About: Flows on a Torus

Systems in $R^{2}$ in which both dimensions are angular, that is, of the form $\dot{\underset{―}{u}} (t) = f (\underset{―}{u} (t)) = f ((\begin{matrix} θ_{1} \\ θ_{2} \end{matrix})),$ 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 $θ_{1}$ , and the tubular angle (ie always rotating about some axis perpendicular to the $z$ -axis) as $θ_{2}$ . Hence, in this way we can embed a two-dimensional system $f (\underset{―}{u} (t))$ in $R^{3}$ . This makes it particularly clear, visually, to see how and when periodic orbits develop.

Consider the simple system $(\begin{matrix} {\dot{θ}}_{a} \\ {\dot{θ}}_{b} \end{matrix}) = (\begin{matrix} p \\ q \end{matrix})$ , 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 $n \in Q$ 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.

Now in the case that $p, q$ are rationally independent, then flows on the torus will be quasiperiodic; never quite completing an orbit, but getting "close". These systems form dense orbits on the surface of the torus.

See the source Code, and resources used:

Linear flow on the torus (Wikipedia)
Torus knot (Wikipedia)
Rational dependence (Wikipedia)
Strogatz's Nonlinear Dynamics and Chaos (See pg. 276, eg)
Hale's Dynamics and Bifurcations (See Chapter 6)