41:null 42:["$","title",null,{"children":"Noise investigation for Kaplan-Yorke | Grant M | Observable"}] 48:T94b,const n = 12; const range = d3.range(0, n); const points = range.flatMap(x => range.map(y => [x, y])); let a = 1.5; let m = 12; let fun = Math.cos; function yorkify(p) { let x = p[0]; let y = p[1]; [x, y] = [(2 * x) % m, a * y + fun(4 * Math.PI * x)]; return [x, y] } const yorke_points = points.map(yorkify); const interps = points.map((p, i) => d3.interpolate(p, yorke_points[i])); // So points don't overlap const jitter = points.map(() => (Math.random() - 0.5) * 0.5 ); const frames = 500; const startButton = html``; const resetButton = html``; const box = document.createElement("div"); display(html`

${startButton}${resetButton}

`); display(box); // Jitter is only applied at the final placement (t = 1) function draw(t) { const yes_jitter = t >= 1 ? 1 : 0; box.replaceChildren(Plot.plot({ title: "Visualization of Kaplan-Yorke mapping.", x: { domain: [0, 12]}, y: { domain: [0, 20]}, color: {scheme: "sinebow", domain: [0, 12]}, grid: true, style: {background: "#FFFAF0"}, caption: "Figure 1. Shows the transform of a point (x, y) to f(x, y). Where f is the Kaplan-Yorke function using a = 1.5 and m = 12. On a point matrix from (0, 0) to (12, 12). Starting positions encoding in a gray. Other colors are based off the y-axis of a point's original position. A horizontal jitter is applied to the final placement to prevent dots from perfectly overlapping for the sake of visualization.", marks: [ Plot.dot(points, {fill: '#a19ab3', r: 6}), Plot.dot(interps.map(f => f(t)), { x: p => p[0], y: (p, i) => p[1] + jitter[i] * yes_jitter, fill: (p, i) => points[i][0], r: 6 }), ] })); } let frame = 0; let runId = 0; async function play() { const id = ++runId; if (frame >= frames) frame = 0; // restart if finished startButton.disabled = true; while (frame <= frames && id === runId) { draw(frame / frames); frame++; await new Promise(r => setTimeout(r, 30)); } if (id === runId) startButton.disabled = false; } function reset() { runId++; // cancels any running animation frame = 0; startButton.disabled = false; draw(0); } startButton.onclick = play; resetButton.onclick = reset; invalidation.then(() => runId++); draw(0);49:T433,// MY FUNCTION!! const set_x = [0, 1.6]; const set_y = [0, 1.6]; function yorke_plot(x_bounds, y_bounds, a, m, fun, {size = 640, pixelSize = 1} = {}) { const f = (x, y) => { let xn = x, yn = y; for (let n = 0; n < 50; ++n) { [xn, yn] = [(2 * xn * yn) % m, a * yn + fun(4 * Math.PI * xn)]; if (Math.abs(yn) > 10) return n; } return 999; }; const make = (w, h, ps) => Plot.plot({ width: w, height: h, x: {ticks: [0, 0.4, 0.8, 1.2, 1.6]}, y: {ticks: [0, 0.4, 0.8, 1.2, 1.6]}, style: {fontSize: "20px"}, color: {...grant_color1, domain: [0, 50]}, marks: [Plot.raster({fill: f, x1: x_bounds[0], x2: x_bounds[1], y1: y_bounds[0], y2: y_bounds[1], pixelSize: ps})] }); const plot = make(size, size, 2); plot.style.cursor = "zoom-in"; plot.addEventListener("click", () => { const big = make(2000, 2000, pixelSize); // high-res version const blob = new Blob([new XMLSerializer().serializeToString(big)], {type: "image/svg+xml"}); window.open(URL.createObjectURL(blob), "_blank"); }); return plot; } 4a:[{"id":"7a4891c35fdbcc691440f67711cf11012e827d51630b3c148c7b5076b9f55ec7d7a58295066fc03916be4d9a262464cb3a058298cc7c14770258e6852411e38a","href":"https://static.observableusercontent.com/files/7a4891c35fdbcc691440f67711cf11012e827d51630b3c148c7b5076b9f55ec7d7a58295066fc03916be4d9a262464cb3a058298cc7c14770258e6852411e38a","name":"IMG_0746.webp","type":"image/webp","lastModified":1792353555435,"size":47808},{"id":"014047c62c0335278359bcd309262449fbf7363ccef5096c68d34a77d3026eb2744b4c3362386662866d765ce6b646140cf27d4f7dfbc2d7c79fd023e34f2aba","href":"https://static.observableusercontent.com/files/014047c62c0335278359bcd309262449fbf7363ccef5096c68d34a77d3026eb2744b4c3362386662866d765ce6b646140cf27d4f7dfbc2d7c79fd023e34f2aba","name":"legend.png","type":"image/png","lastModified":1792425698220,"size":2548}] 46:{"id":"a23fd26d86313027","slug":"noise-investigation-for-kaplan-yorke","title":"Noise investigation for Kaplan-Yorke","thumbnail":"ed351857448081952cbdbadd71306ea7739f1cf744072b2f57c44d4bd7491217","access_level":"public","create_time":"$D2026-10-16T21:07:50.061Z","update_time":"$D2026-10-19T16:56:15.652Z","view_time":null,"first_public_time":"$D2026-10-19T16:56:18.806Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"fbf9eef530f9c2a1","login":"grant-m","name":"Grant M","avatar_url":"https://avatars-next.observableusercontent.com/2ffa116de23fbd0c0705acf3f6c2979c61cbad756e62c014e71a48ebad16d37a","tier":"starter_2024","bio":"The liver is one of the top 5 most essential organs.","following":false},"version":986,"publish_version":986,"role":null,"authored":false,"authors":[{"type":"individual","id":"7dbd1e6e69c36ea1","login":"grant-m","name":"Grant M","avatar_url":"https://avatars-next.observableusercontent.com/2ffa116de23fbd0c0705acf3f6c2979c61cbad756e62c014e71a48ebad16d37a","tier":null,"bio":"The liver is one of the top 5 most essential organs.","following":false}],"members":[{"type":"team","id":"fbf9eef530f9c2a1","login":"grant-m","name":"Grant M","avatar_url":"https://avatars-next.observableusercontent.com/2ffa116de23fbd0c0705acf3f6c2979c61cbad756e62c014e71a48ebad16d37a","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Noise investigation for Kaplan-Yorke","cells":[{"id":1,"value":"# Noise investigation for Kaplan-Yorke","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":41,"value":"### Introduction.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"The Kaplan-Yorke method takes a point (x, y) as input and outputs a different point transformed among the axis. It generally has 2 parameters, a constant scale for the y-axis, a, and a modulo to bound the x-axis, m. Note that m is not usually described as a parameter and usually a fixed value for the range at which some instance of Kaplan-Yorke will operate. This is one of two modifications we make in our investigation, that is to say, we assume that m is a variable. We also assume that the function that governs y (usually cosine), is also modifiable. This results in","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":85,"value":"\\begin{align}\n x_{n+1} &= 2x_ny_n\\% m \\\\\n y_{n+1} &= a*y_{n} + f(4 \\pi x_n).\n\\end{align}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":86,"value":"Though it is rare to apply this method directly. Instead, we are more interested in running this formula over the same point multiple times. In an ideal world, the Kaplan-Yorke method eventually \"loops\", so running it on its own output over and over would eventually produce (x, y). Though this is not the case in practice, eventually, the method produces a point which \"breaks the orbit\", which is to say, one that it will never return to (x, y) given more iterations of Kaplan-Yorke. It is for this reason why Kaplan-Yorke is defined as a representation of a choatic system. When looking at graphs produced by tracking how many iterations it takes to escape the orbit, we define its noise of as the variance as a cluster (of some relative size) of adjacent points. Then, we ask which parameters affect this noise the most (or which parameters result in a more predictable Kaplan-Yorke), which leads us to the following.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3,"value":"### Hypothesis.\n\nThe noise in Kaplan-Yorke sets are influenced more by the parameter a, compared to m or f.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"### Method.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":87,"value":"We begin with a small demonstration of the Kaplan-Yorke method. Note this is only 1 iteration of the method.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"const grant_color1 = {type: \"linear\", range: [\"#08415C\", \"#7FB7BE\", \"#F2E863\", \"#BC13FE\"]}","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"$48","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":77,"value":"$49","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":88,"value":"We then graph the method over 3 variations of a and m. We define columns by their change in m and rows by their change in a. The relative noise is not measured here as it tends to be easier to detect with the human eye as opposed to a function. Which is to say, that functions measuring noise are heavily dependent on a subjective parameter to tell them the size of pixel clusters to observe.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":82,"value":"html`\nKaplan Yorke over both a & m.
\nUsing the cosine function, 50 iterations until escape.
\n

\n ${[0.3, 1, 3].map(m => html`

m = ${m}

`)}\n ${[1.2, 1.5, 2.0].flatMap(a => [ // this is for a\n html`

a = ${a}

`,\n ...[0.3, 1, 3].map(m => yorke_plot(set_x, set_y, a, m, Math.cos))\n ])}\n

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":89,"value":"For fun, we also play with the f parameter. Note the similarities between Cosine and Sine, there should be some intuition as to why these graphs are similar. Though it should be noted that a=1.2 for Sine is particularly noisy. For the same reason, tangent appears very dissimilar. It's also likely that one could deduce what function is used for each particular graph without being told, this is especially apparent in the case of tangent, as traces of the linear version (f(x)), seem to appear in the noise maps of the final version. This is probably due to the contribution of the function to the repeating nature of the function. Though it is unknown how the function contributes the fractilian nature of the plots.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"html`\nKaplan Yorke using Cosine, Sine, and Tagnent.
\n50 iterations until escape. Uses m = 3.
\n

\n ${[\"Cosine\", \"Sine\", \"Tangent\"].map(m => html`

f = ${m}

`)}\n ${[1.2, 1.5, 2.0].flatMap(a => [ // this is for a\n html`

a = ${a}

`,\n ...[Math.cos, Math.sin, Math.tan].map(fun => yorke_plot(set_x, set_y, a, m, fun))\n ])}\n

`","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":61,"value":"### Analysis.\n\nThe evidence gathered here seems to support the hypothesis. A value of a closer to 1 produces significantly more noise, and increasing this parameter appears to decrease this noise significantly. While it seems to be consistent across functions, methods using sine seem to be particularly affected by lower values of a. It should also be noted that m technically does have an reasonable affect on the noise, but it seems to draw \"lines\" across the plot. This is due to how it puts x in the bounds of a range. Thus, increasing the noise in figures is a byproduct of its primary affect. There is no doubt between a correlation between lower values of m and noise, it must be given specific parameters for noise detection which target multiple bounds of m. However, we can observe a to be a parameter which strictly increases the chaos of the system as it gets closer to 1.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"### Procedure.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"#### Figuring out the function (First revision)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"We first attempt a logistic function. This is relatively unique among chaotic systems as it models a two-input, one-output system.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"let re = 10 // gets 'truly random' at above 10. Also explains the red and green hues\n\nfunction logistic2(x, r) {\n let z = x;\n for (let n = 0; n < re; ++n) z = r * z * (1 - z);\n return z;\n}\n\n// display(Plot.raster(database, {x: \"x\", y: \"y\", fill: \"z\", interpolate: \"nearest\"}).plot({color: {scheme: \"turbo\"}}))\ndisplay(Plot.raster({fill: logistic2, x1: 0, x2: 1, y1: 1, y2: 4}).plot({color: {scheme: \"turbo\"}}))","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"This function kind of sucks because it has some pretty strict bounds. It also desperately needs a legend and to move away from a red-green palette.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":45,"value":"I tried LCG too, but it isn't very palpable. Rather, it is perhaps a little too noise-y?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"let init = 0;\nlet lcgA = 1103515245;\nlet lcgC = 12345;\nlet lcgM = 19;\n\n// seed = x position, steps = y position; returns the LCG state after `steps` iterations\nfunction LCG(seed, steps) {\n let x = Math.floor(seed);\n for (let n = 0; n < Math.floor(steps); n++) x = (lcgA * x + lcgC) % lcgM;\n return x;\n}\n\ndisplay(Plot.raster({fill: LCG, x1: 1, x2: 100, y1: 1, y2: 100}).plot({color: {scheme: \"turbo\", legend: true}}))","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"I'm getting the hang of it now, I referenced Mandelbrot sets a bit so it seems like we want to measure how long it takes for points to 'escape' an orbit. It also seems like we can measure it a bit more accurately using how far away the point is from the origin.","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"let duffA = 2.75;\nlet duffB = 0.2;\n\nfunction duffing(x, y) {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n){\n [xn, yn] = [yn, -duffB*xn + duffA*yn - yn**3];\n if (xn**2 + yn**2 > 40) return n;\n }\n}\n\ndisplay(Plot.raster({fill: duffing, x1: -4, x2: 4, y1: -4, y2: 4}).plot({aspectRatio: 1, color: {scheme: \"viridis\", legend: true}}))\n\nfunction duffing_dist(x, y) {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n){\n [xn, yn] = [yn, -duffB*xn + duffA*yn - yn**3];\n if (xn**2 + yn**2 > 40) return Math.hypot(xn, yn);\n }\n}\n\ndisplay(Plot.raster({fill: duffing_dist, x1: -4, x2: 4, y1: -4, y2: 4}).plot({aspectRatio: 1, color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"#### Decided on Kaplan-Yorke map\n\nI think the Kaplan-Yorke method is fairly cool, seems pretty interesting. We will do a few experiments with parameters.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"const yorkeA_1 = 1.2;\nconst mod_test_1 = 81;\n\nfunction new_yorke_1(x: number, y: number): number {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn) % mod_test_1, yorkeA_1 * yn + Math.cos(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return Math.hypot(xn, yn); // escaped\n }\n return 100;\n}\n\ndisplay(Plot.raster({fill: new_yorke_1, x1: 0, x2: 20, y1: 0, y2: 20}).plot({color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"Change the a parameter.","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":48,"value":"const yorkeA_2 = 2.5;\nconst mod_test_2 = 81;\n\nfunction new_yorke_2(x: number, y: number): number {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn) % mod_test_2, yorkeA_2 * yn + Math.cos(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return Math.hypot(xn, yn); // escaped\n }\n return 0;\n}\n\ndisplay(Plot.raster({fill: new_yorke_2, x1: 0, x2: 20, y1: 0, y2: 20}).plot({color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"Change the modulo.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":51,"value":"const yorkeA_3 = 1.2;\nconst mod_test_3 = 30;\n\nfunction new_yorke_3(x: number, y: number): number {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn) % mod_test_3, yorkeA_3 * yn + Math.cos(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return Math.hypot(xn, yn); // escaped\n }\n return 0;\n}\n\ndisplay(Plot.raster({fill: new_yorke_3, x1: 0, x2: 20, y1: 0, y2: 20}).plot({color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"Use tangent instead.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":54,"value":"const yorkeA_4 = 1.2;\nconst mod_test_4 = 81;\nconst math_func_4 = Math.tan;\n\nfunction new_yorke_4(x: number, y: number): number {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn) % mod_test_4, yorkeA_4 * yn + math_func_4(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return Math.hypot(xn, yn); // escaped\n }\n return 0;\n}\n\ndisplay(Plot.raster({fill: new_yorke_4, x1: 0, x2: 20, y1: 0, y2: 20}).plot({color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"Count with iterations rather than point length.","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":57,"value":"const yorkeA_5 = 1.2;\nconst mod_test_5 = 81;\nconst math_func_5 = Math.cos;\n\nfunction new_yorke_5(x: number, y: number): number {\n let xn = x, yn = y;\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn) % mod_test_5, yorkeA_5 * yn + math_func_5(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return n; // escaped\n }\n return 0;\n}\n\ndisplay(Plot.raster({fill: new_yorke_5, x1: 0, x2: 20, y1: 0, y2: 20}).plot({color: {scheme: \"viridis\", legend: true}}))\n","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"#### Draft of Figure.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":65,"value":"FileAttachment(\"IMG_0746.webp\").image()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"#### Code draft","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":81,"value":"// MY FUNCTION!!\nfunction yorke_plot_temp(x_bounds, y_bounds, a, m, fun) {\n\n // Ai used for this\n const f = (x, y) => {\n\n let xn = x, yn = y;\n\n for (let n = 0; n < 80; ++n) {\n [xn, yn] = [(2 * xn* yn) % m, a * yn + fun(4 * Math.PI * xn)];\n if (Math.abs(yn) > 10) return n;\n }\n\n return 0;\n };\n return Plot.raster({fill: f, x1: x_bounds[0], x2: x_bounds[1], y1: y_bounds[0], y2: y_bounds[1]})\n .plot({color: {scheme: \"viridis\", legend: true}});\n}\n\ndisplay(yorke_plot_temp([0, 1.6], [0, 1.6], 1.5, 1.5, Math.cos));","pinned":true,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":59,"value":"### Works Cited.\n\nZoglmann, Kurt. “Kaplan–Yorke Map - the Encyclopedia of Abstractions.” Abstractopedia.Org, 2026, abstractopedia.org/domain-specific/kaplanyorke_map/. Accessed 18 Oct. 2026.\n\nhttps://claude.ai/share/6a5ff77f-fd82-43c6-ab07-f0a19be87cd7\n\nhttps://claude.ai/share/878c29df-e499-4a9e-873b-1417c922ad55","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4a","stdlib":"2"}} 47:"$46:body"