I have a system of ODE's with a steady state solution $x^* \in \mathbb{R}^N$ given by the following $N$ implicit equations:
$$x^*_i = \frac{ \sum_{j=1}^N c_{ij} x^*_j}{a_i + \sum_{j=1}^N c_{ij} x^*_j}.$$
where $0 \leq x^*_i \leq 1$, and the values of $c_{ij}$ and $a_i$ are known.
What would be a good way to compute $x^*$?
In case it helps, this is somewhat similar to the eigenvector centrality equation $x^*_i = \frac{1}{\lambda} \sum_{j=1}^N c_{ij} x^*_j$.