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