An Introduction to Qcurves in the Bifurcation Diagram
This page is a quick introduction to our article "The ShadowCurves of the Orbit Diagram Permeate the Bifurcation Diagram, Too". (Download a PDF copy) This paper appeared in the September 2009 issue of the International Journal of Bifurcation and Chaos, published by World Scientific.
Figure 1 is a bifurcation diagram, showing all attracting and repelling points for f_{c}(x) = x^{2} + c for all periods 1 through 12. We'll call these explicitly drawn curves of periodic points "Pcurves". Each Pcurve is shaped like an elongated backwards letter "C". The Pcurves all reach in from the left side of the figure, turn around (at all the various c's for which a saddlenode or perioddoubling bifurcation occurs) and exit left again. Several Qcurves appear implicitly in Figure 1. We've drawn six of them explictly in Figure 2; their formulas are
Q_{1}(c) = c  Q_{2}(c) = c^{2} + c,  Q_{3}(c) = (c^{2} + c)^{2}+c, and so on, until  Q_{6}(c) = (Q_{5}(c))^{2} + c 
Our contribution to this topic began with the observation that Qcurves also appear implicitly in a computerdrawn bifurcation diagram, like the one in Figure 1. What's more, we found that at any place where Qcurves intersect and are all tangent, the Pcurve which must meet there is also tangent to the Qcurves  see the points marked with t's in Figure 3. On the other hand, at intersections where the Qcurves intersect and no pair are tangent (such as M in Figure 2 or s in Figure 3), the corresponding Pcurve is also tangent to none of the Qcurves, and indeed, its slope is the fixed point of a linear system whose iterates give the slopes of the Qcurves, where the iteration starts with the slope of the lowestnumbered Qcurve at the intersection.
Figure 1. Bifurcation diagram (through period 12). 
Figure 2. The first six Qcurves. 
Figure 3. Q and Pcurve interactions. 
Figure 4. Q and Pcurve interactions up close. 
There are actually infinitely many Qcurves
at any given intersection, so of course we only draw a few. Here's a "zoom" into figure 3
at the asterisk (*). The (*) is on a prime period 5 Pcurve, which bifurcates into
a prime period 10 curve as you move left; another bifurcation from 10 to 20 occurs at the left edge of the picture. Attracting points
on the Pcurves are in orange, repelling in blue.
The six lowestnumbered Qcurves which meet at (*) are drawn; note their subscripts differ by multiples of 5, and they are all tangent to the orange Pcurve at (*). Moving left, notice how the Qcurves alternate going up or down to become tangent to the period 10 Pcurve at M_{3} and M_{8} respectively, with subscripts differing by 10 now. This whole scenario is repeated (using the other Q curve families) in four other places above/below the (*) (see the dotted line in figure 3 above). In particular, the intersection involving Q_{5}, Q_{10}, Q_{15}, ..., "happens" right above the (*), right on the c axis (figure 3). 
Figure 5. Orbit diagram. 
The family of functions f_{c}(x) = x^{2} + c is sometimes referred to as Myrberg's map. Myrberg [1963] was the first to study properties of Qcurves (presumably without the aid of either orbit or bifurcation diagrams). In particular, he was interested in finding relationships among roots of Qcurves and accumulations points of these roots. Using Myrberg's results, Mira [1987] further studied these accumulation points and their relationships with intersecting families of Qcurves. The details of why Qcurves can be seen in the familar orbit diagram (Figure 5) are discussed in Neidinger and Annen [1996]; briefly, we see Qcurves in the orbit diagram because chaotic orbits (or at least, very long periodic orbits!) of 0 "pile up" more above than below Qcurves (or vice versa) and the abrupt difference in densities of points above versus below make the Qcurves visible. 
Figure 6. Bifurcation diagram (again). 
We see the Qcurves in the bifurcation diagram
(Figure 6) for a completely different reason: as the bifurcations
develop from c = 0.25 down to c = 2, the Qcurves are lined with more and more of those tangent intersections with Pcurves. Just to the right of each such intersection, the Pcurve turns around and exits left again. These "turnarounds" make the Qcurves appear due to the "density difference" as there are more Pcurves on the left
than on the right of the Qcurves.
Note that these tangent intersections are near but not at the tip of the involved Pcurves: The tangent intesections occur to the left of any given bifurcation at the instant 0 becomes "superattracting"; for example at (*) and M_{3} and M_{8} in Figure 4 above...see Ross [2009] for details. 
References
Mira, C. [1987]
Chaotic Dynamics: From the OneDimensional Endomorphism to the TwoDimensional Diffeomorphism
World Scientific, Singapore, New Jersey, Hong Kong
Myrberg, P. J. [1958][1959][1963],
"Iteration der Reellen Polynome Zweiten Grades", I, II, III;
Annales Academiae Scientiarum Fennicae, Series A, 256, p.110,
268, p110, 336, p. 118.
Neidinger, R.~D. and Annen III, R.~J. [1996]
"The road to chaos is filled with polynomial curves"
American Mathematical Monthly, 103, 640653.
Ross, C. and Sorensen, J. [2000]
"Will the real bifurcation diagram please stand up,"
College Mathematics Journal, 31, 214.
Ross, C., Odell, M., Cremer, S. [2009],
"The ShadowCurves of the Orbit Diagram Permeate the Bifurcation Diagram, Too",
International Journal of Bifurcation and Chaos 19:9, 30173031.
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© 2011 by Chip Ross Associate Professor of Mathematics Bates College Lewiston, ME 04240 