Fractals - Bifurcation

The logistic map is one of the simplest equations that produces chaos: x_n+1 = r x_n(1−x_n), a single population-growth-style formula with one parameter r. Starting from some x₀ in (0, 1) and iterating repeatedly, the long-term behavior of x_n depends entirely on r:

• For r below about 3, the sequence settles onto a single fixed point.
• Past r = 3 that fixed point splits into a stable 2-cycle, then a 4-cycle, then 8, doubling forever - the period-doubling route to chaos. The ratio between successive period-doubling intervals converges to the Feigenbaum constant (≈ 4.6692), which shows up in countless unrelated chaotic systems.
• Past r ≈ 3.56995 the cycles never stabilize and the map becomes chaotic - though narrow periodic windows (like the period-3 window near r = 3.83) punch clear gaps back out of the chaos.

The main plot is a bifurcation diagram: each vertical slice fixes r, discards a warmup number of iterations to let the sequence settle past its transient, then plots the next points values of x_n. Darker pixels mean the sequence revisited that (r, x) region more often. Scroll or pinch to zoom in on any region - the branching structure keeps repeating at every scale, the same period-doubling cascade in miniature, which is what makes this a fractal rather than just a chart, and the diagram gets recomputed fresh at the new, narrower r/x range as soon as a gesture settles, so zooming in keeps revealing real detail rather than just enlarging the same pixels. The inset in the lower-left corner is that r's cobweb diagram (r = 3.5 until you hover): the parabola is f(x) = rx(1−x), the diagonal is y = x, and the red zig-zag traces x₀ → x₁ → x₂ → … bouncing between the curve and the diagonal - converging to a point, orbiting a cycle, or filling space chaotically, matching whatever the bifurcation diagram shows at that r. Hover anywhere on the diagram to move both the red guideline and the cobweb to that r.

The other buttons draw the same kind of diagram for other one-line maps, those of the bifurcation types of Fractint, the DOS fractal program (the formulas and the starting views are from its miscfrac.c). Each is a rule x → f(x) with a rate r along the horizontal axis; what changes is the shape of f, and with it the cascade.

• Verhulst - x + rx(1−x), the logistic equation stepped as a differential equation: the new population is the old one plus the growth. It goes chaotic at a smaller r and its picture is the logistic one folded over.
• r sin πx and x + r sin πx - a sine in place of the parabola. The first is symmetric about r = 0 and has several chaotic bands; the second climbs through its cascades one after the other.
• Stewart - rx² − 1, a parabola that opens upward, with the cascade turned over.
• May - rx / (1+x)^β, a population model of Robert May's, drawn for negative r as Fractint does.

The cobweb inset follows the map you picked, drawn over the x range of its starting view; where the red path leaves the inset the orbit has left the window. An orbit that runs off to infinity is dropped: nothing is drawn for that r. Scrolling, panning, the sliders and hovering work as on the logistic map.

Fractint lets you put a function inside these maps, so the Logistic, Verhulst, sine and Stewart buttons show a Function fn menu of real functions: identity for the first, second and last, sin for the sine maps, which is the original. Another function moves the diagram, so the x window is fitted to where the orbits go; many choices settle to a single point or run off to infinity and draw little, which is also what the map does. May has no function, but its exponent beta (at least 2, Fractint's 5 to start) is a field of its own. The Seed population field is the value every column starts from; with a long warmup it makes no difference, but with the warmup near zero the early orbit shows as curves laid over the diagram, which is what Fractint's unfiltered maps do.