scholarly journals Spatial dynamics of a nonlocal and time-delayed reaction–diffusion system

2008 ◽  
Vol 245 (10) ◽  
pp. 2749-2770 ◽  
Author(s):  
Jian Fang ◽  
Junjie Wei ◽  
Xiao-Qiang Zhao
Keyword(s):  
Spatial Dynamics ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Delayed Reaction

Bogdanov–Takens Bifurcation of a Class of Delayed Reaction–Diffusion System

2015 ◽  
Vol 25 (06) ◽  
pp. 1550082 ◽  
Author(s):  
Jianzhi Cao ◽  
Peiguang Wang ◽  
Rong Yuan ◽  
Yingying Mei
Keyword(s):  
Center Manifold ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Equivalent Condition ◽  
Zero Eigenvalue ◽  
Delayed Reaction ◽  
Center Manifold Theorem ◽  
Normal Form Method

In this paper, a class of reaction–diffusion system with Neumann boundary condition is considered. By analyzing the generalized eigenvector associated with zero eigenvalue, an equivalent condition for the determination of Bogdonov–Takens (B–T) singularity is obtained. Next, by using center manifold theorem and normal form method, we have a two-dimension ordinary differential system on its center manifold. Finally, two examples show that the given algorithm is effective.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

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