If we let $U_\mu$ be a vector field that associates a direction vector $U_\mu(\pi)$ with each $\pi \in $ unit simplex. Each such vector field is associated with a system of ODEs: $$ \pi'(u) = U_\mu(\pi(u)), \quad \pi(0) = \pi \in \text{Unit Simplex} $$ I am working with the ODE: $$ U_\mu \propto \nabla H(\pi|\mu) =\pi_i'(u) = \log \frac{\pi_i(u)}{\mu_i} - \frac{1}{n} \sum_{j=1}^n \log \frac{\pi_j(u)}{\mu_j} ~~~~~i \in \{1,2,...,n \} $$ with condition: $\pi_i(0) = p_i \in$ Unit Simplex.
Where $H(\pi|\mu) = \sum_{i=1}^n \pi_i \log \frac{\pi_i}{\mu_i}$. I am wondering how I could even show that a solution exists, and if it does, how I would tackle a problem like this given the dependencies of equation $i$ on all $j=1,2,..,n$? Or if this kind of equation can only be solved numerically?