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Add diffusion on each specie to make the stability easier
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Add diffusion on each specie to make the stability easier 

 

 

Making the change of variable z = x -ct in the diffusion coupled system :

 

c˙w=αww21+w2γvw vwk+wλw w+A+ϵ¨w 

c˙v=w21+w2v+ϵ¨v  

c˙A=αAwλA A+D¨A 

 

Or :

˙w=W

˙v=V

˙A=B 

˙W=1/ϵ(αww21+w2+γvw vwk+w+λw wAcW)

˙V=1/ϵ(w21+w2+vcV)  

˙B=1/D(αAw+λA Ac B)   

 

We take ϵ=0.01.

 The bifurcation diagram look a bit different :

  

 

Period :

 

 

Amplitude of w (max - min), level set :

 

 


Stability 

Raw data 

 

 

k=0, black : unstable

k=pi / T, black : unstable

Schematic :

 

 

 

Smoothed version :

 

 

(I)

(II)

 

(III)

 

 

(IV)

 

(V) (Stable)

(VI) (Stable)

 


 

Simulating perturbed stationary solution in the original system 

 

Here is a plot of the norm of the difference between stationary solution w(x,t) and perturbed solution w2(x,t), w2(x,0)=w(x,0)+δψ(x) where ψ(x) is the eigenfunction associated with eigenvalue λ :

 

 

Here is the same plot integrated on space (blue), the red curve is eλt (k = 0):

 

 

If we wait long enough we can also see the perturbation :

 

 

Negative eigenvalue (k=0):

 

 

 

k=pi / T :

 

 

 


 

 

 

 

Spectrum (unstable) :

 

 

Instabilities seems to be in k = pi over T :

 

 


 

\Little circles !

 

 

 

 

 

 

The dynamic of formation of the little circles as alpha decrease :

 

 

 

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