page_title: Linear flow on the torus
hierarchical_section_title: Linear flow on the torus
caption_attribution_description: English: Shows the linear flow on a 2 Torus with the two components being rationally independent showing how as the system continues it becomes a dense orbit.
context_page_description: 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
  which is represented by the following differential equations with respect to the standard angular coordinates:
  The solution of these equations can explicitly be expressed as
  If we represent the torus as we see that a starting point is moved by the flow in the direction ω = at constant speed and when it reaches the border of the unitary n-cube it jumps to the opposite face of the cube.
  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.
context_section_description: 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
  which is represented by the following differential equations with respect to the standard angular coordinates (θ₁, θ₂, ..., θₙ):
  The solution of these equations can explicitly be expressed as
  If we represent the torus as  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.
  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.
caption: Irrational rotation on a 2-torus
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