Suppose we have an ODE in $\mathbb{R}^4$ $$\dot{x}=Dx +E(x)x$$ for a diagonal matrix $D$ with negative entries and a diagonal matrix $E=(-\mu_1(x_1), \mu_1(x_2), -\mu_2(x_3), \mu_2(x_4))$.
for every initial point $x_0$ with nonnegative entries the trajectories are bounded.
$\mu:[0,\infty[$ is concave, postive except at $\mu(0)=0=\mu(\infty)$
Then, every solution (with nonnegative starting point) is asymptotically stable? Can I slightly change the ODE (or its properties) in order to obtain asymptotically stability?
Hint. As $D$ and $E(x)$ are both diagonal, then your system is uncoupled and reduces to 4 independent scalar ODEs of the form $$ x'=-\lambda x+\mu(x)x, $$ where $\lambda>0$ and $\mu(x)$ is continuous and $\mu(0)=0$.
As $\mu(0)=0$, then there exists a $\xi>0$, such that $$ |x|<\xi\Longrightarrow \mu(x)<\frac{\lambda}{2}. $$ Now for $|x_0|<\xi$ the IVP $$ x'=-\lambda x+\mu(x)x, \quad x(0)=x_0, $$ is asymptotically stable. To see this, if $\varphi(t)$ is a solution, then, due to continuity of $\varphi$, there is a $T>0$, such that $|\varphi(t)|<\xi$ is $[0,T]$. Thus, in $[0,T]$ we have that $$ \varphi'(t)-(-\lambda+\mu(\varphi(t)))\varphi(t)=0 \Longrightarrow \varphi(t)=x_0\exp\left(\int_0^t (-\lambda+\mu(\varphi(s)))\,ds\right) $$ and hence, for all $t\in [0,T]$. $$ |\varphi(t)|\le x_0\exp(-\lambda t/2). $$ This implies that $\varphi(t)$ remains bounded by $\xi$, for all $t>0$, and $$ \lim_{t\to\infty} \varphi(t)=0. $$