Given an ODE in $\mathbb{R}^n$
$$\frac{d}{dt}x(t)=b(x(t)).$$
How can the assumption that
$$\langle D b \xi,\xi\rangle \leq - c |\xi|,~~~~~\forall \xi \in \mathbb{R}^n$$
be interpreted? Does this stop the trajectory going too wild in some sense? What would this condition be called? (Here $Db$ is the differential of $b$).
Your condition is that $b'$ has all eigenvalues with real part $\leq -c<0$. (Presuming $c>0$ here).
Linearizing near an equilibrium point $x_0\in \{b(x) = 0\}$ and we get $$\dot{x} = b(x_0) + b'(x_0)x + O(2) = b'(x_0)x + O(2)\ .$$ So in a neighborhood of $x_0$ our system will act like $\dot{x} = b'(x_0)x$ which is asymptotically stable. So I think that if for some $t_1$ $x(t_1)$ is close enough to $\{b=0\}$, it will thereafter approach a stable fixed point.
I'm unsure the conditions needed to guarantee one enters such a region of stability.