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 the 2m cycle or band, at values of the parameter a=am converging geometrically to a "critical" value ac i.e.
am = ac +/- 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 2mth iterate (nf=m) and blowing up the central region (nsc=m).
Another way of saying this is that if we take equivalent values of a, that we call am, in successive 2m cycles, then the ratio
(am - am-1)/(am+1 - am)
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 2m cycles also shows scaling. As we go from a 2 to a 4 to 8 .... the splitting of the orbit at successive transitions becomes smaller and smaller. In fact at a=am the splitting s scales as
sm = B -m
(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 - ac|b
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=am as
m = G 2 -m
Again, this can be rephrased
(a) = G' |a - ac|g
g=log 2/log = 0.45...
Even more amazing, the value of setting the geometric convergence of am to ac is universal, i.e. is independent of the form of the map (as long as it falls into a broad family with the same sort of smooth maximum as the quadratic map). The other scaling exponents , b, and g are also universal. This remarkable fact means that any system showing the onset of chaos by passing through an infinite cascade of period doubling bifurcations will show the same scaling with exactly the same numbers for these universal quantities. We can therefore find the values by such different schemes as iterating a one dimensional map on a computer, or by measuring the properties of a fluid becoming chaotic by this route! The universality has indeed been tested experimentally.
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 ac and the "amplitudes" A, B, B', C, G, G' are not universal i.e. these numbers will be different for different choices of map. Since these parameters set the scale e.g. for x and are changed by simply redefining the variable, it is not surprising they are not universal. It is quantities that set ratios, such as the ratio is control parameter differences (above) equal to , that may be universal. Some values of the nonuniversal quantities for the sine map are set automatically when the sine function is chosen from the drop down list.
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.