Linear flow on the torus

In mathematics, esecially 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

:$mathbb\{T\}^n\; =\; underbrace\{S^1\; imes\; S^1\; imes\; cdots\; imes\; S^1\}\_n$

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

:$frac\{d\; heta\_1\}\{dt\}=omega\_1,\; quad\; frac\{d\; heta\_2\}\{dt\}=omega\_2,quad\; cdots,\; quad\; frac\{d\; heta\_n\}\{dt\}=omega\_n.$

The solution of these equations can explicitly be expressed as

:$Phi\_omega^t(\; heta\_1,\; heta\_2,\; dots,\; heta\_n)=(\; heta\_1+omega\_1\; t,\; heta\_2+omega\_2\; t,\; dots,\; heta\_n+omega\_n\; t)\; mod\; 2pi.$

If we respesent the torus as R^"n"/Z^"n" 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 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 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.

ee also

*Completely integrable system
*Ergodic theory

Bibliography

*