almost linear ODE Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$
$$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$
where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $r^+=max\{0,r\}.$ 
Does this type of ODE have a name? And are there any known criterias for  stability? Has it been studied by anyone in general?
The closest i have found is the "Threshold-Linear networks" studied here for example. I appreciate any reference similar to this system.
 A: Let me change slightly your notations: you want to study the system of ODE
$$
\dot x=Ax+Bx_+,\quad \text{where $x_+=(x_{j,+})_{1\le j\le n}$}\tag{$\ast$}.
$$
A first remark is that the flux $F(x)=Ax+Bx_+$ is globally Lipschitz-continuous, i.e. has a globally bounded (distribution) derivative. Since 
$\Vert B x_+ \Vert\le C\vert x\vert$,
 the general theory of ODE is providing unique solutions for a given initial data and in fact a flow $\phi(t,y)$ so that the unique solution of $(\ast)$ with value $y$ at $t=0$ is $\phi(t,y)$,
$$
\dot\phi(t,y)=A\phi(t,y)+B\phi(t,y)_+, \quad \phi(0,y)=y.
$$
By the way, that flow is also Lipschitz-continuous with respect to the $y$ variables.
A second remark is that you can get rid of the linear part by conjugation: setting
$
x=e^{tA} X,
$ 
you get
$$
e^{tA} \dot X+Ae^{tA} X=Ae^{tA} X+B(e^{tA} X)_+, \quad\text{i.e}\quad
\dot X=e^{-tA}B(e^{tA} X)_+.
$$
A third remark is that for $n=1$, you can separate the variables and get formally
$$
\frac{dx}{ax+bx_+}=dt\quad\text{i.e.}\quad \frac{dX}{X_+}=bdt\quad\text{i.e.}\quad
\ln (X_+(t)/X_+(0))=bt\quad\text{i.e.}\quad
X_+(t)=e^{bt}X_+(0).
$$
In fact we see that in one dimension
$ 
\phi(t,y)=e^{ta}e^{bt} y\mathbf 1_{\mathbb R_+}(y)+
\mathbf 1_{\mathbb R_-}(y)e^{ta} y.
$
Now for studying stability, you may for instance start with the case where $A$ is a self-adjoint real-valued matrix: you can diagonalize $A$ (and thus $e^{tA}$) in an orthonormal basis of $\mathbb R^n$. Since the matrix $e^{tA}$ is diagonal and positive, you get that
$$
(e^{tA} X)_+=e^{tA} X_+,
$$
and thus the system is
$
\dot X=e^{-tA}Be^{tA} X_+.
$
You need to study the spectrum of the matrix $e^{-tA}Be^{tA}$ and say if the matrix $B$ commutes with $A$, you are reduced to study the stability of 
$
\dot X=BX_+,
$
and we may note that 
$$
\frac{d}{dt}\Vert X\Vert^2=2\langle B X_+, X\rangle\le C\Vert X\Vert^2,
$$
and Gronwall's inequality gives you $\Vert X(t)\Vert^2\le e^{Ct}\Vert X(0)\Vert^2$. 
