I have a set of real non-linear equations as following:
\begin{equation} y_0 = f(y_0,y_1) \\ y_1 = g(y_0,y_1,y_2) \\ y_2 = g(y_1,y_2,y_3) \\ \vdots \\ y_{n-1} = g(y_{n-2},y_{n-1},y_n) \\ y_n = f(y_{n-1},y_n) \\ \end{equation}
In other words, there are limited number of variables each a function of itself, previous variable, and next variable. Using an iterative computer simulation the solution of this system is found and the convergence is achieved. Is there any analytic method to find the solution?
$$ g(y_0,y_1,y_2) = \frac{1}{\sqrt{2\pi}}\frac{e^{-(y_0 + y_1)^2/8} - e^{-(y_1 + y_2)^2/8}}{Q(\frac{y_0 + y_1}2) - Q(\frac{y_1 + y_2}2)} $$ and $$ f(y_0,y_1) = -\frac{1}{\sqrt{2\pi}}\frac{e^{-(y_0 + y_1)^2/8}}{1 - Q(\frac{y_0 + y_1}2)} $$ where $Q(.)$ is the Q-function.