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Revision as of 17:37, 12 June 2008 editDrbreznjev (talk | contribs)16,660 editsm WikiProject Stub sorting← Previous edit Revision as of 04:35, 13 June 2008 edit undoMichael Hardy (talk | contribs)Administrators210,266 edits \dotsNext edit →
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A '''linear flow on the torus''' is a ] on the ''n''-dimensional ] A '''linear flow on the torus''' is a ] on the ''n''-dimensional ]

:<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math> :<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math>

which is represented by the following differential equations with respect to the standard angular coordinates (θ<sub>1</sub>, θ<sub>2</sub>, ..., θ<sub>''n''</sub>): which is represented by the following differential equations with respect to the standard angular coordinates (θ<sub>1</sub>, θ<sub>2</sub>, ..., θ<sub>''n''</sub>):

:<math>\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.</math> :<math>\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.</math>

The solution of these equations can explicitly be expressed as The solution of these equations can explicitly be expressed as

:<math>\Phi_\omega^t(\theta_1, \theta_2, ..., \theta_n)=(\theta_1+\omega_1 t, \theta_2+\omega_2 t, ..., \theta_n+\omega_n t) \mod 2\pi.</math> :<math>\Phi_\omega^t(\theta_1, \theta_2, \dots, \theta_n)=(\theta_1+\omega_1 t, \theta_2+\omega_2 t, ..., \theta_n+\omega_n t) \mod 2\pi.</math>


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.

Revision as of 04:35, 13 June 2008

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,...,\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.

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.

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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