Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

2020 ◽  
pp. 55-112 ◽  
Author(s):  
Hermen Jan Hupkes ◽  
Leonardo Morelli ◽  
Willem M. Schouten-Straatman ◽  
Erik S. Van Vleck
Keyword(s):  
Pattern Formation ◽  
Traveling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations

Stripes or spots? Nonlinear effects in bifurcation of reaction—diffusion equations on the square

1991 ◽  
Vol 434 (1891) ◽  
pp. 413-417 ◽  
Keyword(s):  
Pattern Formation ◽  
General Pattern ◽  
Reaction Diffusion ◽  
Nonlinear Effects ◽  
Neural Nets ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Two Dimensional ◽  
Linearized Model ◽  
Selection Of

Bifurcation to spatial patterns in a two-dimensional reaction—diffusion medium is considered. The selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model. The absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns. Examples that include neural nets and more general pattern formation equations are considered.

IMAGE PROCESSING AND SELF-ORGANIZING CNN

2005 ◽  
Vol 15 (09) ◽  
pp. 2939-2958 ◽  
Author(s):  
MAKOTO ITOH ◽  
LEON O. CHUA
Keyword(s):  
Image Processing ◽  
Pattern Formation ◽  
Vector Fields ◽  
Reaction Diffusion ◽  
Neural Field ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Self Organization ◽  
Field Equations ◽  
Neural Field Equations

CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields.

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