Misplaced Pages

Linear flow on the torus

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

This is an old revision of this page, as edited by 130.86.98.142 (talk) at 18:04, 21 October 2016 (Tagged suggested merge). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Revision as of 18:04, 21 October 2016 by 130.86.98.142 (talk) (Tagged suggested merge)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)
It has been suggested that Irrational winding of a torus be merged into this article. (Discuss) Proposed since October 2016.

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.

Irrational Rotation on a 2 Torus

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.
Stub icon

This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it.

Categories: