Completed Proof For Incommensurate Lissajous Curves/Bowditch Curves Are Dense In The Rectangle












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$begingroup$


A Lissajous curve, or a Bowditch curve, is given by the parametric equations



$x(t)=Asin(ω_{x}t + phi)$



$y(t)=Bsin(ω_{y}t+δ)$,



Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$



I have seen:



Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle



But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?










share|cite|improve this question









$endgroup$

















    3












    $begingroup$


    A Lissajous curve, or a Bowditch curve, is given by the parametric equations



    $x(t)=Asin(ω_{x}t + phi)$



    $y(t)=Bsin(ω_{y}t+δ)$,



    Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$



    I have seen:



    Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle



    But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?










    share|cite|improve this question









    $endgroup$















      3












      3








      3


      1



      $begingroup$


      A Lissajous curve, or a Bowditch curve, is given by the parametric equations



      $x(t)=Asin(ω_{x}t + phi)$



      $y(t)=Bsin(ω_{y}t+δ)$,



      Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$



      I have seen:



      Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle



      But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?










      share|cite|improve this question









      $endgroup$




      A Lissajous curve, or a Bowditch curve, is given by the parametric equations



      $x(t)=Asin(ω_{x}t + phi)$



      $y(t)=Bsin(ω_{y}t+δ)$,



      Now if $frac{omega_{x}}{omega_{y}}$ is irrational, and $phi$ and $delta$ are fixed, then the set $mathcal{L} = (x(t), y(t) | -infty < t < infty)$ is supposedly dense in the rectangle $R =[-A,A]$x$[-B,B]$



      I have seen:



      Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle



      But a completed proof was not given. This problem is frequently mentioned in the literature, but a reference to a full proof is never given. Does anyone have such a reference, or could someone provide a proof here?







      real-analysis complex-analysis analysis proof-explanation ergodic-theory






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      asked May 23 '17 at 19:17









      The DudeThe Dude

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          You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.






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            $begingroup$

            You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.






            share|cite|improve this answer









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              $begingroup$

              You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.






              share|cite|improve this answer









              $endgroup$
















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                2








                2





                $begingroup$

                You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.






                share|cite|improve this answer









                $endgroup$



                You can write $x$ and $y$ in complex number with the exponentials (easiest way): $mathbf rleft(tright)=left(Ae^{ω_{x}t + phi},Be^{ω_{y}t + delta}right)$. Then proof it by induction, suppose that there is a period $Tneq0$ so that $mathbf rleft(tright)=mathbf rleft(t+Tright)$.







                share|cite|improve this answer












                share|cite|improve this answer



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                answered Jan 18 at 2:02









                David Garrido GonzálezDavid Garrido González

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