Nonlinear matrix differential equation

Question

I want to solve the equilibrium of the following differential equation:

$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$

which is essentiall in matrix notation:

$\dot{\mathbf{x}} = A\mathbf{x} + \mathrm{diag}(\mathbf{x)}B\mathbf{x}$ with
$x\in \mathbb{R}^n$ and $A,B\in \mathbb{R}^{n\times n}$.

I wondered if you had any idea how to approach the nonlinear part?
I found the paper (1) which gives some hints for approximations, but essentially it is of no help. Maybe you know how to deal with it?

Thanks in Advance!

(1) Elliot W.Montroll: On coupled Rate Equations with Quadratic Nonlinearities

http://www.jstor.org/stable/61810?seq=1#page_scan_tab_contents

Comments

Are you looking for explicit solutions? They probably don't exist except in special cases. 

Or for numerical methods? Equlibrium means that you just have a nonlinear system of equations. Thus a damped Newton's iteration (or in the high-dimensional case a Broyden method) would be adequate.