation Conditions ◽  
Diffusion Systems ◽  
Film Preparation ◽  
Self Organizing

Chemical organization in reaction-diffusion systems offers a strategy for the generation of materials with ordered morphologies and structural hierarchy. Periodic structures are formed by either molecules or nanoparticles. On the premise of new directing factors and materials, an emerging frontier is the design of systems in which the precipitation partners are nanoparticles and molecules. We show that solvent evaporation from a suspension of cellulose nanocrystals (CNCs) and l-(+)-tartaric acid [l-(+)-TA] causes phase separation and precipitation, which, being coupled with a reaction/diffusion, results in rhythmic alternation of CNC-rich and l-(+)-TA–rich rings. The CNC-rich regions have a cholesteric structure, while the l-(+)-TA–rich bands are formed by radially aligned elongated bundles. The moving edge of the pattern propagates with a finite constant velocity, which enables control of periodicity by varying film preparation conditions. This work expands knowledge about self-organizing reaction-diffusion systems and offers a strategy for the design of self-organizing materials.

Turing and Benjamin–Feir instability mechanisms in non-autonomous systems

2020 ◽  
Vol 476 (2238) ◽  
pp. 20200003 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Autonomous Systems ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Time Dependent ◽  
Model Parameters ◽  
Time Varying ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Dependent Model ◽  
Spatio Temporal

The Turing and Benjamin–Feir instabilities are two of the primary instability mechanisms useful for studying the transition from homogeneous states to heterogeneous spatial or spatio-temporal states in reaction–diffusion systems. We consider the case when the underlying reaction–diffusion system is non-autonomous or has a base state which varies in time, as in this case standard approaches, which rely on temporal eigenvalues, break down. We are able to establish respective criteria for the onset of each instability using comparison principles, obtaining inequalities which involve the in general time-dependent model parameters and their time derivatives. In the autonomous limit where the base state is constant in time, our results exactly recover the respective Turing and Benjamin–Feir conditions known in the literature. Our results make the Turing and Benjamin–Feir analysis amenable for a wide collection of applications, and allow one to better understand instabilities emergent due to a variety of non-autonomous mechanisms, including time-varying diffusion coefficients, time-varying reaction rates, time-dependent transitions between reaction kinetics and base states which change in time (such as heteroclinic connections between unique steady states, or limit cycles), to name a few examples.

scholarly journals A Nonlocal Eigenvalue Problem and the Stability of Spikes for Reaction–Diffusion Systems with Fractional Reaction Rates

2003 ◽  
Vol 13 (06) ◽  
pp. 1529-1543 ◽  
Author(s):  
Juncheng Wei ◽  
Matthias Winter
Keyword(s):  
Eigenvalue Problem ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Stability ◽  
Nonlocal Eigenvalue Problem ◽  
Fractional Reaction

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction–diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray–Scott system, the hypercycle of Eigen and Schuster, angiogenesis, and the generalized Gierer–Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters.

Noise-induced suppression of periodic travelling waves in oscillatory reaction–diffusion systems

2010 ◽  
Vol 466 (2119) ◽  
pp. 1903-1917 ◽  
Author(s):  
Michael Sieber ◽  
Horst Malchow ◽  
Sergei V. Petrovskii
Keyword(s):  
Travelling Waves ◽  
Natural Populations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Oscillatory Reaction ◽  
Stochastic Forcing ◽  
Spatial Direction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

Ecological field data suggest that some species show periodic changes in abundance over time and in a specific spatial direction. Periodic travelling waves as solutions to reaction–diffusion equations have helped to identify possible scenarios, by which such spatio-temporal patterns may arise. In this paper, such solutions are tested for their robustness against an irregular temporal forcing, since most natural populations can be expected to be subject to erratic fluctuations imposed by the environment. It is found that small environmental noise is able to suppress periodic travelling waves in stochastic variants of oscillatory reaction–diffusion systems. Irregular spatio-temporal oscillations, however, appear to be more robust and persist under the same stochastic forcing.

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