Let us assume that we have the following nonlinear system, depicted in the above figure:
$\ddot{x_{11}} = a\cdot Sigm(x_{31}-x_{21})-b\cdot\dot{x_{11}}-c\cdot x_{11}$
$\ddot{x_{21}} = d\cdot Sigm(x_{11})-k\cdot\dot{x_{21}}-g\cdot x_{21}$
$\ddot{x_{31}} = a\cdot(Sigm(x_{11})+p_1)-b\cdot\dot{x_{31}}-c\cdot x_{31}$
and let us assume that we want to couple it with an identical one:
$\ddot{x_{12}} = a\cdot Sigm(x_{32}-x_{22})-b\cdot\dot{x_{12}}-c\cdot x_{12}$
$\ddot{x_{22}} = d\cdot Sigm(x_{12})-k\cdot\dot{x_{22}}-g\cdot x_{22}$
$\ddot{x_{32}} = a\cdot(Sigm(x_{12})+p_2)-b\cdot\dot{x_{32}}-c\cdot x_{32}$
Info about it:
all $x$ represent potential, $\dot{x}$ could represent current for constant capacitance
$p_1$ is the output of the second system which ends up as input for the first
$p_2$ is the output of the first system which ends up as input for the second
and $Sigm(x)$ is a sigmoid function
The Question:
How could we express the occurring system after the coupling and rewrite its equations, using the relationship potential-current between the state variables and their derivatives?