Preliminaries
Let $A = [0, 1]^N$. Consider a dynamical system $\dot{x} = f(x)$, where $x = x(t) : \mathbb{R} \to \mathbb{R}^N$ and $f : A \to \mathbb{R}^N$, where each $f_i$ is in the class $\mathcal{C}^2(A).$
Suppose that the following holds true:
For any initial condition $x(0) \in A$, then the corresponding solution $x(t)$ belongs to the set $A$ for all $t > 0$.
The problem
Which are the conditions so that any solution starting in the set $A$ converges (asymptotically) to a steady state in the space $A$?
In other words, which are the conditions so that
$$\forall x_0 \in A ~\exists y \in A : \lim_{t \to +\infty} x(t) = y.$$
Maybe the question can be restated also as follows: which are the conditions so that the system does not exhibit cyclic solution like centers of limit cycle, or other behavior like chaos?
Are there any theoretical results that can help me?