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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 the torus. The first case is what happens when the components of ω are rationally dependent while the second case occurs when they are rationally independent. 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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