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| In [[mathematics]], especially in the area of [[mathematical analysis]] known as [[dynamical systems theory]], a '''linear flow on the torus''' is a [[flow (mathematics)|flow]] on the ''n''-dimensional [[torus]]
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| :<math>\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n</math>
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| 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>):
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| :<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>
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| The solution of these equations can explicitly be expressed as
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| :<math>\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.</math>
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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.
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| For a linear flow on the torus either all orbits are [[periodic orbit|periodic]] or all orbits are [[dense set|dense]] on a subset of the ''n''-torus which is a ''k''-torus. When the components of ω are [[rational dependence|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.
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| ==See also==
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| *[[Completely integrable system]]
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| *[[Ergodic theory]]
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| *[[Quasiperiodic motion]]
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| ==Bibliography==
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| * {{cite book | author=Anatole Katok and Boris Hasselblatt | title= Introduction to the modern theory of dynamical systems | publisher= Cambridge | year= 1996 | isbn=0-521-57557-5}}
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| [[Category:Dynamical systems]]
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| [[Category:Ergodic theory]]
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| {{mathanalysis-stub}}
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