Linear flow on the torus

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

$\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_n$

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

$\frac{d\theta_1}{dt}=\omega_1, \quad \frac{d\theta_2}{dt}=\omega_2,\quad \cdots, \quad \frac{d\theta_n}{dt}=\omega_n.$

The solution of these equations can explicitly be expressed as

$\Phi_\omega^t(\theta_1, \theta_2, \dots, \theta_n)=(\theta_1+\omega_1 t, \theta_2+\omega_2 t, \dots, \theta_n+\omega_n t) \mod 2\pi.$

If we respesent the torus as Rⁿ/Zⁿ 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.

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

• Completely integrable system
• Ergodic theory
• Quasiperiodic motion

Bibliography

• Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5.