Renormalization Group

The geometric scaling and universality can both be understood from the "renormalization group theory" developed by Feigenbaum. This page presents a brief introduction.

• change a_m to a_m+1
• functional composition: look at a higher iterate of the map function f: change f ^2m to f ^2m+1
• rescale: multiply the coordinates by an additional power of ;

leads to a graph that appears unchanging as m is increased. You can check this using the drop down list boxes and setting nf =nsc=m . Alternatively, set a=a_c , and then look at successive

• functional composition: look at a higher iterate of the map function f: change f ^2m to f ^2m+1
• rescale: multiply the coordinates by an additional power of

Again as m is increased the graph appears to approach another fixed curve. These asymptotic curves are universal (e.g. the same for the quadratic or sine maps, as you can roughly see from the applet).

We define an operator T that accomplishes the functional composition and rescaling by a factor (as yet unknown). Acting on any function f :

T[f](x)=- f(f(-x/))

Note that T operates on (i.e. changes) the function f . In the context of the map functions for example

T[f ^2m] gives f ^2m+1 (rescaled)

Now the scaling and universality follow from two hypotheses:

1. The operation T has a fixed point solution g(x) for a particular value of = 2.502807876..., i.e.

   T[g]=g

   or

   g(x) = - g(g(-x/))

2. Linearizing about the fixed point, there is a single unstable direction (eigenvector) with an expansion rate (eigenvalue) that is 4.6692016.. This turns out to fix the value of in the geometric approach of a_m to a_c .

Notice that and and the function g are defined completely independently of any starting map function, and indeed are defined by purely abstract mathematical operations. This yields the universality of the period doubling cascade route to chaos. It is a remarkable result that these abstract mathematical constructions lead to numbers that can be measured in actual experiments on chaotic fluid and other systems.

Further details (laTeX file).