A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type

2018 ◽  
Vol 20 (2) ◽  
pp. 297-336
Author(s):  
Marie Henry ◽  
Danielle Hilhorst ◽  
Cyrill Muratov
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Multiple Scale ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Inhibitor Type ◽  
Activator Inhibitor Type ◽  
Activator Inhibitor

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.

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