Let's consider the full system :

(1) $latex \displaystyle{ \Large \frac{d}{dt} z_j = (\mu +i \omega) z_j - z_j |z_j|^2 + e^{i \theta} A(t,x_j)   }$  

(2) $latex \displaystyle{ \Large A_t = \alpha ~ \sum_j^N z_j~ \delta(x-x_j) ~-\gamma A } + D A_{xx}$

In a frame rotating at a frequency $latex \displaystyle { \large \Omega }$, the system writes :

(1') $latex \displaystyle{ \Large \frac{d}{dt} z_j = (\mu +i (\omega-\Omega) z_j - z_j |z_j|^2 + e^{i \theta} A(t,x_j)   }$  

(2') $latex \displaystyle{ \Large A_t = \alpha ~ \sum_j^N z_j~ \delta(x-x_j) ~-(\gamma + i\Omega ) A } + D A_{xx}$

or :

$latex \displaystyle{ \Large \frac{d}{dt} z_j = F(z_j,A)   }$  

$latex \displaystyle{ \Large A_t = G(z_j,A) }$

We cannot compute the jacobian of this system directly because F is not differentiable in the complex sense. (ie. the condition $latex \displaystyle{ \Large \frac{\partial F(x,y,A)}{\partial x}+i\frac{\partial F(x,y,A)}{\partial y} = 0 }$ is not fulfilled )

$latex  \displaystyle {\Large L= \left[\begin{array}{cccc} \mu-(x^{2}+y^{2})-2x^{2} & -(\omega-\Omega)-2xy & \cos(\theta) & -\sin(\theta)\\ (\omega-\Omega)-2xy & \mu-(x^{2}+y^{2})-2y^{2} & \sin(\theta) & \cos(\theta)\\ \alpha\sum_{j}\delta(\xi-\xi_{j}) & 0 & -\gamma+D\partial_{xx} & \Omega\\ 0 & \alpha\sum_{j}\delta(\xi-\xi_{j}) & -\Omega & -\gamma+D\partial_{xx} \end{array}\right] }$

eigenvector belonging to the kernel of the Jacobian : 

If (Z*, A*) is a stable state, then $latex\displaystyle { \large \sigma( \displaystyle { \small \frac{-\pi}{2} }) (Z^*,A^*) }$ is an eigenvector belonging to the kernel of the Jacobian, where $latex \displaystyle { \large \sigma( \displaystyle { \small \frac{-\pi}{2} }) }$ is a rotation matrix...

If Z* and A* are complex numbers, then it becomes easy to write the rotation as $latex \displaystyle { \large e^{\frac{-\pi}{2}} (Z^*,A^*) }$

We descretize the system and write $latex \displaystyle {Z_i}$ the coordinates of oscillator $latex \displaystyle {i}$ at position $latex \displaystyle {x_i}$, and $latex \displaystyle {A_i=A(x_i)}$