Why is solving non-linear recurrence relations "hopeless"? I came across a non-linear recurrence relation I want to solve, and most of the places I look for help will say things like "it's hopeless to solve non-linear recurrence relations in general." Is there a rigorous reason or an illustrative example as to why this is the case? It would seem to me that the correct response would be "we just don't know how to solve them," or "there is no solution using elementary functions," but there might be a solution in the form of, say, an infinite product or a power series or something.
Just for completion, the recurrence relation I'm looking at is (slightly more than just non-linear, and this is a simplified version):
$p_n = a_n b_n\\ a_n = a_{n-1} + c \\ b_n = b_{n-1} + d$
And $a_0 > 0, b_0 > 0, c,d$ fixed constants 

A rather well-known example is the Mandelbrot set. Consider a sequence $(a_n)$ defined by $a_0=0$ and $a_{n+1}=a_n^2+c$, for some given parameter $c$, and ask the seemingly innocuous question: 

For which values of $c$ is the sequence $(a_n)$ bounded? 

One may note that the recursion is simply quadratic rather than affine (the degree of $a_n$ in the formula for $a_{n+1}$ is 2 instead of 1). The answer, or rather, some elements of the answer, have opened an entire mathematical field.
