Hey abscissa.

Although I don't know the answer to your question, I'm wondering whether there is a result for uniqueness in terms of the Jacobian of the matrix for the ODE system.

Usually when it comes to properties like this involving systems of equations, the determinant, eigenvectors and eigen-values will usually always play some role in determining the answer for these kinds of questions.

I haven't had a class in DE's for many years but I do remember that these played a role in convergence and uniqueness.