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 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 ( θ 1 , θ 2 , … , θ n ) : {\displaystyle \left(\theta _{1},\theta _{2},\ldots ,\theta _{n}\right):} 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 \ldots ,\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 .