The pattern of period-doubling is quantitatively the same for a very large class of these 1D maps, including all the ones shown here (Logistic, Sine, and Quadratic). This phenomenon is known as universality, and its discovery by Mitchell Feigenbaum in the 1970s was a watershed moment in chaos theory. He found that the rate at which the bifurcations (the splitting points) occur is governed by a universal constant. The ratio of the distances between successive bifurcation points approaches the first Feigenbaum constant, δ (delta), which is approximately 4.669.... This means that no matter what specific equation you use (as long as it belongs to the same general class), the road to chaos follows the exact same rhythmic spacing. It's a fundamental constant of nature, like π or e.
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The Logistic Map
This plot, called a bifurcation diagram, shows the long-term behavior of a simple iterated function. For each parameter value on the horizontal axis, the function is run hundreds of times to let it settle, and then the subsequent values are plotted as vertical points. This reveals where the function converges to a single value, oscillates between several values, or enters a state of chaos, where it never repeats.
Equation
x_new = c * x * (1 - x)
The Sine Map
A classic iterative map that behaves very similarly to the Logistic Map, demonstrating a universal period-doubling cascade into chaos. Its bifurcation diagram is formed by smooth, flowing sine curves, offering a slightly different aesthetic.
Equation
x_new = c * sin(π * x)
The Quadratic Map (x² + c)
This powerful map directly relates to the famous Mandelbrot set. The diagram you see here is a one-dimensional slice of the Mandelbrot set along the real axis, showing how its behavior changes from stable to chaotic as the parameter 'c' varies.
Equation
x_new = x² + c