The demonstrations to this point should have illustrated the following conclusion:

The map shows similar sorts of behavior (instability, "superstable" cycle, band-merging etc.), except involving the2cycle or band, at values of the parameter^{m}a=aconverging geometrically to a "critical" value_{m}ai.e._{c}

a_{m}= a_{c}+/- C^{ -m}

with = 4.6692016091...

This sort of geometrical convergence is known as "scaling".

This is easy to show using the drop down boxes: set a value of
your choice for *C* in the text box, and vary *m* with the
drop-down list. The similarity of the behavior may be shown by
looking at the *2 ^{m}th* iterate (

Another way of saying this is that if we take equivalent values of
*a*, that we call *a _{m}*, in successive

(a_{m} - a_{m-1})/(a_{m+1} -
a_{m})

approaches a fixed value for large *m*. This asymptotic value
of the ratio is just in the formula above.

The separation of points within the *2 ^{m}* cycles
also shows scaling. As we go from a

s_{m} = B ^{ -m}

with

= 2.502807876...

(This is an oversimplification: the splitting actually depends on
which point around the orbit is considered - the quoted value is for
the point nearest the center *x=0.5*.) Another way of saying
this is

*s(a) = B' |a - a _{c}|^{b}*

with

b = log /log = 0.595...

This puts the scaling in a form reminiscent of the scaling at a thermodynamic phase transition.

The orbit splitting scaling has already been shown in the
demonstration, since the *nsc* parameter blows up the central
region by the amount ^{ nsc}.

There are many other scaling phenomena that cannot be demonstrated
with the present applet. For example the *Lyapunov exponent
*
that determines the stability of the periodic orbits, or
sensitive dependence on initial
conditions in the chaotic state, scales at *a=a _{m}*
as

_{m} = G 2^{ -m}

Again, this can be rephrased

(a) = G' |a - a_{c}|^{g}

with

g=log 2/log = 0.45...

Even more amazing, the value of setting the geometric
convergence of *a _{m}* to

You can crudely verify the universality here by changing to the
*sine map* in the drop down list, and observing the same set of
phenomena (geometric convergence, rescaling etc.).

Note that *a _{c}* and the "amplitudes"

A small complication arising from the fact that the rescaling amplitude

- "superstable rescaled 1 cycles" i.e.
*Superstable*and*"-"*in the list boxes,*a=a*,_{m}*nf=m*,*nsc=n*, use*scale=0.88*; - "superstable rescaled 2 cycles" i.e.
*Superstable*and*"-"*in the list boxes,*a=a*,_{m}*nf=m-1*,*nsc=n-1*, use*scale=1.05*; - "instability of the rescaled
*2*cycles", i.e.^{m}*Instability*and*"-"*in the list boxes,*a=a*,_{m}*nf=m*,*nsc=n*, use*scale=1.01*; - "bandmerging" for
*a > a*, i.e._{c}*Bandmerging*and*"+"*in the list boxes, and*a=a*,_{m}*nf=m*,*nsc=n*, use*scale=1.14*.

In all cases these values are chosen so that the rescaled map curve and the diagonal intersect at the origin, and then the iterations will remain within the (rescaled) unit box. For the sine map, the values will be slightly different.

Last modified Saturday, September 7, 1996

Michael Cross