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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+w2−γvw vwk+w−λw w+A+ϵ¨w −c˙v=w21+w2−v+ϵ¨v −c˙A=αAw−λA A+D¨A
Or : ˙w=W ˙v=V ˙A=B ˙W=1/ϵ(−αww21+w2+γvw vwk+w+λw w−A−cW) ˙V=1/ϵ(−w21+w2+v−cV) ˙B=1/D(−αAw+λA A−c B)
We take ϵ=0.01. The bifurcation diagram look a bit different :
Period :
Amplitude of w (max - min), level set :
StabilityRaw 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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