2b:null 2c:["$","title",null,{"children":"Swarmalator | Mario Delgado | Observable"}] 31:T625,md` The model is then simplified in this notebook to equations ***3a*** and ***4a*** following the paper's methodology: > ...We first consider identical swarmalators so that ${tex`\omega _i`} = ω and ${tex`{\bf v} _i = {\bf v}`}. Further, we assume propulsion with constant magnitude and direction ${tex`{\bf{v}} = v_0\hat n`} where ${tex`\hat n`} is a constant vector (we relax these simplifications later). Then by a > choice of reference frame we can set ω = ${tex`v _0`} = 0 without loss of generality. Finally, by rescaling time and space > we set ${tex`A = B = 1`}. This leaves us with a system with two parameters (${tex`\color{red}J`},${tex`\color{green}K`})... ${tex.block`{\dot {\bf x}}_i = \frac{1}{\color{blue}N}\left[ {\sum_{j \ne i}^{\color{blue}N} \frac{{{\bf{x}}_j - {\bf{x}}_i}}{{\left| {{\bf{x}}_j - {\bf{x}}_i} \right|}}\left( {1 + {\color{red}J} \,{\rm{cos}}\left( {\theta _j - \theta _i} \right)} \right) - \frac{{{\bf{x}}_j - {\bf{x}}_i}}{{\left| {{\bf{x}}_j - {\bf{x}}_i} \right|^2}}} \right] \tag{3a}`} ${tex.block`\dot \theta _i = \frac{\color{green}K}{\color{blue}N}\sum_{j \ne i}^{\color{blue}N} \frac{{{\rm{sin}}\left( {\theta _j - \theta _i} \right)}}{{\left| {{\bf{x}}_j - {\bf{x}}_i} \right|}}. \tag{4a}`} With ${tex`\omega = {\bf{v}}_0 = 0`} and ${tex`A = B = 1`} the system of ${tex`\color{blue}N`} nodes can be controlled with the two parameters ${tex`\color{red}J`} and ${tex`\color{green}K`} using ${tb`[@Fil](https://observablehq.com/@fil)`}'s ${tb`[visualization](https://observablehq.com/@fil/swarmalator)`}: `32:[] 2f:{"id":"da4460965b9b4953","slug":"swarmalator","title":"Swarmalator","thumbnail":"cad05b71174a185da7fec73125917021394e907581cbdebf66465e5c9134c338","access_level":"public","create_time":"$D2020-12-22T15:33:19.465Z","update_time":"$D2023-02-06T16:25:59.211Z","view_time":null,"first_public_time":"$D2021-01-03T11:10:13.339Z","fork":{"id":"a03057c2a2ecadf2","slug":"swarmalator","title":"Swarmalator","owner":{"id":"da2a32154638c457","login":"fil","name":"Fil"}},"stars":2,"starred":false,"owner":{"type":"team","id":"f57b9f8a19d28778","login":"mariodelgadosr","name":"Mario Delgado ","avatar_url":"https://avatars-next.observableusercontent.com/2d2982ad09c4ca9dfd6570810ae00301e0db6ef798c4f7025ea03d4325d4807d","tier":"starter_2024"},"version":516,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"1d3988d482a8ec0f","login":"mariodelgadosr","name":"Mario Delgado ","avatar_url":"https://avatars-next.observableusercontent.com/2d2982ad09c4ca9dfd6570810ae00301e0db6ef798c4f7025ea03d4325d4807d","tier":null}],"members":[{"type":"individual","id":"1d3988d482a8ec0f","login":"mariodelgadosr","name":"Mario Delgado ","avatar_url":"https://avatars-next.observableusercontent.com/2d2982ad09c4ca9dfd6570810ae00301e0db6ef798c4f7025ea03d4325d4807d","tier":null,"role":"editor"}],"comments":[{"id":"302c53d37372315a","reply_to_id":null,"content":"Wow these equations are beautiful! They are also quite daunting (lots of symbols for simple computations), and would certainly benefit from plain language explanations. I encourage you to publish this notebook, and I will link to it from mine—which I'd rather keep focused on the visual demo.","create_time":"$D2020-12-23T08:54:34.237Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"45a379fcfcb14253","login":"fil","name":"Fil","avatar_url":"https://avatars-next.observableusercontent.com/d5b1777907117a2bdf2a07eb98600d7a9beb21639fde42b9753e95c88c7731a6","tier":null}}],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Swarmalator","cells":[{"id":0,"value":"md`# ${tb`[Swarmalator](https://observablehq.com/@fil/swarmalator)`}\n\nObservable simulation for\n> “Oscillators that sync and swarm,”\n> Kevin P. O’Keeffe, Hyunsuk Hong & Steven H. Strogatz,\n> Nature Communications, volume 8, Article number: 1504 (2017) \n> ${tb`[https://www.nature.com/articles/s41467-017-01190-3](https://www.nature.com/articles/s41467-017-01190-3)`}\n\n(See also ${tb`[http://usediscretion.blogspot.fi/2017/01/the-swarmalator.html](http://usediscretion.blogspot.fi/2017/01/the-swarmalator.html)`} and this ${tb`[block](https://bl.ocks.org/Fil/2cec616f64cda2cdb0e7302c9b1be3a5)`}.)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":266,"value":"md`\n\nEquations ***3*** and ***4***, as modeled in the paper *${tb`[Oscillators that sync and swarm](https://www.nature.com/articles/s41467-017-01190-3)`}*, are referenced in this notebook. \n\nThey represent changes in position (${tex`{\\dot {\\bf x}}_i`}) and phase (${tex`{\\dot \\theta _i}`}) for the nodes that will be visualized:\n\n${tex.block`{\\dot {\\bf x}}_i = {\\bf{v}}_i + \\frac{1}{N}\\left[ {\\mathop {\\sum}\\limits_{j \\ne i}^N \\frac{{{\\bf{x}}_j - {\\bf{x}}_i}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|}}\\left( {A + J\\,{\\rm{cos}}\\left( {\\theta _j - \\theta _i} \\right)} \\right) - B\\frac{{{\\bf{x}}_j - {\\bf{x}}_i}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|^2}}} \\right] \\tag{3}`}\n\n${tex.block`\\dot \\theta _i = \\omega _i + \\frac{K}{N}\\mathop {\\sum}\\limits_{j \\ne i}^N \\frac{{{\\rm{sin}}\\left( {\\theta _j - \\theta _i} \\right)}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|}}. \\tag{4}`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":297,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":461,"value":"sliders = html`\n

${viewof controls_N}

${viewof controls_J}

${viewof controls_K}

\n${md`${tex.block`{\\dot {\\bf x}}_i = \\frac{1}{\\color{blue}${controls_N}}\\left[ {\\sum_{j \\ne i}^{\\color{blue}${controls_N}} \\frac{{{\\bf{x}}_j - {\\bf{x}}_i}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|}}\\left( {1 + {\\color{red}${controls_J}} \\,{\\rm{cos}}\\left( {\\theta _j - \\theta _i} \\right)} \\right) - \\frac{{{\\bf{x}}_j - {\\bf{x}}_i}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|^2}}} \\right] \\tag{3a*}`}\n\n${tex.block`\\dot \\theta _i = \\frac{\\color{green}${controls_K}}{\\color{blue}${controls_N}}\\sum_{j \\ne i}^{\\color{blue}${controls_N}} \\frac{{{\\rm{sin}}\\left( {\\theta _j - \\theta _i} \\right)}}{{\\left| {{\\bf{x}}_j - {\\bf{x}}_i} \\right|}}. \\tag{4a*}`}\n`}\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"schema = {\n const canvas = this || DOM.canvas(width, height),\n context = canvas.getContext(\"2d\");\n\n if (!canvas.nodes) canvas.nodes = [];\n\n yield canvas;\n\n const nodes = canvas.nodes;\n\n while (true) {\n control_nodes(nodes);\n swarmalator(nodes);\n draw(context, nodes);\n yield canvas;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"swarmalator = function(nodes) {\n nodes.forEach((d, i) => {\n d.vx = d.vy = 0;\n });\n var n = nodes.length;\n nodes.forEach((d, i) => {\n nodes.forEach((e, j) => {\n if (j <= i) return;\n var e = nodes[j];\n var dx = d.x - e.x,\n dy = d.y - e.y,\n df = d.f - e.f,\n dist2 = dx * dx + dy * dy,\n dist = 0.0001 + Math.sqrt(dist2);\n\n var mu = ((1 + controls_J * Math.cos(df)) * dist - 1) / dist2 / n;\n\n d.vx -= dx * mu;\n d.vy -= dy * mu;\n e.vx += dx * mu;\n e.vy += dy * mu;\n d.f -= ((Math.sin(df) / dist) * controls_K) / n;\n e.f += ((Math.sin(df) / dist) * controls_K) / n;\n });\n\n d.x += d.vx;\n d.y += d.vy;\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":467,"value":"md`🢇 Click cell to expand code for ***controls_N***` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"viewof controls_N = slider({\n value: 50,\n min: 3,\n step: 1,\n max: 200,\n title: md`${tex`N`}`,\n description: \"Number of nodes\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":477,"value":"md`🢇 Click cell to expand code for ***controls_J***` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":27,"value":"viewof controls_J = slider({\n value: 1,\n min: 0,\n max: 2,\n title: md`${tex`J`}`,\n description:\n \"The parameter J measures the extent to which phase similarity enhances spatial attraction.\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":480,"value":"md`🢇 Click cell to expand code for ***controls_K***` ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"viewof controls_K = slider({\n value: -0.1,\n min: -1,\n max: 0.02,\n title: md`${tex`K`}`,\n description: \"The parameter K is the phase coupling strength\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"function draw(context, nodes) {\n const r = (0.25 * height) / Math.sqrt(nodes.length);\n context.clearRect(0, 0, width, height);\n\n nodes.forEach(node => {\n context.beginPath();\n context.arc(X(node.x), Y(node.y), r, 0, 2 * Math.PI);\n context.fillStyle = F(node.f);\n context.fill();\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":129,"value":"function control_nodes(nodes) {\n if (nodes.length < controls_N)\n nodes.push({\n x: 1 - 2 * Math.random(),\n y: 1 - 2 * Math.random(),\n f: Math.random() * 2 * Math.PI\n });\n\n if (nodes.length > controls_N) nodes.pop();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":52,"value":"M = Math.min(640, width) - height","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"margin = 10","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"X = d3\n .scaleLinear()\n .domain([-2, 2])\n .range([margin + M / 2, Math.min(640, width) - margin - M / 2])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"Y = d3.scaleLinear().domain([-2,2]).range([margin, height - margin])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":48,"value":"F = d3.scaleSequential(d3.interpolateRainbow).domain([0, 2 * Math.PI])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"height = width / 2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":234,"value":"d3 = require(\"d3-scale\", \"d3-color\", \"d3-scale-chromatic\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"import { slider, checkbox } from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":390,"value":"import {tb} from \"@mariodelgadosr/target-blank\"\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"