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If we respesent the torus as '''R'''<sup>''n''</sup>/'''Z'''<sup>''n''</sup> we see that a starting point is moved by the flow in the direction ω=(ω<sub>1</sub>, ω<sub>2</sub>, ..., ω<sub>''n''</sub>) at constant speed and when it reaches the border of the unitary ''n''-cube it jumps to the opposite face of the cube. If we respesent the torus as '''R'''<sup>''n''</sup>/'''Z'''<sup>''n''</sup> we see that a starting point is moved by the flow in the direction ω=(ω<sub>1</sub>, ω<sub>2</sub>, ..., ω<sub>''n''</sub>) 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 ] or all orbits are ] on a subset of the ''n''-torus which is a ''k''-torus. When the components of ω are ] 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 ] of the flow on an edge of the unit square is an ] 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. For a linear flow on the torus either all orbits are ] or all orbits are ] on a subset of the ''n''-torus which is a ''k''-torus. When the components of ω are ] 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 ] of the flow on an edge of the unit square is an ] 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== ==See also==

Revision as of 00:10, 23 June 2013

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

T n = S 1 × S 1 × ⋯ × S 1 ⏟ n {\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, ..., θn):

d θ 1 d t = ω 1 , d θ 2 d t = ω 2 , ⋯ , d θ n d t = ω 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

Φ ω t ( θ 1 , θ 2 , … , θ n ) = ( θ 1 + ω 1 t , θ 2 + ω 2 t , … , θ n + ω n t ) mod 2 π . {\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, ..., ωn) 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

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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