Add diffusion on each specie to make the stability easier

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

$latex \displaystyle{ \Large -c \dot{w} = \alpha_w \frac{w^2}{1 + w^2} - \gamma_{vw} ~ v \frac{w}{k + w} - \lambda_w ~w + A + \epsilon \ddot{w}}$

$latex \displaystyle{ \Large -c \dot{v} = \frac{w^2}{1 + w^2} - v + \epsilon \ddot{v}}$

$latex \displaystyle{ \Large -c \dot{A} = \alpha_A w - \lambda_A ~A + D \ddot{A}}$

Or :

$latex \displaystyle{ \Large \dot{w} = W}$

$latex \displaystyle{ \Large \dot{v} = V}$

$latex \displaystyle{ \Large \dot{A} = B }$

$latex \displaystyle{ \Large \dot{W} = 1/\epsilon ( - \alpha_w \frac{w^2}{1 + w^2} + \gamma_{vw} ~ v \frac{w}{k + w} + \lambda_w ~w - A - c W )}$

$latex \displaystyle{ \Large \dot{V} = 1/\epsilon( -\frac{w^2}{1 + w^2} + v - c V )}$

$latex \displaystyle{ \Large \dot{B} = 1/ D ( -\alpha_A w + \lambda_A ~A - c~B )}$

We take $latex \displaystyle{ \Large \epsilon = 0.01}$.

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), $latex \displaystyle{ \Large w_2(x,0) = w(x,0) + \delta \psi(x) }$ where $latex \displaystyle{ \Large \psi(x) }$ is the eigenfunction associated with eigenvalue $latex \displaystyle{ \Large \lambda }$ :