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.