Find a pair of analytic functions? For $ 0 \leq x $ ,
Find Pairs of analytic functions $f,g$ such that
 
$$f(x+1) = 2 f(x)^2 + 3 g(x)^2 + 4 f(x) + 5 g(x) + 6$$
$$g(x+1) = f(x)^2 + 7 g(x)^2 + 8 f(x) + 9 g(x) + 10$$
hold simultanously.
I know some stuff about complex dynamics and I know people who know alot more than me about it.
But for these types of equations I am stuck and so are they.
How to handle this ?
 A: Suppose you have one of the constant solutions. In this case, they are complex numbers, and there are four constant solutions. Then pick one of the solutions and move the fixed points to zero. Then temporarily ignore the $()2$ terms assuming they don't matter when f and g are small enough. Then near the fixed points of zero we have:
$$f(x+1)\approx a_1 f(x)+a_2 g(x);\;\;\;g(x+1)≈b_1 f(x)+b_2 g(x);$$
Next, assume for small x that $g(x) \approx k f(x)$.  This holds exactly while we temporarily ignore the $()^2$ terms.
$$f(x+1)\approx a_1 f(x)+a_2 k f(x);\;\;\;g(x+1)≈b_1 f(x)+b_2 k f(x);$$
edit I assume both f and g have the same periodicity, and have the same $\lambda$ scaling factor.  This seems to be the key to get a Koenig's style exponential solution.  Otherwise, the approximation $g(x) \approx k f(x)$ does not hold even when the absolute value of f and g are small. If there were no $()^2$ terms, then the solution for $S_f$ and $S_g$ below would not have any $x^2$ or higher frequency terms.
$$\lambda = a_1 +a_2 k = b_1 +b_2 k ;\;\;\; k = \frac{a_1-b_1}{b_2-a_2};\;\;\;\lambda=\frac{b_2 a_1 - b_1 a_2}{b_2-a_2}$$
Then we formally solve for the two Schroeder equations using $\lambda$ and $k$ from above, where $\lambda$ is the multiplier at the pair of fixed points.  This works so long as $|\lambda| \ne 1$ although well behaved irrational multipliers with $|\lambda|=1$ work too.
$$S_f(x) = x + c_2 x^2 + c_3 x^3 + ... + c_n x^n;\;\;\;S_g(x) = k x + d_2 x^2 + d_3 x^3 + ... + d_n x^n;\;\;\;$$
$$S_f(\lambda x) = a_1 S_f(x) + a_2 S_g(x) + 2 S_f(x)^2 + 3 S_g(x)^2 $$
$$S_g(\lambda x) = b_1 S_f(x) + b_2 S_g(x) +   S_f(x)^2 + 7 S_g(x)^2 $$
I haven't worked through the details to show the joint formal solution exists, but I think it does.  Then the desired solution to the OP's question is
$$f(x) = S_f(\lambda ^ x)\;\;\; g(x) = S_g(\lambda ^ x)$$
