The motivation for the question is about whether you can have periodic orbits without a stationary point around which to orbit? A weaker (but more formal) statement would be:
Suppose that $\underline{\dot{x}} = f(\underline{x})$ defines an n-dimensional autonomous system of ODEs for $f : [ 0,1 ]^{n} \rightarrow [0,1]^{n}$ with $f_{i}$ polynomial in $x_{1},...,x_{n}$ for all $i \in \{1,...,n\}$. Suppose that there is a periodic orbit that is parameterised by $g : [0,1] \rightarrow [0,1]^{n} ; t \mapsto (g_{1}(t),...,g_{n}(t))$. Does there always exist $\underline{x'}$ such that $f(\underline{x'}) = \underline{0}$ and for all $i \in \{1,...,n \}$ $\min_{t} g_{i}(t) \leq x_{i}' \leq \max_{t} g_{i}(t)$?
Any help would be appreciated.
Edit: For the system that I have in mind, a fixed point is one such that: 1) $f_{i}(\underline{x}) = 0$ for $x_{i} \in (0,1)$. 2) $f_{i}(\underline{x}) \leq 0$ for $x_{i} = 0$. 3) $f_{i}(\underline{x}) \geq 0$ for $x_{i} = 1$. Does the statement hold, but with $f(\underline{x'}) = \underline{0}$ replaced with the previous three conditions? Alternatively, does the statement hold with the hypercube replaced with $\mathbb{R}^{n}$?