Linear flow on the torus

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a linear flow on the torus is a flow on the n-dimensional torus $${\displaystyle \mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}}$$, which is represented by the following differential equations with respect to the standard angular coordinates ${\displaystyle \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):}$ $${\displaystyle {\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \ldots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}}$$.

The solution of these equations can explicitly be expressed as $${\displaystyle \Phi _{\omega }^{t}(\theta _{1},\theta _{2},\dots ,\theta _{n})=(\theta _{1}+\omega _{1}t,\theta _{2}+\omega _{2}t,\dots ,\theta _{n}+\omega _{n}t){\bmod {2}}\pi }$$.

If we represent the torus as ${\displaystyle \mathbb {T} ^{n}=\mathbb {R} ^{n}/\mathbb {Z} ^{n}}$ we see that a starting point is moved by the flow in the direction ${\displaystyle \omega =\left(\omega _{1},\omega _{2},\ldots ,\omega _{n}\right)}$ at constant speed and when it reaches the border of the unitary ${\displaystyle n}$-cube it jumps to the opposite face of the cube.

For a linear flow on the torus, all orbits are either periodic or dense on a subset of the ${\displaystyle n}$-torus, which is a ${\displaystyle k}$-torus. When the components of ${\displaystyle \omega }$ are rationally independent all the orbits are dense on the whole space. This can be easily seen in the two-dimensional case: if the two components of ${\displaystyle \omega }$ are rationally independent, the Poincaré section of the flow on an edge of the unit square is an irrational rotation on a circle and therefore its orbits are dense on the circle, as a consequence the orbits of the flow must be dense on the torus.