BIFURCATIONS OF THE DEGREETWO STANDARD FAMILY OF CIRCLE MAPS
By
WILLIAM CHRISTOPHER STRICKLAND
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2007
by Lemma 5, c is in the flat spot of T. By the discussion in the previous section,
a(() = and since c is in the basin of x, we also have a( ) = m. E
2.7 The (b, w) Diagram for the DegreeTwo Standard Family
The degreetwo standard family (24) has exactly two critical points in the interval
[0, 1] and thus the circle map gb,w has two critical points. The critical points of gb,w are the
solutions to g'b(x) = 0, and they depend on b but not w. I denote them as cl(b) and c2(b)
and solve for them as follows:
0 = gb(x) = 2 + 2b7 cos(27x)
2
2b= cos(27x)
2bx7
arccos() 27 arccos( )
= x = cl(b), 2 = x = c2(b)
2xr 2xr
The notes so far indicate the importance of studying the value of a at a critical point.
Note that we get a different a for each (b, w). I indicate this by writing ab,, and for each
(b, w), I define:
s(b, w) = ab,w(Cl(b)) and 2(b, w) = b,w(c2(b)). (222)
Note that Lemma 6 ,i that if gb,, has an attracting periodic point i of type (n, m),
then sj(b, w) = for j = 1 or 2, or both, but not necessarily the converse.
Remark 2. As will become evident in the numerical results, we can r, i,,i.;l. things about
the relationship between si and s2. In particular, si(b, 1 w) = s2(b, w), allowing us to
study si and generalize to 2*.
Now conversely, let f be a degreen function: f(x + 1) = f(x) + n. For each y E S1,
pick some r E R such that p(r) = y and define f(y) = p(f(r)). The actual choice of r does
not matter since p(f(r + k)) = p(f(r) + kn) = p(f(r)), so this function is well defined. We
now have that p(f(x)) = f(p(x)) by definition. D
2.3 The Degree2 Case
From now on, we will assume that f : R +I R is degree 2 and that f: S1 S1 is its
corresponding circle map. From Lemma 3, we may assume that
f(x) = 2x + y(x) (23)
with 7 periodic. The most important degree2 map for us is the degree2 standard family:
gbw(x) =2x + w + bsin(2rx) (24)
The following theorem is essential in our discussion of gb,w(X) [4].
Theorem 1 (Boyland). Let f : R + R be a continuous, degree2 map and 1. 7;,'.: F, by:
f"
F,: (25)
Then there exists a continuous, degreeone map a : R + R with F, + a unifoi ,,l ;
Furthermore, if 7 is as in (23), then:
F,(x) x + 7(f 1(x)) (2 6)
j=1
and
a(x) x + f )) (27)
2J
j= 1
In addition, a o f = d o a where d(x) = 2x, i.e. the following l.:,rram commutes:
f R
a a (28)
R d R
 RKD
Figure 326. gb,w(X) in region X, when b = 0.6 and w = 0.2.
Figure 327. gb,w(x) in region Y, when b = 0.6 and w = 0.8.
compact and orbits of f often come back near themselves (recur). We need to understand
the connection between periodic orbits of f and their lifted orbits of f.
Definition 8. Assume that x E S1 has a periodn orbit under f and that x E R is a lift of
. Now since f (() = then using (22), f"(x) = x + m for some mn c Z. In this case,
we r ;, that x is a 1,',' (n, m) point for f.
Note that if I chose a different lift of x, I would get a different m. In fact, using
Lemma 1 for an arbitrary k e Z,
f"(x + k) f"(x) + 2"k = x + m + 2"k = (x + k) + m + (2" 1)k (221)
and so x + k is of type (n, (2" 1)k + m).
This means that the type of a periodn point x E S1 under f is only defined modulo
2" 1.
If x has type (n, m) for a degreetwo f and a is as in Theorem 1, then using diagram
(28),
(f"(x)) = a( + m)
dn(a(x)) = a(x) + m
2'a(x) = a(x) + m
which implies that a(x) = 2 If I choose another lift (x + k) of 7, I get that
(2" l)k +m m
a(x+ k) ( 1)k k +
2" 1 2 1'
and so a(x) a(x + k) mod 1. So on the circle we have that if x is a periodn point, then
2 ) = e S1 for some 0 < m < 2" 1.
2.6 Flat Spots in the Graph of a
Definition 9. A flat spot for the 'j'i,, of a is a closed interval I so that a(xi) = a(x2)
for all x X2 E I, and I is the maximal interval with this 1", 1 .,/;, i.e. f is not constant
on iw,1 interval J with J / I and I C J.
1 1
0.8
0.6
0.4
0.2
0
0
Figure 35. s(b, w).
1
0.8
0.6
0.4
0.2
0
.4 0.6 0.8 1 1.2
0.4 0.6 0.8 1 1.2
Figure 36. s2(b, w).
CHAPTER 1
INTRODUCTION
This work is a study of nonlinear dynamical systems. Heuristically, a nonlinear
system is one whose behavior cannot be described using only the parts that comprise
it. Systems of this type are of particular interest because they are the most prevalent
in nature and are often difficult to understand. Looking at the wide variety of fractals
obtained from iterating the degreen standard family, one can immediately see that the
dynamics of this system are much more than just the sum of its parts.
The one dimensional dynamics that I will discuss have the added feature that they
act on the circle and not just the real line. Often we use the lift, the real line version
or the function, to better understand what is happening on the circle where output has
a tendency to overlap. Once this step is accomplished and the theory properly built
up, the natural way to seek understanding of the function is to find some sort of map
that describes how it behaves when the values of its parameters change. Some standard
references for one dimensional dynamics are [1], [2] and [3].
This strategy is exactly what is used in this paper in which I describe the bifurcation
diagram for the degree2 standard family. I begin by explaining more about this family
and discuss some of its particular details before moving on to the specific dynamical
systems studied here. After establishing the relationship between the family on the circle
map and the family on the real line, I will rely heavily on numerical results to draw
conclusions about how changes in the parameters effect the long term behavior found by
iterating the function.
2x+1.05sin(2 7x) +0.1
1.5
1.25
0.75
0.5
0.25
0.2 0.4 0.6 0.8 1
0.25
0.5
Figure 331. ab,w(x) in area theta when b = 1.05 and w = 0.1.
2 x+ 0.97 sin(2 x x) +0.83
2.5
2.25
2
1.75
1.5
1.25
0.2 0.4 0.6 0.8 1
0.75
0.5
Figure 332. ab,w(x) in area U when b = 0.97 and w = 0.83.
2 x+0.97 sin(27Tx) +0.17
1.5
1.25
0.75
0.5
0.25
0.2 0.4 0.6 0.8 1
0.25
0.5
Figure 333. ab,w (x) in area V when b = 0.97 and w = 0.17.
Figure 37. Key for 35 and 36.
0.5
0.25
Figure 38. 3D image for 35.
3 ,
0.5 1 1.5 2 2.5
Figure 39. s(b, w) with b ranging from to and w ranging from 0 to 3.
7T 7T
Figure 324. gb,w(x) when b = 1.235 and w = 0.5 with three fixed point lines.
2 x + 0.8 sin(2 x)
Figure 325. ab,w(x) when b = 0.8 and w = 0.
Finally, both areas U and V lie in the first iteration flat spot of one of the critical
points while sitting in a later iteration flat spot of the other. The first iteration flat spot
seems to have the effect of swallowing the other, so that the result is the graphs shown in
Figure 332 and Figure 333.
Lemma 5.
1. If I is an interval with f(I) C I, then I is contained in a flat spot of a.
2. If I C S1 is an interval with f (I) C I for some n > 0, then I is in a flat spot of u.
3. If I C S1 is the basin of an attracting periodic point for f, then I is in a flat spot of
0a.
Proof:
1. Let I be an interval,v I [a,b]. Then by induction, ff"() C I C [a, b]. So for
all x C I, a < f'(x) < b and thus < < f. Now by the squeeze theorem,
limn. fn 0, so for all x C I, a(x) 0. As a result, I is contained in a flat spot
of a.
2. The proof is the same as for 1, just add in the bars denoting that you are on the
circle.
3. By definition, I is the maximal interval on which lim f (x) = J(x) = o, where Xo is
an attracting periodic point. By part two of this lemma, I is a flat spot of 5. o
Remark 1. An example of a flat spot that does not come from the basin of a periodic
point is a tril j.':,,, region; a region in which the points cannot escape. The f,l.it.:. of a
flat spot I... ,, 1,]:,,j to a = 0 or a = 1 is i. /l;, that: a trapping region that does not go
around the circle (see numerical results for .. ..:'' examples and descriptions).
Lemma 6.
1. If x is precritical and f"(x) = c, a critical point, then for all j > 0, F]+j(x) = 0, with
F, as 1, 1. ,lin Theorem 1.
2. If f has an attracting periodic point x of type (n, m) with 0 < m < (2" 1), then for
some critical point T, (d) = and further, c is contained in a flat spot of a.
Proof:
1. Since f"(x) = c, F, and thus FT = 0. F1+i = f() and since f(x) is a real
number, F+, = 0. By induction, F' + = 0.
2. We have that S(f) < 0, so by Singer's Theorem there exists a critical point T
with w(() = o(), the orbit of x. This fact means that c is in the basin of x and
Proof: First, I will show that (26) holds by induction.
Base case: If j 1, then we have that
Fl(x) f(x) 2x + 7(x) +
2 2 2
Induction step: Suppose that (26) holds for n = k and consider k + 1:
f k+ () f(fk(x)) 2 fk(x) + (fk(x))
Fk+(x) 2k+1 2k+1 2k+1
fk(x) (fk(x)) Fk (f ())
2k 2k+1 2k+1
By the induction hypothesis,
Fk+I(X) + k fl + (f ))
2 i2k+1
j=1
k+1 _ ))
2j
j=1
(29)
(210)
(211)
Now to show that limn,,, Fn converges uniformly, consider 7 from (26). 7 is
degreezero, so 7(x + 1) = 7(x). Since 7 is periodic and continuous, there exists M with
I7(x) < M for all x E R. As a result, for each j E N:
(212)
M converges, so by the Weierstrass M test [5], FT converges uniformly.
All that remains is to check that a o f d o a when d(x) = 2x:
f"( f(r))
a (f (x)) = lim Fn(f (x)) = lim
n*oo n2oo n
fn+1(.) f'(x)
lim = lim 2 f (X 2a(x) = d(a(x))
noo 2n noO 2n
All of the maps in (28) project to the circle and so we also get
S1 f S1
S1 S1
(213)
(214)
7(f J(x)) < M
23i 2i
0.4
0.2
01
0.2
0.4
0.4 0.6 0.8 1 1.2
Figure 311. Bifurcation diagram with the colors running through running seven times the
normal mod 1. Once again, a graphics program was used to create the
picture, and I have included code which verifies the placement of the line.
Figure 312. Graph of gb,w when b = 1 and the critical point is a fixed point. The fixed
point line mod 1 has been included.
the pixels of the most prominent colors under different resolutions, I have arrived at the
following statement:
Conjecture 1. In the complete plot of s(b, w) and s2(b,w) (that is, when b r,",. from
to .,I ),.i)l the areas of i(. ,, *ui,,.:., colors is equivalent.
This conjecture is further supported by the second part of Lemma 6 in C'! Ilpter 2.
For different parameter values, cl (c2) could be attracted to different points in the same
periodic orbit of gb,w. As long as the period of the attractor is the same, it is natural to
expect that the areas for attraction in the parameter space be equal. Consider 37 and
find the position of the most prominent green and blue in the parameter space as an
example. Green is found at 2 and blue is at 4, which means that green corresponds to
the value 1/3 and blue to 2/3 on S1. These are exactly the values possible when gb,w has
an attracting periodic point of type (2, m) for 0 < m < (2" 1).
3.3 Characterizing the Bifurcation Diagram
Now that we have generated these density plots for si(b, w) and S2(b, w), the obvious
question to ask is what the features of the plot tell us. As mentioned at the end of
C'!i lpter 2, si(b, 1 w) = 2(b, w) so in this section, I will only talk about si(b, w) since
the behavior of s2(b, w) is similar. Figure 310 shows si(b, w) again, with some of the more
interesting features labeled.
3.3.1 Fixed Points and Saddle Points
The fixed point line in 310 represents the w value (mod 1) for which a given b value
makes the critical point a fixed point of a. This line was found numerically by solving the
equation:
ci(b) 2c1(b) + w + bsin(27rci(b)) (32)
We can immediately see that using the parameters to shift the fixed point slightly
away from the critical point does not affect the value of ab,w(cl). The critical point still
converges to the same place in the graph, the fixed point of gb,,. By Lemma 4 in C'!i lpter
2007 William C'! il, 1!1. ir Strickland
The ideas behind point B have also been discussed. B is special in that the horizontal
lines through the left and right fixed point form a sort of box with the function gb,w. It is
the threshold for both critical points having an escape hatch. Point S represents the place
where gb,w has a double saddle point, since it is the intersection between both saddle point
lines (Figure 323).
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0.2 0.4 0.6 0.8 1
Figure 323. gb,w(x) when b = 0.726 and w = 0 with two fixed point lines.
Another intersection occurs at B. At this point, the large value of b in gb,w(x) has
expanded the graph to just touch three fixed point lines (Figure 324). The area A should
also be familiar, as it di,'1,v behavior like the area alpha in 310 for both critical points.
ab,w of this region is di .1' '.1 in Figure 325.
Referring back to Figure 323, we can also imagine the defining characteristics
for areas X, Y, and Z. In X, we have the situation where the top most fixed point line
intersects gb,w, and the other fixed point line is too low to pl ,i a role. Area Y is just the
opposite, and area Z places both critical points between the fixed point lines.
The plots of ab,,(x) will vary depending on the choice of b and w in these regions,
since later iterations of gb,w(x) have a pronounced effect on the limit. This is also the case
for areas nu, mu, lambda, and theta. Below are the ab,w plots for nu, lambda, and theta
respectively, mu is simply a 180 degree rotation of nu.
diagram. As we shall see when discussing the point phi in 310, the threshold of this effect
does become visible if the bifurcation diagram of the second critical point was shown on
top of the diagram for the first critical point. Now in Figure 316, we see that beta lies
right on the threshold the first critical point escaping.
1
0.75
0.5
0.25
.2 0.4 0.6 0.8 1
0.25
0.5
0.75
1
Figure 316. Graph of gb,w, approximately at parameter point 3 in 310.
Above point beta on the diagram, the critical point escapes the trapping region on
the first iteration of gb,w. It is still possible that critical point may enter the region under
a later iteration, but it now has an escape hatch where parts of the curve rise above the
top of the trapping box. The green lines in Figure 317 show how the critical point maps
below the trapping region in the first iteration.
Conjecture 2. In iw.;, plot that falls within area ,Iaiia,, the x values in the interval
represented by the bottom of the tr'j''.:'r region form a cantor set of points that never
escape the region. This statement should be ..,; to prove using a code space.
At point delta, gb,w is on a threshold of no longer using the curve to fall below the
trapping region, as seen in Figure 318.
For area epsilon, iterations of gb,w sometimes take the critical point into the trapping
region while others allow it to escape. Because we must examine ab,w to determine the
result, the graph of gb,w in Figure 319 is somewhat unenlightening.
326 gb,w(x) in region X, when b = 0.6 and w = 0.2. .................. .. 35
327 gb,w(x) in region Y, when b = 0.6 and w = 0.8. .................... .. 35
328 gb,w(x) in region Z, when b = 0.4 and w = 0. ................ 36
329 ab,w(x) in area nu when b = 0.85 and w = 0.3. .................... .. 36
330 ab,((x)in area lambda when b = 1.05 and w = 0.5. ................ 36
331 ab,w(x) in area theta when b= 1.05 and w= 0.1. ................ 37
332 ab,,w(x) in area U when b = 0.97 and w = 0.83. .................... .. 37
333 ab,w(x) in area V when b =0.97 and w 0.17. .................... 37
3.4 Characterizing the Double Bifurcation Diagram
Now we shall examine different areas of the density plot formed when the bifurcation
diagram of the second critical point is placed on the bifurcation diagram of the first
critical point. These areas are labeled in Figure 321. Obviously, some of these areas
contain concepts we have already examined in the previous section. Area F, for example,
has already been shown under gb,w in Figure 314. Figure 322 is the a(x) of the same
point in the parameter space (mod 1 for w).
I
0.4 0.6 0.8 1 1.2
Figure 321. Double Bifurcation Diagram. The bifurcation diagram of the second critical
point on top of the bifurcation diagram of the first critical point.
2x+ 0.6 sin(2 x) +0.6
0.75
0.5
0.25
0.2 0.4
Figure 322. ab,w,(x ) when b = 0.6 and w = 0.6.
0.6 0.8 1
CHAPTER 2
THEORY
2.1 Degreen Maps of the Real Line
Definition 1. Let f : X X be a function. We /. I;,'. the kt iterate off, or fk, as
fofofo ... o f(x) k times.
Definition 2. Ifn E Z, a function f : R + R is said to be degree n when f(x + 1)
f(x) + n for all x e R.
Lemma 1. If f : R + R is degree n and m E Z, then
1. f(x + m) = f(x) + nm.
2. fk is degree nk, i.e. fk(x + 1) f(x) + k
Proof: We can assume m / 0. Suppose that m E N, and proceed by induction
on m. The base case is m 1, and by definition we have our result. Now suppose the
lemma holds for m < m. Then f(x + m + 1) = f((x + 1) + m) = f(x + 1) + nm
f(x) + nm + n = f(x) + n(m + 1). We can quickly check that the lemma holds for m E Z
as well. Suppose m < 0 and let k = ml. Then f(x) = f((x k) + k) = f(x k) + kn, and
so f(x) k = f(x k). Thus f(x) + mn = f(x + m).
Proof of 2: Proceed by induction. If k = 1, f(x + 1) = f(x) + n = f(x) + n1 which
proves the base case. Now assume the lemma is true for k = m. Then fm+l(x + 1) =
f(f m(x + 1)) = f(f(x) + nm) by the induction hypothesis. Now f(f(x) + n"m)
f(x) + n nm = f(x) l. +
Example 1. The following are some examples of degree n functions:
1. f(x) = nx is degree n.
2. The degreen standard fi n,,:l; for parameters w, b E R:
g(x) = nx + w + bsin(27rx) (21)
3. A periodic function of period 1 (y(x + 1) = 7(x)) is the same as a degreezero map.
4. If 7 is degree zero, then f(x) = nx + 7(x) is degree n.
1
0.75
0.5
0.25
.2 0. 0.6 0.8 1
0.25
0.5
0.75
1
Figure 313. Graph of gb,w at parameter point s in 310.
Finally, the graph of point a in 310 simply shows a fixed point that is not a saddle
point but still left of the critical point. I have omitted it for the sake of brevity.
3.3.2 Trapping Regions and Other Effects
As explained in C'! lpter 2, the flat spots in ab,w are not only the result of attracting
fixed points but can also occur from trapping regions. In the following diagrams, trapping
regions for the critical point will be represented by red boxes. Figure 314 is a typical
graph of gb,w with a trapping region taken from the middle of the red region of the
bifurcation diagram. Note that using the fixed point line, we can visually see that the
critical point never maps outside the box.
Figure 315 is the graph of a typical point in the region alpha of the bifurcation
diagram. For the next few graphs, I will also include the horizontal line through the fixed
point which defines the base of the trapping region.
Since the line connecting the leftmost fixed point intersects gb,"(x), the curve
provides a way that the critical point might escape if the box was extended to the right.
Notice that this effect could not happen in Figure 314 because the horizontal line
connecting the leftmost fixed point never intersects the curve. Since this line is dependent
on the position of the second critical point, it is not directly visible on the bifurcation
REFERENCES
[1] L. S. Block and W. A. Coppel, Dwii.i in one dimension, vol. 1513 of Lecture Notes
in Mathematics, SpringerVerlag, Berlin, 1992.
[2] W. de Melo and S. van Strien, Onedimensional l;;,i/,,ii, vol. 25 of Ergebnisse der
Mathematik und ihrer G, .q, 1.: /..1 (3) [Results in Mathematics and Related Areas
(3)], SpringerV. i1 i: Berlin, 1993.
[3] L. Alsedh, J. Llibre, and M. Misiurewicz, Combinatorial ;i,. ii. and entropy in
dimension one, vol. 5 of Advanced Series in Nonlinear Dl,,,ii. World Scientific
Publishing Co. Inc., River Edge, NJ, second edition, 2000.
[4] P. Boyland, "Semiconjugacies to angledoubling," Proc. Amer. Math. Soc., vol. 134,
no. 5, pp. 12991307 (electronic), 2006.
[5] W. Rudin, Principles of Mathematical A,...u..: McGrawHill, Inc., New York, 1976.
[6] D. Singer, "Stable orbits and bifurcation of maps of the interval," SIAM J. Appl.
Math., vol. 35, no. 2, pp. 260267, 1978.
is continuous at x. Thus, by continuity, there exists 6 > 0 such that for all ly xl < 6,
If'(y) < 1. Now by the Mean Value Theorem, there exists c such that c E [x, y] and
I f()f( I f'(c) < 1. Thus If(y) x = If(y) f(x)l = If'(c,)1 y x We claim that
yX
Mf(y) X  f'(ci) ly Xy (215)
i= 1
for some ci E [x, y] with If'(ci)I < 1 and prove the claim by induction.
Base case: There is nothing to prove when n = 1.
Induction step: Assume that the claim is true for n = k. Then Ifk(y) xl < 6 by
hypothesis, so (fk+l)/(y) < 1 and by the Mean Value Theorem,
Sfk+(y) fk+( = If(Ck+l) < 1 (2 16)
fk(y) fk(x)
for some ICk+1 XI < Ifk(y) xI. Now we have that Ifk+l(y) fk+l(x)l fk+l(y) xl
If'(Ck+i1) Ifk(y) fk(x)l If'(Ck+)l Ifk(y) xl. By the induction hypothesis,
k k+1
Ifk+1() X_ (cCk+1) f'(ci) y XI f'(i) y x (217)
i=1 i=1
which proves the claim.
Now we claim that as limit If'(ci)l I 1. By way of contradiction, assume the
contrary. Then by the continuity of f', there exists z such that Iz x y < ly x < 6 and
If'(z) = 1. But then Iz x < 6, which implies that If'(z)l < 1, a contradiction.
As a result, we have that there is a number R < 1 such that Vc, If'(ci)l < R. Let N
be such that RN < Then for all n > N, we have that:
i ff(y) x = f'(ci)l y x1 < Ry x < 6= (218)
i= 1
So when n  o, If"(y) x = 0 and thus f"f(y)  x. D
2.4.2 Critical Points
A point c E X is called a critical point for f if f'(c) = 0.
0.8
0.6
0.4
0.2
0.2 0.4 D0. 1
0.2
0.4
Figure 317. Graph of gb,w in area 7 in the bifurcation diagram. w
create the plot.
0.8 was used to
Figure 318. Graph of gb,w approximately at point 6 in the bifurcation diagram.
Figure 319. Graph of gb,w in the area c of the bifurcation diagram. w = 0.65 was used to
make this particular plot.
Finally, phi is situated on the line in the bifurcation diagram that marks where the
horizontal line through the right fixed point intersects the first critical point. As discussed
earlier for the horizontal line through the left fixed point, this threshold has particular
significance for the second critical point, and I have graphed it in Figure 320.
1.2
1
0.8
0.6
0.4
0.2
0.2 0.4 0.6 .8 1
0.2
Figure 320. Graph of gb,w approximately at point ( in the bifurcation diagram. A
horizontal line through the right fixed point has been included.
LIST OF FIGURES
Figure page
31 gb,w when w = 0.5 and b = 0.75. ............... ....... 19
32 gb,, when w = 0.2 and b = 0.2. ............... ...... 20
33 a when b = 0.87 and w = 0.3. ............... ...... 20
34 a when b 0.87 and w 0. .................. .. ...... 21
35 si(b,w). ...... ....... ................... .. ..22
36 2(b, w). ..... ......... ................. .. .. 22
37 Key for 35 and 36 ................ ........... .. 23
38 3D im age for 35. .. .. .. ... .. .. .. ... ... ... .. ... .. 23
39 sl(b,w) with b ranging from 1 to and w ranging from 0 to 3.. . . 23
310 si(b,w) with w ranging from 0.5 to 0.5. .............. ...... 25
311 Bifurcation diagram with the colors running through running seven times the
normal mod 1 .................. ................. .. 26
312 Graph of gb,w when b = 1 and the critical point is a fixed point. . ... 26
313 Graph of gb,w at parameter point s in 310. ............... .. .. 27
314 Graph of gb,w when b = 0.6 and w = 0.6 ................ .... 28
315 Graph of gb,w at parameter point a in 310. ................ .... 28
316 Graph of gb,w, approximately at parameter point 3 in 310. ........... .29
317 Graph of gb,w in area 7 in the bifurcation diagram. .............. 30
318 Graph of gb,w approximately at point 6 in the bifurcation diagram . .... 30
319 Graph of gb,w in the area c of the bifurcation diagram. ............ ..31
320 Graph of gb,w approximately at point Q in the bifurcation diagram. . 31
321 Double Bifurcation Diagram. .................. ...... 32
322 ab,w(X) when b = 0.6 and w = 0.6. .................... ...... 32
323 gb,w(x) when b = 0.726 and w = 0 with two fixed point lines. ......... ..33
324 gb,w(x) when b = 1.235 and w = 0.5 with three fixed point lines. . ... 34
325 ab,w(x) when b = 0.8 and w = 0. .................. ........ .. 34
In the situation shown in the diagram (214) (or (28)), the map d is called a factor
of the map f and conversely, the map f is called an extension of the map d. The map J is
a ',,,'... ,' i,, ;i from (or of) f to d.
2.4 Some Dynamical Systems
2.4.1 Periodic Points and Orbits
Definition 3. The orbit or tr i. /.., '; of a point x is all iterates of x under f and is
written as o(x, f). That is, o(x, f) {f"(x) : n E N}.
In this section the space X is either the real line R, the interval [0, 1] or the circle S1,
and f, g : X X are differentiable functions in the class C3.
Definition 4. A point x with f(x) = x is called a fixed point of f. For a map g, the set
Fix(g) denotes the set of all fixed points of g. If for some n > 0, f"(x) = x and n is the
least such integer for which that holds, then x is called a periodic point of least period n or
a periodn point. The orbit o(x, f) of a periodic point x is called a periodic orbit. When x
is a periodn point, its orbit o(x, f) contains ,,. /l;, n elements. Note that for a periodn
point x, x e Fix(fkn) for all k e N.
Definition 5. The fixed point x for f is attracting if there is an open interval I containing
x, and y E I implies that f"(y)  x as n  oo. The 1., . such interval that contains
x is called the basin of x. The point is said to be a onesided attractor if I is as above, but
instead of x being in the interior of I, it is one of the endpoints. A fixed point x for f is
repelling if there is an open interval I containing x, and y E I implies that for some n > 0,
f"f(y) I. A periodn point x is said to be an attractor, onesided attractor, or repeller
if x meets the same fI;./,:'.,n for fn. Attractors and repellers are also sometimes called
stable and unstable, /' 1.. ,/; .
Lemma 4. Assume that x is a fixed point for f. If If'(x) < 1, then x is an attractor and
if f'(x) > 1, then x is a repeller.
Proof: I will show that if If'(x) < 1, then x is an attractor. The proof that if
f'(x) > 1 then x is a repeller is similar. Let E > 0. Since the derivative exists at x, f
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
BIFURCATIONS OF THE DEGREETWO STANDARD FAMILY OF CIRCLE MAPS
By
William C!i iI. 11h!. r Strickland
August 2007
C(!i ': Philip Boyland
Major: Mathematics
In this research I parameterize the fractals generated by lim,,, f(xr)/2' where
g(x) is the degree2 standard family 2x + w + b sin(27rx) of circle maps iterated n times.
Beginning with an exploration of degreen maps and their projections onto the circle, I
introduce some theory on lim.. g for continuous, degree2 maps f and present key
points from dynamical systems. Next, I explore the phenomenon of large flat spots in
the graphs of many of the fractals and present the motivation for taking the values of
'(x)/22 at the critical points of g(x), and using the results as a bifurcation diagram for
the family of fractals. After building up this theory, I present numerical results for the
fractals and use these results to explain the trapping effects in T(x)/2' and their relation
to flat regions in the bifurcation diagram. Finally, I conclude with a characterization of
the bifurcation diagram of lim ,,.T(x)/2' involving first one, and then both critical
points.
1
0.8
0.6
0.4
0.2
Figure 314. Graph of gb,w, when t
0.2 0.4 0.6 0.8 1
 0.6 and w = 0.6
.2 0.4 0.6 0.8 1
Figure 315. Graph of gb,w at parameter point a in 310. w
the plot.
1.2 was used to generate
ACKNOWLEDGMENTS
I wish to thank my advisor, Philip Boyland, for his constant support and patience
during the researching and writing of this thesis. The time which he committed to this
project has been a critical part of its success, and I am extremely grateful for all that he
has taught me.
Lemma 2. The map f : R  R is degree n if and only if there exits a degreezero map 7
with f(x) = x + 7(x).
Proof: Let f be degree n. Then f(x + 1) = f(x) + n. We want to show that
f(x) = nx + 7(x), so define 7(x) = f(x) nx. We just need to check that 7 is degreezero:
7(x + 1) f(x + 1) n(x + 1) = f(x) + n x n f(x) nx = 7(x). Now conversely, let
7(x) be a degreezero map with f(x) = nx + 7(x). Then f(x + 1) = n(x + 1) + 7(x + 1) =
nx + n + 7(x) = f(x) + n. So f(x) is degree n. D
2.2 Covering Space Point of View
The circle is denoted S1 and treated as the real numbers mod 1, i.e. R/Z, or as the
unit circle in the complex plane {z E C : z = 1}. I will often use both of paradigms.
The covering projection is the map p : R  S1 defined alternately as p(x) = x [xI,
i.e. the fractional part of x, or else p(x) = exp(27ix). The map f : R + R is said to be a
lift of a circle map if there is a map / : S1 S1 so that p o f = f o p, i.e. the following
diagarm commutes:
R R
(22)
S1 f S1
In the situation shown in diagram (22), the map f is called the lift of the map f and
conversely, the map f is called the projection of the map f. A point x with p(x) = is
called a lift of T. Note that if x is a lift of i, then so is x + n for all n E Z.
Lemma 3. A function f : R + R is the lift of a circle map if and only if it is degreen for
some n E Z.
Proof: Suppose that f is the lift of a circle map. Note that for x, x' E R, p(x) =
p(x') if and only if x = x' + n for some n E Z. Now since p(x + 1) p(x), f(p(x +
1) = f(p(x)) and by the commuting diagram (22), p(f(x + 1) p(f(x)). Thus
(f(x + 1) f(x)) E Z and so, by continuity, there exists n such that f(x + 1) f(x) = n.
ab,w changes with its other parameter by using animations or 3D plots. In addition, all of
ab,w can be viewed by using an animation of 3D plots.
g [x] (2x+ 0.87 sin(2 x))
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0.2 0.4 0.6 0.8 1
Figure 34. a when b = 0.87 and w = 0.
Unfortunately, all these v,v of viewing the ab,w family are quite cumbersome and
do little to tell us what is actually causing the plots to look the way they do. Given the
importance of the critical points shown in chapter 2, a natural solution to figuring out the
trends in vb,w, is to look at the behavior of the gb,w critical points as we vary b and w.
3.2 The Bifurcation Diagram of the Standard Family
As stated in C'! ipter 2, gbw has exactly two critical points whenever b > 1, and so
there are two bifurcation diagrams, si(b, w) and s2(b, w). Graphed as a density plot, these
bifurcation diagrams are shown in 35 and 36. If x e S1, the color value for x in Figure
35 and Figure 36 can be found by the radius 0 = 27x on the circle in Figure 37.
To better visualize what the colors mean, I have also included the 3D version of the
bifurcation diagram in Figure 35 as Figure 38.
Of course, even though we must take b > , there is no reason why we must limit
ourselves to b < 1 + , which is the upper bound for b in all the previous plots. Figure 39
is a plot of s1(b, w) in which the dimensions of the plot are 3x3.
Recall that since 1s(b, w) is on the circle, 2x + (w + n) + b sin(27x) 2x +w + b sin(27x)
mod 1 which accounts for the vertical repetition in 39. 39 also dispels the illusion that
there is noticeably more blue in si(b, w) than green. Similarly, the nonred colors appear
more even when we expand the range of b in s2(b, w). Using a graphics program to count
BIOGRAPHICAL SKETCH
William (C!i I, p! i r Strickland was born on April 16, 1983 in Houston, Texas. He
grew up in Oxford, Mississippi and graduated from Oxford High School in 2001. He
earned his B.A in French and his B.S. in mathematics with a minor in physics from
the University of Mississippi in 2005. (C'! Ii 1!r then entered graduate school at the
University of Florida in order to continue his studies in mathematics. He completed his
M.S. in 2007.
Figure 328. gb,w(x) in region Z, when b = 0.4 and w =
2
1.75
1.5
1.25
1
0.75
0.5
0.25
2 x+ 0.85 sin(2 Tx)
Figure 329. ab,w(x) in area nu when b
0.85 and w
2 x+ 1.05 sin(2 T x)
0.5
Figure 330. ab,,(x)in area lambda when b
1.05 and w = 0.5.
Gb,,n (X)
2.5
1.5
2"
(31)
Figure 32. gbw when w = 0.2 and b = 0.2.
Since Gb,w,10 is a good approximation of Kb,w under most of the graphical resolutions
used in this paper, the reader may assume that Gb,w,10 was used to produce results for Tb,w
unless stated otherwise. With this note in mind, here is an example of a:
g[x] (2 x+ 0.87 sin (2 x) +0.3)
1.75
1.5
1.25
1
0.75
0.5
0.25
0.2 0.4 0.6 0.8 1
Figure 33. a when b = 0.87 and w = 0.3.
While this plot is obviously quite complicated, if we merely reset w to 0, we get the
plot shown in Figure 34. Now there are obvious flat spots, and the process used to create
the plot is clear. If we are willing to hold either b or w fixed, we can directly observe how
0
0.2.
0.4 0.6 0.8 1 1.2
Figure 310. si(b, w) with w ranging from 0.5 to 0.5. While this image was created using
a graphics program to label the bifurcation diagram and combine it with the
fixed point line in a size ratio preserving way, code can be found in the index
which places the line on the bifurcation diagram, verifying that the placement
is correct.
2, the fixed point is an attractor as long as the derivative of the point is less than 1, so we
could conjecture that the parameter line on which the derivative of the fixed point equals
1 (making the fixed point a saddle point) would form a boundary for the flat spot of the
critical point. Indeed, plotting this "saddle point line" on the bifurcation diagram shows
that it exactly follows the largest boundary for the red flat region (see Figure 311).
In examining 311, it is important to recall that the image was actually produced
using Gb,w,10 instead of ab,w and that there is only a limited amount of precision available
in density plots of this nature. The line appears further away from the colors as b
decreases, but the width of the colors increases in this direction as well. One can easily
imagine that if a true plot of ab,w was available, the colors would reach the saddle point
line for all values of b greater than 1.
To further validate this claim, we will now examine gb,w at the points s, a, and f listed
in 310. Point f is obvious and shown in Figure 312.
The graph in Figure 313 for point s uses w = 1.265.
TABLE OF CONTENTS
page
ACKNOW LEDGMENTS ................................. 3
LIST OF FIGURES .................................... 5
A B ST R A CT . . . . . . . . .. . 7
CHAPTER
1 INTRODUCTION .................................. 8
2 T H EO RY . . . . . . . . . 9
2.1 Degreen Maps of the Real Line ........... ............ 9
2.2 Covering Space Point of View ......................... 10
2.3 The Degree2 Case ............................... 11
2.4 Some Dynamical Systems ........................... 13
2.4.1 Periodic Points and Orbits ......... ............. 13
2.4.2 Critical Points .................. ........... .. 14
2.4.3 Schwarzian Derivative .................. ....... .. 15
2.5 Periodic Orbits of DegreeTwo Maps ................ .. .. 15
2.6 Flat Spots in the Graph of a .......... . . .... 16
2.7 The (b, w) Diagram for the DegreeTwo Standard Family . ... 18
3 NUMERICAL RESULTS .................. ........... .. 19
3.1 Plots of the Standard Family .................. ..... .. 19
3.2 The Bifurcation Diagram of the Standard Family . ...... 21
3.3 C(! i .:terizing the Bifurcation Diagram ............. .. .. 24
3.3.1 Fixed Points and Saddle Points . ......... 24
3.3.2 Trapping Regions and Other Effects ................. .. 27
3.4 C(! i i'terizing the Double Bifurcation Diagram .............. .. 32
REFERENCES .............................. .. ..... 38
BIOGRAPHICAL SKETCH ........... ..... . .... .. 39
CHAPTER 3
NUMERICAL RESULTS
3.1 Plots of the Standard Family
We will now begin to look specifically at the family of maps b,w = limo(gl,/2"') of
the standard family gb,w = 2x + w + bsin(27rx) on the circle. To begin, 31 is a plot of gb,w
when b = 0.75 and w = 0.5 with y = x + 1 included:
3
2.5
2
1.5
1
0.5
0.2D.4. D. 8 1
Figure 31. gb,w when w = 0.5 and b = 0.75.
The position of w and b in gb,w gives us the result that changing w has the effect of
shifting the plot along the yaxis while changing b has the effect of increasing or decreasing
the size of the bumps in the graph. To illustrate this effect, Figure 32 is gb,w when both b
and w have the value of 0.2. Again, I have plotted y = x + 1 with the graph.
Since gb,w is on the circle, the intersections of y = x + n with gb,w represent the fixed
points of b,b,Vn E Z. In both Figure 31 and Figure 32, we can visualize all intersections
with only y = x + 1 since the graphs on R2 do not cross y =x or y x + 2. Comparing
these graphs, we also notice that the number of fixed points in gb,w can change with the
parameters. Now as in Theorem 3.1 of C! .lpter 2, define Gb,w,,(X) as follows in equation
(31):
It is convenient to avoid crowding the superscripts and write Df(x) for f'(x). The
chain rule implies that
Df(x) = Df (f" (x))Df (f2(x)) ... Df(x). (219)
Thus the derivative of the nth iterate is the product of the derivatives along the orbit. A
point y is called precritical if for some n, f"(y) = c, a critical point. By (219) this implies
that DfI(y) 0 for j > n.
2.4.3 Schwarzian Derivative
Definition 6. For a function f as above, the Schwarzian derivative, S(f), is 1. fi., as
f"' 3 ,f" 2
SW 2 f) f (220)
An easy calculation shows that the degreetwo standard family (24) is not injective if
and only if b > 1/7, and when b > 1/7, S(gbw)(x) < 0 for all x.
Definition 7. For an iterated function f and a point x in the domain of f, we f,.': the
omega limit of x as all the points y such that for some subsequence ni, limit f"' (x) = y.
That is,
u(x) = {y : lim f"'(x) = y for some ni  oo}
With this definition, I proceed with a fundamental result by Singer [6]:
Theorem 2 (Singer). If f : X + X ,i/r. S(f)(x) < 0, then for each attracting
periodn point xo there is a critical point c in the basin of xo and hence u(c) is equal to the
orbit of xo.
Thus the degreetwo standard family can have at most two attracting periodic points
because it has exactly two critical points.
2.5 Periodic Orbits of DegreeTwo Maps
Now assume as above that f : R + R is degree two and it projects to the map of the
circle f : S1 + S1. The main dynamic object of interest is the circle map, as the circle is

PAGE 1
1
PAGE 2
2
PAGE 3
Iwishtothankmyadvisor,PhilipBoyland,forhisconstantsupportandpatienceduringtheresearchingandwritingofthisthesis.Thetimewhichhecommittedtothisprojecthasbeenacriticalpartofitssuccess,andIamextremelygratefulforallthathehastaughtme. 3
PAGE 4
page ACKNOWLEDGMENTS ................................. 3 LISTOFFIGURES .................................... 5 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2THEORY ....................................... 9 2.1DegreenMapsoftheRealLine ........................ 9 2.2CoveringSpacePointofView ......................... 10 2.3TheDegree2Case ............................... 11 2.4SomeDynamicalSystems ........................... 13 2.4.1PeriodicPointsandOrbits ....................... 13 2.4.2CriticalPoints .............................. 14 2.4.3SchwarzianDerivative .......................... 15 2.5PeriodicOrbitsofDegreeTwoMaps ..................... 15 2.6FlatSpotsintheGraphof 16 2.7The(b;w)DiagramfortheDegreeTwoStandardFamily .......... 18 3NUMERICALRESULTS .............................. 19 3.1PlotsoftheStandardFamily ......................... 19 3.2TheBifurcationDiagramoftheStandardFamily .............. 21 3.3CharacterizingtheBifurcationDiagram .................... 24 3.3.1FixedPointsandSaddlePoints .................... 24 3.3.2TrappingRegionsandOtherEects .................. 27 3.4CharacterizingtheDoubleBifurcationDiagram ............... 32 REFERENCES ....................................... 38 BIOGRAPHICALSKETCH ................................ 39 4
PAGE 5
Figure page 31gb;wwhenw=0:5andb=0:75. ........................... 19 32gb;wwhenw=0:2andb=0:2. ........................... 20 33whenb=0:87andw=0:3. ............................ 20 34whenb=0:87andw=0. ............................. 21 35s1(b;w). ........................................ 22 36s2(b;w). ........................................ 22 37Keyfor 35 and 36 ................................. 23 383Dimagefor 35 ................................... 23 39s1(b;w)withbrangingfrom1 .......... 23 310s1(b;w)withwrangingfrom0:5to0:5. ...................... 25 311Bifurcationdiagramwiththecolorsrunningthroughrunningseventimesthenormalmod1. .................................... 26 312Graphofgb;wwhenb=1andthecriticalpointisaxedpoint. ......... 26 313Graphofgb;watparameterpointsin 310 ..................... 27 314Graphofgb;wwhenb=0:6andw=0:6 ..................... 28 315Graphofgb;watparameterpointin 310 .................... 28 316Graphofgb;w,approximatelyatparameterpointin 310 ............ 29 317Graphofgb;winareainthebifurcationdiagram. ................ 30 318Graphofgb;wapproximatelyatpointinthebifurcationdiagram. ........ 30 319Graphofgb;wintheareaofthebifurcationdiagram. .............. 31 320Graphofgb;wapproximatelyatpointinthebifurcationdiagram. ....... 31 321DoubleBifurcationDiagram. ............................ 32 322b;w(x)whenb=0:6andw=0:6. .......................... 32 323gb;w(x)whenb=0:726andw=0withtwoxedpointlines. ........... 33 324gb;w(x)whenb=1:235andw=0:5withthreexedpointlines. ......... 34 325b;w(x)whenb=0:8andw=0. ........................... 34 5
PAGE 6
................... 35 327gb;w(x)inregionY,whenb=0:6andw=0:8. ................... 35 328gb;w(x)inregionZ,whenb=0:4andw=0. .................... 36 329b;w(x)inareanuwhenb=0:85andw=0:3. ................... 36 330b;w(x)inarealambdawhenb=1:05andw=0:5. ................. 36 331b;w(x)inareathetawhenb=1:05andw=0:1. ................. 37 332b;w(x)inareaUwhenb=0:97andw=0:83. ................... 37 333b;w(x)inareaVwhenb=0:97andw=0:17. ................... 37 6
PAGE 7
7
PAGE 8
1 ],[ 2 ]and[ 3 ].ThisstrategyisexactlywhatisusedinthispaperinwhichIdescribethebifurcationdiagramforthedegree2standardfamily.Ibeginbyexplainingmoreaboutthisfamilyanddiscusssomeofitsparticulardetailsbeforemovingontothespecicdynamicalsystemsstudiedhere.Afterestablishingtherelationshipbetweenthefamilyonthecirclemapandthefamilyontherealline,Iwillrelyheavilyonnumericalresultstodrawconclusionsabouthowchangesintheparameterseectthelongtermbehaviorfoundbyiteratingthefunction. 8
PAGE 9
Denition1.
PAGE 10
2{2 ),themapfiscalledtheliftofthemap 2{2 ),p(f(x+1)=p(f(x)).Thus(f(x+1)f(x))2Zandso,bycontinuity,thereexistsnsuchthatf(x+1)f(x)=n. 10
PAGE 11
3 ,wemayassumethat 4 ]. 2{3 ),then: 2j(2{6)and 2j(2{7)Inaddition,f=dwhered(x)=2x,i.e.thefollowingdiagramcommutes: 11
PAGE 12
2{6 )holdsbyinduction.Basecase:Ifj=1,thenwehavethat 2=2x+(x) 2=x+ 2{6 )holdsforn=kandconsiderk+1:Fk+1(x)=fk+1(x) 2k+1=f(fk(x)) 2k+1=2fk(x)+(fk(x)) 2k+1 2k+(fk(x)) 2k+1=Fk(x)+(fk(x)) 2k+1(2{10)Bytheinductionhypothesis, 2j+(fk(x)) 2k+1=x+k+1Xj=1(fj1(x)) 2j(2{11)Nowtoshowthatlimn!1Fnconvergesuniformly,considerfrom( 2{6 ).isdegreezero,so(x+1)=(x).Sinceisperiodicandcontinuous,thereexistsMwithj(x)j
PAGE 13
2{14 )(or( 2{8 )),themap 2.4.1PeriodicPointsandOrbits Denition3.
PAGE 14
.ThenforallnN,wehavethat: ="(2{18)Sowhenn!1,jfn(y)xj=0andthusfn(y)!x.2 14
PAGE 15
2{19 )thisimpliesthatDfj(y)=0forj>n. Denition6. 2f00 2{4 )isnotinjectiveifandonlyifb>1=,andwhenb>1=,S(gbw)(x)<0forallx. 6 ]: 15
PAGE 16
2{2 ),fn(x)=x+mforsomem2Z.Inthiscase,wesaythatxisatype(n;m)pointforf.NotethatifIchoseadierentliftofx,Iwouldgetadierentm.Infact,usingLemma 1 foranarbitraryk2Z, 1 ,thenusingdiagram( 2{8 ),
PAGE 17
2n
PAGE 18
5 2{4 )hasexactlytwocriticalpointsintheinterval[0;1]andthusthecirclemap 0=g0bw(x)=2+2bcos(2x)2 2b=cos(2x)arccos(1 2=x=c1(b);2arccos(1 2=x=c2(b)Thenotessofarindicatetheimportanceofstudyingthevalueofatacriticalpoint.Notethatwegetadierentforeach(b;w).Iindicatethisbywritingb;wandforeach(b;w),Idene: 6 saysthatif
PAGE 19
31 isaplotofgb;wwhenb=0:75andw=0:5withy=x+1included: Figure31. Thepositionofwandbingb;wgivesustheresultthatchangingwhastheeectofshiftingtheplotalongtheyaxiswhilechangingbhastheeectofincreasingordecreasingthesizeofthebumpsinthegraph.Toillustratethiseect,Figure 32 isgb;wwhenbothbandwhavethevalueof0.2.Again,Ihaveplottedy=x+1withthegraph.Since 31 andFigure 32 ,wecanvisualizeallintersectionswithonlyy=x+1sincethegraphsonR2donotcrossy=xory=x+2.Comparingthesegraphs,wealsonoticethatthenumberofxedpointsin 3{1 ): 19
PAGE 20
2n(3{1) Figure32. Since Figure33. Whilethisplotisobviouslyquitecomplicated,ifwemerelyresetwto0,wegettheplotshowninFigure 34 .Nowthereareobviousatspots,andtheprocessusedtocreatetheplotisclear.Ifwearewillingtoholdeitherborwxed,wecandirectlyobservehow 20
PAGE 21
Figure34. Unfortunately,allthesewaysofviewingthe 35 and 36 .Ifx2S1,thecolorvalueforxinFigure 35 andFigure 36 canbefoundbytheradius=2xonthecircleinFigure 37 .Tobettervisualizewhatthecolorsmean,Ihavealsoincludedthe3DversionofthebifurcationdiagraminFigure 35 asFigure 38 .Ofcourse,eventhoughwemusttakeb1 39 isaplotof 39 39 alsodispelstheillusionthatthereisnoticeablymoreblueins1(b;w)thangreen.Similarly,thenonredcolorsappearmoreevenwhenweexpandtherangeofbins2(b;w).Usingagraphicsprogramtocount 21
PAGE 22
Figure36. 22
PAGE 23
Keyfor 35 and 36 Figure38. 3Dimagefor 35 Figure39. 23
PAGE 24
37 andndthepositionofthemostprominentgreenandblueintheparameterspaceasanexample.Greenisfoundat2 310 showss1(b;w)again,withsomeofthemoreinterestingfeatureslabeled. 310 representsthewvalue(mod1)forwhichagivenbvaluemakesthecriticalpointaxedpointof.Thislinewasfoundnumericallybysolvingtheequation: 4 inChapter 24
PAGE 25
2,thexedpointisanattractoraslongasthederivativeofthepointislessthan1,sowecouldconjecturethattheparameterlineonwhichthederivativeofthexedpointequals1(makingthexedpointasaddlepoint)wouldformaboundaryfortheatspotofthecriticalpoint.Indeed,plottingthis"saddlepointline"onthebifurcationdiagramshowsthatitexactlyfollowsthelargestboundaryfortheredatregion(seeFigure 311 ).Inexamining 311 ,itisimportanttorecallthattheimagewasactuallyproducedusingGb;w;10insteadofb;wandthatthereisonlyalimitedamountofprecisionavailableindensityplotsofthisnature.Thelineappearsfurtherawayfromthecolorsasbdecreases,butthewidthofthecolorsincreasesinthisdirectionaswell.Onecaneasilyimaginethatifatrueplotofb;wwasavailable,thecolorswouldreachthesaddlepointlineforallvaluesofbgreaterthan1 310 .PointfisobviousandshowninFigure 312 .ThegraphinFigure 313 forpointsusesw=1:265. 25
PAGE 26
Bifurcationdiagramwiththecolorsrunningthroughrunningseventimesthenormalmod1.Onceagain,agraphicsprogramwasusedtocreatethepicture,andIhaveincludedcodewhichveriestheplacementoftheline. Figure312. Graphofgb;wwhenb=1andthecriticalpointisaxedpoint.Thexedpointlinemod1hasbeenincluded. 26
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Graphofgb;watparameterpointsin 310 Finally,thegraphofpointain 310 simplyshowsaxedpointthatisnotasaddlepointbutstillleftofthecriticalpoint.Ihaveomitteditforthesakeofbrevity. 314 isatypicalgraphofgb;wwithatrappingregiontakenfromthemiddleoftheredregionofthebifurcationdiagram.Notethatusingthexedpointline,wecanvisuallyseethatthecriticalpointnevermapsoutsidethebox.Figure 315 isthegraphofatypicalpointintheregionalphaofthebifurcationdiagram.Forthenextfewgraphs,Iwillalsoincludethehorizontallinethroughthexedpointwhichdenesthebaseofthetrappingregion.Sincethelineconnectingtheleftmostxedpointintersectsgb;w(x),thecurveprovidesawaythatthecriticalpointmightescapeiftheboxwasextendedtotheright.NoticethatthiseectcouldnothappeninFigure 314 becausethehorizontallineconnectingtheleftmostxedpointneverintersectsthecurve.Sincethislineisdependentonthepositionofthesecondcriticalpoint,itisnotdirectlyvisibleonthebifurcation 27
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Graphofgb;wwhenb=0:6andw=0:6 Figure315. Graphofgb;watparameterpointin 310 .w=1:2wasusedtogeneratetheplot. 28
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310 ,thethresholdofthiseectdoesbecomevisibleifthebifurcationdiagramofthesecondcriticalpointwasshownontopofthediagramfortherstcriticalpoint.NowinFigure 316 ,weseethatbetaliesrightonthethresholdtherstcriticalpointescaping. Figure316. Graphofgb;w,approximatelyatparameterpointin 310 Abovepointbetaonthediagram,thecriticalpointescapesthetrappingregionontherstiterationofgb;w.Itisstillpossiblethatcriticalpointmayentertheregionunderalateriteration,butitnowhasanescapehatchwherepartsofthecurveriseabovethetopofthetrappingbox.ThegreenlinesinFigure 317 showhowthecriticalpointmapsbelowthetrappingregionintherstiteration. 318 .Forareaepsilon,iterationsofgb;wsometimestakethecriticalpointintothetrappingregionwhileothersallowittoescape.Becausewemustexamineb;wtodeterminetheresult,thegraphofgb;winFigure 319 issomewhatunenlightening. 29
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Graphofgb;winareainthebifurcationdiagram.w=0:8wasusedtocreatetheplot. Figure318. Graphofgb;wapproximatelyatpointinthebifurcationdiagram. 30
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Graphofgb;wintheareaofthebifurcationdiagram.w=0:65wasusedtomakethisparticularplot. Finally,phiissituatedonthelineinthebifurcationdiagramthatmarkswherethehorizontallinethroughtherightxedpointintersectstherstcriticalpoint.Asdiscussedearlierforthehorizontallinethroughtheleftxedpoint,thisthresholdhasparticularsignicanceforthesecondcriticalpoint,andIhavegrapheditinFigure 320 Figure320. Graphofgb;wapproximatelyatpointinthebifurcationdiagram.Ahorizontallinethroughtherightxedpointhasbeenincluded. 31
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321 .Obviously,someoftheseareascontainconceptswehavealreadyexaminedintheprevioussection.AreaF,forexample,hasalreadybeenshownundergb;winFigure 314 .Figure 322 isthe(x)ofthesamepointintheparameterspace(mod1forw). Figure321. DoubleBifurcationDiagram.Thebifurcationdiagramofthesecondcriticalpointontopofthebifurcationdiagramoftherstcriticalpoint. Figure322. 32
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323 ). Figure323. AnotherintersectionoccursatB.Atthispoint,thelargevalueofbingb;w(x)hasexpandedthegraphtojusttouchthreexedpointlines(Figure 324 ).TheareaAshouldalsobefamiliar,asitdisplaysbehaviorliketheareaalphain 310 forbothcriticalpoints.b;wofthisregionisdisplayedinFigure 325 .ReferringbacktoFigure 323 ,wecanalsoimaginethedeningcharacteristicsforareasX,Y,andZ.InX,wehavethesituationwherethetopmostxedpointlineintersectsgb;w,andtheotherxedpointlineistoolowtoplayarole.AreaYisjusttheopposite,andareaZplacesbothcriticalpointsbetweenthexedpointlines.Theplotsofb;w(x)willvarydependingonthechoiceofbandwintheseregions,sincelateriterationsofgb;w(x)haveapronouncedeectonthelimit.Thisisalsothecaseforareasnu,mu,lambda,andtheta.Belowaretheb;wplotsfornu,lambda,andthetarespectively.muissimplya180degreerotationofnu. 33
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Figure325. Finally,bothareasUandVlieintherstiterationatspotofoneofthecriticalpointswhilesittinginalateriterationatspotoftheother.Therstiterationatspotseemstohavetheeectofswallowingtheother,sothattheresultisthegraphsshowninFigure 332 andFigure 333 34
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Figure327. 35
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Figure329. Figure330. 36
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Figure332. Figure333. 37
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[1] L.S.BlockandW.A.Coppel,Dynamicsinonedimension,vol.1513ofLectureNotesinMathematics,SpringerVerlag,Berlin,1992. [2] W.deMeloandS.vanStrien,Onedimensionaldynamics,vol.25ofErgebnissederMathematikundihrerGrenzgebiete(3)[ResultsinMathematicsandRelatedAreas(3)],SpringerVerlag,Berlin,1993. [3] L.Alseda,J.Llibre,andM.Misiurewicz,Combinatorialdynamicsandentropyindimensionone,vol.5ofAdvancedSeriesinNonlinearDynamics,WorldScienticPublishingCo.Inc.,RiverEdge,NJ,secondedition,2000. [4] P.Boyland,\Semiconjugaciestoangledoubling,"Proc.Amer.Math.Soc.,vol.134,no.5,pp.1299{1307(electronic),2006. [5] W.Rudin,PrinciplesofMathematicalAnalysis,McGrawHill,Inc.,NewYork,1976. [6] D.Singer,\Stableorbitsandbifurcationofmapsoftheinterval,"SIAMJ.Appl.Math.,vol.35,no.2,pp.260{267,1978. 38
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WilliamChristopherStricklandwasbornonApril16,1983inHouston,Texas.HegrewupinOxford,MississippiandgraduatedfromOxfordHighSchoolin2001.HeearnedhisB.AinFrenchandhisB.S.inmathematicswithaminorinphysicsfromtheUniversityofMississippiin2005.ChristopherthenenteredgraduateschoolattheUniversityofFloridainordertocontinuehisstudiesinmathematics.HecompletedhisM.S.in2007. 39
