Lorenz Attractor

DEFINITION

ẋ = σ(y − x)
ẏ = x(ρ − z) − y
ż = xy − βz

NOTES

Lorenz was running a truncated weather model, restarted it from a rounded printout, and got a completely different forecast. The butterfly shape is the set the trajectory settles onto — bounded, never repeating, never crossing itself. This is the origin of the phrase "sensitive dependence on initial conditions".

PROVENANCE

Origin

E. N. Lorenz, "Deterministic Nonperiodic Flow", J. Atmos. Sci. 20, 1963

Standing

Public domain — a system of differential equations

Constants

σ=10, ρ=28, β=8/3 is the classic parameter set

Source

doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

HOUDINI · VEX

The same published mathematics as a Detail Wrangle body. Paste it into a Wrangle with Run Over set to Detail; every constant is the published value plus a tweak channel, so Create Spare Parameters gives a slider that starts where the paper does.

// FORMA — PL. 07 · LORENZ ATTRACTOR — Edward Lorenz, 1963
//   ẋ = σ(y − x)
//   ẏ = x(ρ − z) − y
//   ż = xy − βz
// Paste into a Detail Wrangle (Run Over: Detail), no inputs needed.
// Written from the published mathematics, not adapted from any code.
// Constants arrive at their published values. Press the node's Create
// Spare Parameters button and every tweak becomes a slider — starting
// at 0, the published figure, and moving in the constant's own units.
// https://forma-gen.com/#plate=lorenz

float p_sigma = 10 + chf('sigma_tweak');      // σ — Prandtl · live 4 .. 20
float p_rho   = 28 + chf('rho_tweak');        // ρ — Rayleigh · live 14 .. 60
float p_beta  = 2.667 + chf('beta_tweak');    // β — geometry · live 0.5 .. 5

// The plate's own colour: FORMA's ATTRACTORS accent as a cosine ramp,
// brightest near t = 0 and t = 1, near-black around t = 0.5.
vector forma_ramp(float t){
  return set(
    0.0956 + 0.1039 * cos(6.28318530718 * (t + 0)),
    0.4041 + 0.4392 * cos(6.28318530718 * (t + 0.05)),
    0.46 + 0.5 * cos(6.28318530718 * (t + 0.1)));
}

// waived: view — the azimuth the finished attractor is viewed from, and Houdini has a camera of its own
// Euler integration matching the plate: dt = 0.005, 18000 steps, the first
// 400 discarded — the walk toward the attractor is not the attractor.
int   forma_steps = 18000;
int   forma_skip  = 400;
float forma_dt    = 0.005;

// Lorenz's z is conventionally the vertical; mapped onto Houdini's y-up.
float x = 0.1, y = 0.0, z = 0.0;
int prim = addprim(0, "polyline");
for (int i = 0; i < forma_steps; i++){
    float dx = p_sigma * (y - x);
    float dy = x * (p_rho - z) - y;
    float dz = x * y - p_beta * z;
    x += dx * forma_dt;  y += dy * forma_dt;  z += dz * forma_dt;
    if (i <= forma_skip) continue;
    int pt = addpoint(0, set(x, z, y));
    // colour sweeps the bright lobe of the ramp along the path
    float u = float(i - forma_skip) / float(forma_steps - forma_skip);
    setpointattrib(0, "Cd", pt, forma_ramp(0.8 + 0.3 * u));
    addvertex(0, prim, pt);
}

AFTER EFFECTS · EXPRESSION

The same published mathematics as a Shape Layer path expression. Paste it onto a Path property; every constant is the published value plus a Slider Control named _tweak, so a bare paste already draws the figure and each slider moves one constant in its own units. Trim Paths is the comet.

// FORMA — PL. 07 · LORENZ ATTRACTOR — Edward Lorenz, 1963
//   ẋ = σ(y − x)
//   ẏ = x(ρ − z) − y
//   ż = xy − βz
// After Effects port — paste onto a Shape Layer's Path property
// (Contents › Shape › Path). Written from the published mathematics, not
// adapted from any code. Constants arrive at their published values; add a
// Slider Control (Effect › Expression Controls) named <k>_tweak and that
// constant moves in its own units, starting at 0 — the published figure.
// The plate's comet and its reveal are Trim Paths; the stroke colour is
// FORMA's ATTRACTORS accent, #35E0FF. Animation runs on time.
// https://forma-gen.com/#plate=lorenz

// A missing slider reads 0, so a bare paste already draws the figure.
function forma_tweak(n){ try { return effect(n)("Slider"); } catch (e){ return 0; } }
var p_sigma = 10 + forma_tweak("sigma_tweak");    // σ — Prandtl · live 4 .. 20
var p_rho   = 28 + forma_tweak("rho_tweak");      // ρ — Rayleigh · live 14 .. 60
var p_beta  = 2.667 + forma_tweak("beta_tweak");  // β — geometry · live 0.5 .. 5
var p_view  = 0 + forma_tweak("view_tweak");      // view azimuth ° · live -180 .. 180

// The frame: the plate's W × H canvas is this comp, with the origin at the
// layer's anchor; canvas y already runs down, as After Effects' does.
var forma_W = thisComp.width, forma_H = thisComp.height, forma_t = time;
var forma_phase = 0.05998828588053584;   // this plate's own fixed phase, as the page has it
function forma_pt(x, y){ return [x - forma_W / 2, y - forma_H / 2]; }

// Euler integration of the Lorenz system, the plate's own 18,000 steps of
// dt = 0.005 from (0.1, 0, 0), the first 400 discarded — the walk toward the
// attractor is not the attractor. The z axis is lowered by 0.72ρ so the wings
// sit centred. Then the page's view: the x–y plane turns at 0.13 rad/s and z
// is the screen vertical, scaled by 1/56 of the shorter side. Every second
// point is emitted — 8,800 vertices is what one After Effects path carries
// comfortably; the integration itself is untouched.
var x = 0.1, y = 0, z = 0, dt = 0.005;
var DEG = Math.PI / 180;
var rot = forma_t * 0.13 + p_view * DEG, co = Math.cos(rot), si = Math.sin(rot);
var s = Math.min(forma_W, forma_H) / 56, cx = forma_W / 2, cy = forma_H / 2;
var pts = [];
for (var i = 0; i < 18000; i++){
  var dx = p_sigma * (y - x);
  var dy = x * (p_rho - z) - y;
  var dz = x * y - p_beta * z;
  x += dx * dt; y += dy * dt; z += dz * dt;
  if (i > 400 && (i & 1) === 0){
    var zz = z - p_rho * 0.72;
    pts.push(forma_pt(cx + (x * co + y * si) * s, cy - zz * s));
  }
}
createPath(pts, [], [], false);