About: Linear flow on the torus

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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 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.

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rdfs:label	Linear flow on the torus (en) トーラス上の線型フロー (ja)
rdfs:comment	数学の、特に力学系理論として知られる解析学の分野において、n-次元トーラス上の線型フロー（せんけいフロー、英: linear flow）とは、標準的な角度座標 (θ1, θ2, ..., θn) に関する次の微分方程式によって表現されるフローのことを言う：この方程式の解は次の様に陽的に表現される：トーラスを Rn/Zn と表すなら、始点はフローによって ω=(ω1, ω2, ..., ωn) の方向に一定速度で移動されることが分かる。またそのフローがユニタリ n-立方体の境界に到達した場合は、その反対側の面に移動される。トーラス上の線型フローに対して、すべての軌道は周期的であるか、n-次元トーラスの部分集合で k-次元トーラスであるようなものの上で稠密である。ω の成分が有理独立であるなら、すべての軌道は全空間で稠密である。これは二次元の場合には簡単に分かる：すなわち、ω の二つの成分が有理独立であるなら、単位正方形の辺上でのフローのポアンカレ切断面は、円上の無理回転であり、したがってその軌道は円上で稠密で、結果としてトーラス上で稠密となる。 (ja) 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. (en)
foaf:depiction
dcterms:subject	Topological spaces Lie groups Dynamical systems Ergodic theory General topology
Wikipage page ID	17905842 (xsd:integer)
Wikipage revision ID	1090801417 (xsd:integer)
Link from a Wikipage to another Wikipage	Monomorphism Dynamical systems theory Lie group Continuous function Mathematical analysis Mathematics Quasiperiodic motion Kronecker foliation Submanifold Line (geometry) Dense set Closed curve Topological spaces Topology Torus Hausdorff space Irrational number Irrational rotation Lie groups Ergodic theory Flow (mathematics) Quotient space (topology) Group homomorphism Dynamical systems Ergodic theory General topology Torus knot Manifold Injective function Rational number Unit square Subspace topology Rational dependence Simple curve Poincaré section Periodic orbit Injective mapping Completely integrable system Dense subspace
sameAs	Linear flow on the torus Linear flow on the torus Linear flow on the torus Linear flow on the torus Linear flow on the torus
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thumbnail	wiki-commons:Special:FilePath/Irrational_Rotation_on_a_2_Torus.png?width=300
has abstract	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. (en) 数学の、特に力学系理論として知られる解析学の分野において、n-次元トーラス上の線型フロー（せんけいフロー、英: linear flow）とは、標準的な角度座標 (θ1, θ2, ..., θn) に関する次の微分方程式によって表現されるフローのことを言う：この方程式の解は次の様に陽的に表現される：トーラスを Rn/Zn と表すなら、始点はフローによって ω=(ω1, ω2, ..., ωn) の方向に一定速度で移動されることが分かる。またそのフローがユニタリ n-立方体の境界に到達した場合は、その反対側の面に移動される。トーラス上の線型フローに対して、すべての軌道は周期的であるか、n-次元トーラスの部分集合で k-次元トーラスであるようなものの上で稠密である。ω の成分が有理独立であるなら、すべての軌道は全空間で稠密である。これは二次元の場合には簡単に分かる：すなわち、ω の二つの成分が有理独立であるなら、単位正方形の辺上でのフローのポアンカレ切断面は、円上の無理回転であり、したがってその軌道は円上で稠密で、結果としてトーラス上で稠密となる。 (ja)
prov:wasDerivedFrom	wikipedia-en:Linear_flow_on_the_torus?oldid=1090801417&ns=0
page length (characters) of wiki page	6771 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Linear_flow_on_the_torus
is Link from a Wikipage to another Wikipage of	Index of physics articles (L) Foliation Irrational winding of a torus Rational dependence Linear flows on the torus Irrational cable on a torus Irrational winding on a torus
is Wikipage redirect of	Irrational winding of a torus Linear flows on the torus Irrational cable on a torus Irrational winding on a torus
is foaf:primaryTopic of	wikipedia-en:Linear_flow_on_the_torus

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