Nonlinear matrix differential equation 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
 A: Sorry if my notation will be a bit sloppy. I assume you want to find $x$ such that
$$Ax+diag(x)Bx=Ax+diag(Bx)x=0.$$
I dont know if there is an explicit expression but we can certainly do Newton. Linearising we get
$$Ax+diag(x)Bx+(A+diag(x)B+diag(Bx))dx+O(|dx|^2)=0.$$
Hence the Newtoniteration is
$$\phi(x)=x-(A+diag(x)B+diag(Bx))^{-1}(Ax+diag(x)Bx).$$
I tested the iteration in Matlab and it seemed to converge:
n=5;
A=eye(n);
B=eye(n);
x=-rand(n,1);
for i=1:10
    (A*x+diag(x)Bx)
    x=x-(A+diag(x)B+diag(Bx))(A*x+diag(x)Bx);
end
x
However I am not sure if it will converge to the correct equilibrium. There can be many equilibria, for example if $A=B=I$ any vector $x=(x_1,\dots,x_n)$ with $x_i \in \{-1,0\}$ is an equilibrium.
A: Of course $x=0$ is always a solution.  There may be others.  I suspect the number of (complex) solutions is generically $2^n$.  I don't know how many of these can be real.
If $n$ is not too big, you might find the solutions by using methods involving polynomial ideals (Groebner bases, regular chains, ...).  
For example, it might amuse you to try 
$$ A = \pmatrix{-2 & -1 & -1\cr -2 & -1 & 2\cr 2 & -2 & -2\cr},\ B = \pmatrix{1 & 2 & -1\cr 0 & 2 & -2\cr -2 & -2 & 1\cr}$$
where there are $8$ real solutions.  
