Systems in $\mathbb{R}^2$ in which both dimensions are angular, that is, of the form $$\dot{\underline{u}}(t) = f(\underline{u}(t)) = f(\begin{pmatrix}\theta_1\\\theta_2\end{pmatrix}),$$ are quite convenient to plot on the surface of a torus. Here, we denote the central angle (ie lying in the $x-y$ plane) as $\theta_1$, and the tubular angle (ie always rotating about some axis perpendicular to the $z$-axis) as $\theta_2$. Hence, in this way we can embed a two-dimensional system $f(\underline{u}(t))$ in $\mathbb{R}^3$. This makes it particularly clear, visually, to see how and when periodic orbits develop.
Consider the simple system $\begin{pmatrix}\dot{\theta}_a\\\dot{\theta}_b\end{pmatrix} = \begin{pmatrix} p\\ q \end{pmatrix}$, where we have a completely decoupled system that has clear linear solutions. Despite the simplicity, we can still have rather remarkable visuals from this system. In the case that $p$ and $q$ are rationally dependent (that is, $p \neq n \cdot q$ where $n \in \mathbb{Q}$ for any $n$), then we will always have a periodic orbit.
More geometrically, in the case that $\gcd(p, q) = 1$ (even if $p, q$ are not integers!) the solutions that arise form a unknot (try turning off the opacity of the torus under display settings to see this more "clearly"). Otherwise, some kind of non-trivial torus knot is formed; more specifically, a $(p, q)-$torus knot (hence the suggestion notation). For instance, given $(p, q) = (2, 3)$, we have a trefoil knot.
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