scholarly journals Double Hopf Bifurcation in Delayed reaction–diffusion Systems

2019 ◽  
Vol 32 (1) ◽  
pp. 313-358 ◽  
Author(s):  
Yanfei Du ◽  
Ben Niu ◽  
Yuxiao Guo ◽  
Junjie Wei
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Double Hopf Bifurcation ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2011 ◽  
pp. 205-213
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this chapter, we establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

scholarly journals Travelling fronts in time-delayed reaction–diffusion systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180127 ◽  
Author(s):  
Jan Rombouts ◽  
Lendert Gelens ◽  
Thomas Erneux
Keyword(s):  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Delay Systems ◽  
Wave Shape ◽  
Delayed Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Speed Up ◽  
Travelling Fronts ◽  
Time Delayed Feedback

We review a series of key travelling front problems in reaction–diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368 , 483–493 ( doi:10.1098/rsta.2009.0228 )) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

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.

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