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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 $x^*_i = \frac{1}{\lambda} \sum_{j=1}^N c_{ij} x^*_j$.

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    $\begingroup$ Can't you rewrite the system as a $F(x^*) = 0$ by putting everything that's on the RHS on the LHS? In this case, the Jacobian seems not too hard to compute exactly and Newton method will converge very quickly $\endgroup$ – G. Gare Feb 11 at 7:15
  • $\begingroup$ Thanks for your suggestion. I edited the question because I had forgotten to say that the $x_i^*$ value must be between 0 and 1. Would Newton method work under these constraints? $\endgroup$ – rpa Feb 11 at 17:35
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    $\begingroup$ Then the problem becomes more complicated. I found this answer scicomp.stackexchange.com/questions/26598/… which presents some method, but no guarantee of convergence whatsoever is given. Moreover, you should prove that there exists a solution in the hypercube $[0, 1]^N$, which I don't see as trivial in this moment. Otherwise, the problem to solve is an optimisation problem (for which you could still employ Newton) $\endgroup$ – G. Gare Feb 12 at 9:53

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