The Lorenz attractor comes from a 1963 simplified model of atmospheric convection: three
coupled ordinary differential equations relating a state (x, y, z) through three parameters
σ, ρ, β:
dx/dt = σ(y−x) dy/dt = x(ρ−z)−y dz/dt = xy−βz
The classic values σ = 10, ρ = 28, β = 8/3 produce the iconic
butterfly-shaped strange attractor: the trajectory never settles to a point or a repeating
cycle, never crosses itself, and never leaves a bounded region - it spirals around one wing, unpredictably
kicks over to the other, and repeats forever without exact repetition. Two trajectories starting a hair's
breadth apart diverge exponentially, the sensitive dependence on initial conditions that gave
chaos theory its "butterfly effect" name (Lorenz's own 1972 talk title). The attractor itself is a fractal:
it has a non-integer dimension of roughly 2.06 - more than a surface, less than a solid.
Below ρ ≈ 24.74 (with σ = 10, β = 8/3) the chaos vanishes:
trajectories spiral down to one of two stable fixed points instead. Try dragging ρ below that threshold to
watch the butterfly collapse into a simple spiral.
The trajectory is integrated with fourth-order Runge–Kutta and rendered by the same
point-cloud renderer built for the 3D IFS page - a dense point sequence
along the curve, projected and density-shaded exactly like that chaos-game point cloud.
Color by time tints the trajectory by how far along the sequence each point falls, split into
256 evenly sized chunks along d3's Turbo gradient (near-black at the very start, through
blue, green, and yellow, to dark red at the end). Unlike IFS's coloring, which tints by which of a handful of
transformations produced a point
- where a cyclical rainbow palette is fine, since there's no inherent order to "which vertex" - time has a direction, so a plain hue wheel would make the beginning and end look like the same color. The
sequential gradient makes the spiral-and-switch motion visible all at once: each wing's loops fan out through
the gradient, and a visible color jump marks exactly where the trajectory kicks over to the other wing.
The other buttons are the "orbit" fractals of Fractint, the DOS fractal program (the formulas are those of its lorenz.c and its default parameters). Every one is a short rule applied again and again to a point, and the picture is where the point lands. The number of points and the sliders work as for the Lorenz attractor; a rule whose point runs off to infinity (try a large a in Hénon) stops there and shows what it had.
- Rössler - a flow like Lorenz's, but with a single fold: the trajectory spirals outward in a plane and is thrown up and back, once per turn. Integrated the same way.
- Hopalong, Martin, Chip, Quadruptwo, Threeply - maps of the plane of the form x′ = y − f(x), y′ = a − x, with a different f each (a square root, a sine, products of logarithms and arctangents). Hopalong and Martin are Barry Martin's, made known by A. K. Dewdney in Scientific American; Chip, Quadruptwo and Threeply are from Michael Peters's HOP program. The pictures are rings and webs with a surprising amount of symmetry.
- Hénon - the classic map x′ = 1 + y − ax², y′ = bx, whose attractor is a curve folded over and over, a Cantor set across it.
- Gingerbreadman - x′ = 1 − y + |x|, y′ = x, a chaotic sea around islands of order, with no parameters.
- Pickover - a map in three dimensions with four parameters, so it can be turned like the Lorenz attractor.
- Icons - Michael Field and Martin Golubitsky's symmetric maps of the plane; the degree is the symmetry (five-fold at 5), and ω = 0 adds mirror symmetry. The buttons under the sliders are the 23 settings of Fractint's
icons.par with its names (Sand Dollar, Kachina Dolls, French Glass, 36 Spoke Wheel and so on); the page starts the orbit where Fractint does and skips its first 2000 points. The page keeps |z|² as the height of each point, so Fractint's icons3d is the same picture turned: drag it, and the symmetric web is a surface.
- Lorenz 1-, 3- and 4-lobe - Fractint's variants of the Lorenz equations whose attractor has one, three or four wings, written from its source as flows. They are drawn looking down the z axis, where the lobes show as a comma, a three-armed and a four-armed pinwheel.
- Latoocarfian - Clifford Pickover's map from Chaos in Wonderland, named for the beings of Ganymede whose forms it was said to draw: x′ = sin(by) + c sin(bx), y′ = sin(ax) + d sin(ay). It is the one type added in Fractint 20. Each of the four sins is a menu (fn1 to fn4) of real functions; all four sin is the original, and Fractint's own examples use mixtures such as sin, atan, sin, conj. The starting values are Pickover's, and the buttons add the four examples of Fractint's
fract200.par (sin, sin, sin, sinh and so on). Small moves of a, b, c and d change the web completely.
- Kam torus - a series of orbits of the area-preserving map x′ = x cos a − (y − x²) sin a, y′ = x sin a + (y − x²) cos a, each started a little farther out (the step) until the stop value, so nested rings and islands appear around the center. The starting radius of each orbit is the height of its points, so turning the picture shows the orbits stacked like layers (Fractint's kamtorus3d). Fractint's defaults are a step of 0.05 and 150 points an orbit; the page starts with more of both.
- Inverse Julia - the Julia set of z² + c drawn without any escape test: go backward, z → ±√(z − c), choosing the sign at random, and the orbit never leaves the set. Move c and the outline changes shape.
- Inverse Julia, breadth first and depth first - Fractint's modified inverse iteration: the two preimages of every point are kept only while the pixel they fall in has fewer than max hits points. That spreads the points evenly over the set, where the random walk above piles them up wherever it happens to linger. Breadth first works through the points in the order found; depth first follows the newest, so the set first grows as one long path.
- Popcorn, Dynamic, Dynamic (implicit) and Mandelcloud - "clouds": many short orbits started from a grid of points, not one long one. Popcorn is Clifford Pickover's x′ = x − h sin(y + tan 3y), y′ = y − h sin(x + tan 3x), written as Fractint 20 does with four functions and a constant C in place of the 3: sin, tan, sin, tan and C = 3 are the original, and the menus take complex functions, so log or sqrt of a negative number gives an imaginary part that feeds the other coordinate. Dynamic steps the cyclic system x′ = −f(y), y′ = f(x) with f(k) = sin(k + a fn1(bk)), fn1 being sin until you pick another function; the implicit version reuses the new y, which makes the orbits more complex. Mandelcloud iterates z² + c from every point with c equal to that point, until it leaves the circle of radius √2. The sliders set the length of an orbit and the size of the grid; the starts are visited in a scattered order, so any number of points shows the whole window.
The flat maps are shown straight on; drag to turn them. Pick a button to reset its sliders to the starting values.