scholarly journals Instability of pulses in gradient reaction–diffusion systems: a symplectic approach

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170187 ◽  
Author(s):  
M. Beck ◽  
G. Cox ◽  
C. Jones ◽  
Y. Latushkin ◽  
K. McQuighan ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Maslov Index ◽  
Homoclinic Orbits ◽  
Reaction Diffusion Equation ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Symmetry Argument ◽  
Constant State ◽  
The Stability ◽  
Gradient Nonlinearity

In a scalar reaction–diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state’s critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction–diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have non-zero Maslov index, and hence be unstable. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

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.

scholarly journals Stability of reaction–diffusion systems with stochastic switching

2019 ◽  
Vol 24 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Lijun Pan ◽  
Jinde Cao ◽  
Ahmed Alsaedi
Keyword(s):  
Direct Method ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Fubini Theorem ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Stochastic Switching ◽  
The Stability ◽  
Independent And Identically Distributed

In this paper, we investigate the stability for reaction systems with stochastic switching. Two types of switched models are considered: (i) Markov switching and (ii) independent and identically distributed switching. By means of the ergodic property of Markov chain, Dynkin formula and Fubini theorem, together with the Lyapunov direct method, some sufficient conditions are obtained to ensure that the zero solution of reaction–diffusion systems with Markov switching is almost surely exponential stable or exponentially stable in the mean square. By using Theorem 7.3 in [R. Durrett, Probability: Theory and Examples, Duxbury Press, Belmont, CA, 2005], we also investigate the stability of reaction–diffusion systems with independent and identically distributed switching. Meanwhile, an example with simulations is provided to certify that the stochastic switching plays an essential role in the stability of systems.

scholarly journals On Classical Solutions for A Kuramoto–Sinelshchikov–Velarde-Type Equation

Algorithms ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 77 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Thermal Conduction ◽  
Type Equation ◽  
Reaction Diffusion ◽  
Classical Solutions ◽  
Well Posedness ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
The Cauchy Problem ◽  
The Stability ◽  
Plane Flame

The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

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.

Analysis of bifurcations in reaction–diffusion systems with no-flux boundary conditions: the Sel'kov model

1995 ◽  
Vol 125 (2) ◽  
pp. 413-438 ◽  
Author(s):  
J. E. Furter ◽  
J. C. Eilbeck
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Singularity Theory ◽  
Reaction Diffusion Equation ◽  
Reaction Diffusion Systems ◽  
Local Bifurcation Analysis ◽  
Diffusion Systems ◽  
Flux Boundary Conditions ◽  
Complex Picture ◽  
The One

A plot of the bifurcation diagram for a two-component reaction-diffusion equation with no-flux boundary conditions reveals an intricate web of competing stable and unstable states. By studying the one-dimensional Sel'kov model, we show how a mixture of local, global and numerical analysis can make sense of several aspects of this complex picture. The local bifurcation analysis, via the power of singularity theory, gives us a framework to work in. We can then fill in the details with numerical calculations, with the global analytic results fixing the outline of the solution set. Throughout, we discuss to what extent our results can be applied to other models.

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