Deterministic Chaos

The Univrsal Function

Here are the pictures of the universal function at the superstable 2-cycle, 4-cycle, and 8-cycle.

These pictures hint at the fractal nature of the limiting graph as s -> s_infinity - that is, of the universal function.

Comparing these pictures, we see at each period-doubling the horizontal segments of the previous graph are replaced by shrunken (sometimes vertically reflected) copies of the 2-cycle graph.

(At the scale of our pictures, this is clearest comparing the 2-cycle and 4-cycle graphs.)

As additional evidence for the fractal character of this universal function, note the red part of the picture is a scaled (and reflected) version of the blue part.

Return to the universal Function.