The fact that this same Feigenbaum ratio keeps showing up in
the attractor of unrelated functions is known as
"universality" . Does this mean that the instructions
on where to bifurcate the attractor are not contained in the
The notion of universality is at the heart of the science of
chaos. What we have seen is just an easily visible tip of a large
iceberg, most of which is submerged in a sea of complexity. In
studying as we have the simplest sort of nonlinear equations we
have glimpsed a structure that also controls real physical
systems, like flow turning turbulent and metal being magnetized,
at the onset of chaotic behavior. If the rules are independent of
the function, then applying the rules does not require that we
necessarily understand the function details.
As in the section on iterations and attractors, you now have
an opportunity to explore the attractors of all three functions.
Take some time to play around with the
Phase-Control Map Research
This completes the discussion on phase-control maps. In the next lesson we will go back and pick up another thread of the chaos story, involving sets in the complex plane.
Are there any questions?