I have the following iterative mapping: $$x_{n+1} = (N-n)^{-1} \frac{x_n}{f(x_n)} \left(C - \sum_{i=1}^n f(x_i)\right)$$ defined for $n \leq N$ and where $C > 0$ is some constant. I am trying to analyze the potential convergence to fixed points or instability of such a system, in particular when we assume the function $f(x)$ is nonnegative and monotonic increasing.
By making the simplifying assumption that $f(x) = x^k$ for some $k$, we can derive convergence to a fixed point for $k$ restricted to a small region of the parameter space as well as instability and chaos in line with the logistic map as we increase, but it is not clear to me how to analyze this iterative map for general $f(x)$? Or potentially just $L$-Lipschitz $f(x)$?