Lorenz System

The System

Lorenz system is a system of (non-linear) ordinary differential equations. \begin{aligned} \dot{x} &= \sigma(y-x)\\ \dot{y} &= x(\rho-z)-y\\ \dot{z} &= xy-\beta z \end{aligned}

Here \sigma,\rho,\text{ and }\beta are system parameters.

Evolution Projected on Y-Z Plane

Plot.plot({
  height: 500,
  widht: 500,
  x: {
    domain: [-30, 30],
    grid: true
  },
  y: {
    domain: [0, 50],
    grid: true
  },
  marks: [
    Plot.line(lorenzData.slice(0, n), {
      x: "y",
      y: "z",
      strokeWidth: 0.5,
      stroke: "red"
    }),
    Plot.ruleX([0]),
    Plot.ruleY([0])
  ]
})

Time-series from the Z-axis

Plot.plot({
  x: {
    domain: [0, 4000]
  },
  marks: [
    Plot.line(lorenzData.slice(0, n), {
      x: "n",
      y: "z"
    }),
    Plot.ruleX([1500, 3500], { stroke: "red" }),
    Plot.ruleY([0])
  ]
})

Takens’ Embedding

viewof delay = Inputs.number({ value: 7, label: tex`Delay\ (\tau)` })
Plot.plot({
  marks: [
    Plot.dot(
      delayTS(
        lorenzData.map((d) => d.x),
        delay
      ).slice(0, n),
      {
        stroke: "red",
        r: 1
      }
    )
  ]
})

delayTS = function (ts, delay = 1, dim = 2) {
  return ts.map((d, i) => {
    const out = [];
    for (let j = 0; j < dim; j++) out.push(ts[i + j * delay]);
    return out;
  });
}

lorenz = function (
  [x, y, z] = [0, 0, 0],
  n = 100,
  { sigma = 10, r = 28, b = 8 / 3 } = {}
) {
  let F = function (t, [x, y, z]) {
    return [sigma * (y - x), r * x - x * z - y, x * y - b * z];
  };

  let s = new odex.Solver(3);
  let flow = [];

  for (let i = 0; i < n; i++) {
    [x, y, z] = s.solve(F, 0, [x, y, z], 0.01).y;
    flow.push({ n: i, x: x, y: y, z: z });
  }
  return flow;
}

lorenzData = lorenz([1, 1, 1], 4000)

odex = import("https://cdn.skypack.dev/odex");

n = {
  let i = 0;
  while (i < 4000) {
    yield Promises.delay(10, ++i);
  }
}