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

In the section on scaling and universality the following combination of operations

- change
*a*to_{m}*a*_{m+1} - functional composition: look at a higher iterate of the map
function
*f*: change*f*to^{ 2m}*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

- functional composition: look at a higher iterate of the map
function
*f*: change*f*to^{ 2m}*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:

- 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/))* - 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*to_{m}*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.

[Return to Home Page]

Last modified 18 August, 2009

Michael Cross