◀ Navigate through the Chaos pages ►

# Universality

We see the same sort of behavior independent of the details of the map we use.

## Brief Explanation

Any function that maps a range into itself — and the range from 0 to 1 is as good as any — can be meaningfully iterated. As the shape of the function changes, so does the bifurcation diagram. Even the chaos is chaotic out there, which is to say there are a lot of differerent-looking bifurcation diagrams. Here we use the slider just just to manipulate the function.

## Try It Yourself

Slider for setting height

### Interesting functions

• The usual logistic map.
• Like the logistic map but higher-order: essentially x²∙(1-x)².
• A circle that flattens to an ellipse as it gets smaller.
• The logistic map but with added wiggles.
• The simplest up-and-down function.
• Not 2∙tent(x) but tent(tent(x)).
• One pure wave of the sine function.
• A function with opposite curvature. Kinda disappointing.

## Remarks

This is the most interesting page of the sequence, if only for the reason that it suggests there is so much more to see! Here are a wide variety of phenomena arising from the particulars of the iterated function. So much similarity, and so much difference. Why does the quartic map’s attractor collapse to zero at the end? The discontinuities in the attractor of the wiggly map show how changes in curvature lead to the abrupt changes that René Thom classified as elementary catastrophes. Even the simplest of up-and-down functions, the tent map, yields strangeness when iterated.

The example of the “double” tent map — really tent(tent(x)) — gives an idea of how intricate functions become when you compose (iterate) them. This is why the behavior, and logistic diagrams, get so complicated: you are not just dealing with f(x), but f(f(f(...(x)...))). And that function is not so simple! The next page in this sequence will look at these shapes for many steps of nesting of our favorite example, the logistic map.

◀ Navigate through the Chaos pages ►