Fractals - Logistic Map

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. Click anywhere on the diagram to zoom in — 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. 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 whole page was created with Claude Code. These were the most significant prompts that shaped this page:

1. Create a new fractal page about the logistic map equation, following the layout of the existing fractals.
2. Make the cobweb diagram always on, and move it into the lower-left corner of the bifurcation diagram instead of below it.