Nonlinear matrix differential equation

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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?

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

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

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