Completed Proof For Incommensurate Lissajous Curves/Bowditch Curves Are Dense In The Rectangle
$begingroup$
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?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
$endgroup$
add a comment |
$begingroup$
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?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
$endgroup$
add a comment |
$begingroup$
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?
real-analysis complex-analysis analysis proof-explanation ergodic-theory
$endgroup$
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
real-analysis complex-analysis analysis proof-explanation ergodic-theory
asked May 23 '17 at 19:17
The DudeThe Dude
315112
315112
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2293852%2fcompleted-proof-for-incommensurate-lissajous-curves-bowditch-curves-are-dense-in%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
$endgroup$
add a comment |
$begingroup$
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 $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
$endgroup$
add a comment |
$begingroup$
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 $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
$endgroup$
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 $mathbf rleft(tright)=mathbf rleft(t+Tright)$.
answered Jan 18 at 2:02
David Garrido GonzálezDavid Garrido González
212
212
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2293852%2fcompleted-proof-for-incommensurate-lissajous-curves-bowditch-curves-are-dense-in%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown