Analytically solving a 2 ODE system with one nonlinear coupling, $\frac{\text{d}y}{\text{d}\tau} = - \mu y +\Delta [(1-z)-Ezy ]$ Can anyone help solve this non-linear ODE? I've been attempting to solve it using asymptotics, however I have found this approach falls apart under many parameter regimes of interest, I was wondering if there is a way to solve the full system analytically.
The problem can be defined as the following:
\begin{align}
%%% vitreous %%%
\frac{\text{d}y}{\text{d}\tau} &=
- \mu y
%
+\Delta
\left[
(1-z)-Ezy
\right],
\\ 
%%% vitreal network %%%
\frac{\text{d}z}{\text{d}\tau} &=
(1-z)
-Ezy.
\end{align}
Where $z(0)=y_0\in(0,1)$ and $z(0)=z_0\in(0,1)$, and
$$0<\mu\ll E,\Delta$$
$$E,\Delta \sim O(1)$$
This is roughly what the system looks like when numerically solved, the 2 rightmost panels are dimensional $y,z$ from middle to right. (note, $E=$1/eps)

This looks quite similar to the Lorenz system.  I wonder if it can be transformed to that system.  In any case, a closed-form solution is not to be expected.  You may get chaotic behaviour in some parameter regimes.
EDIT: Depending on parameter values, there may be stable fixed points: your screenshot may be evidence of approach to some fixed point.  In particular, $x=0, y=0, z=1$ is always a fixed point.  Depending on the parameters, this may be stable or unstable.
In particular, this is likely to depend on $AD-BC$ which is the determinant of the Jacobian matrix of the linearlization at that fixed point.
