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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 (θ₁, θ₂, ..., θ_n):

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

The solution of these equations can explicitly be expressed as

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

If we respesent the torus as R/Z 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.

A linear flow on the torus is such that either all orbits are periodic or all orbits are dense on a subset of the n-torus whihc is a 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 Poincare 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

Bibliography

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

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