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

$\mathbb {T} ^{n}=\underbrace {S^{1}\times S^{1}\times \cdots \times S^{1}} _{n}$

which is represented by the following differential equations with respect to the standard angular coordinates (θ₁, θ₂, ..., θ_n):

${\frac {d\theta _{1}}{dt}}=\omega _{1},\quad {\frac {d\theta _{2}}{dt}}=\omega _{2},\quad \cdots ,\quad {\frac {d\theta _{n}}{dt}}=\omega _{n}.$

The solution of these equations can explicitly be expressed as

$\Phi _{\omega }^{t}(\theta _{1},\theta _{2},\dots ,\theta _{n})=(\theta _{1}+\omega _{1}t,\theta _{2}+\omega _{2}t,\dots ,\theta _{n}+\omega _{n}t)\mod 2\pi .$

If we respesent the torus as Rⁿ/Zⁿ we see that a starting point is moved by the flow in the direction ω=(ω₁, ω₂, ..., ω_n) at constant speed and when it reaches the border of the unitary n-cube it jumps to the opposite face of the cube.

For a linear flow on the torus either all orbits are periodic or all orbits are dense on a subset of the n-torus which is a k-torus. When the components of ω 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 ω are rationally independent then 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.

See also

• Completely integrable system
• Ergodic theory
• Quasiperiodic motion

Bibliography

• Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5.