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Bifurcations of the Degree-Two Standard Family of Circle Maps

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Title:
Bifurcations of the Degree-Two Standard Family of Circle Maps
Creator:
Strickland, William C
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (39 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Boyland, Philip L.
Committee Members:
Keesling, James E.
Gopalakrishnan, Jayadeep
Graduation Date:
8/11/2007

Subjects

Subjects / Keywords:
Boxes ( jstor )
Critical points ( jstor )
Dynamical systems ( jstor )
Fractals ( jstor )
Graduate schools ( jstor )
Induction assumption ( jstor )
Mathematics ( jstor )
Periodic orbit ( jstor )
Real lines ( jstor )
Saddle points ( jstor )
Mathematics -- Dissertations, Academic -- UF
attraction, basin, bifurcations, circle, covering, diagram, dynamical, family, fractals, lift, orbits, parameter, periodic, region, schwarzian, space, standard, systems, trapping
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Mathematics thesis, M.S.

Notes

Abstract:
In this research I parameterize the fractals generated by the limit as n goes to infinity of the degree-2 standard family 2x + w + b*sin(2pi*x) of circle maps (g bar) iterated n times and divided by two to the nth power (g bar resized). Beginning with an exploration of degree-n maps and their projections onto the circle, I introduce some theory on the limit as n goes to infinity of continuous, degree-2 maps iterated n times and divided by two to the nth power 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 g bar resized at critical points of g bar 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 the g bar resized and their relation to flat regions in the bifurcation diagram. Finally, I conclude with a characterization of the bifurcation diagram of the limit of g bar resized involving first one, and then both critical points. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2007.
Local:
Adviser: Boyland, Philip L.
Statement of Responsibility:
by William C Strickland.

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BIFURCATIONS OF THE DEGREE-TWO 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 Degree-Two Standard Family

The degree-two standard family (2-4) 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)). (2-22)

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 degree-n 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 Degree-2 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) (2-3)

with 7 periodic. The most important degree-2 map for us is the degree-2 standard family:


gbw(x) =2x + w + bsin(2rx) (2-4)

The following theorem is essential in our discussion of gb,w(X) [4].

Theorem 1 (Boyland). Let f : R -+ R be a continuous, degree-2 map and 1. 7;,'.: F, by:

f"
F,-: (2-5)

Then there exists a continuous, degree-one map a : R -+ R with F, -+ a unifoi ,,l ;

Furthermore, if 7 is as in (2-3), then:

F,-(x) x + 7(f 1(x)) (2 6)
j=1

and

a(x) x + f )) (2-7)
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 (2-8)
R d R
----- RKD






























Figure 3-26. gb,w(X) in region X, when b = 0.6 and w = 0.2.


Figure 3-27. 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 period-n orbit under f and that x E R is a lift of

-. Now since f (() = then using (2-2), 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 (2-21)

and so x + k is of type (n, (2" 1)k + m).

This means that the type of a period-n point x E S1 under f is only defined modulo

2" 1.

If x has type (n, m) for a degree-two f and a is as in Theorem 1, then using diagram

(2-8),


(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 period-n 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 3-5. 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 3-6. s2(b, w).









CHAPTER 1
INTRODUCTION

This work is a study of non-linear dynamical systems. Heuristically, a non-linear

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 degree-n 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 degree-2 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 3-31. 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 3-32. 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 3-33. ab,w (x) in area V when b = 0.97 and w = 0.17.






























Figure 3-7. Key for 3-5 and 3-6.


0.5


0.25


Figure 3-8. 3D image for 3-5.


3 ,


0.5 1 1.5 2 2.5


Figure 3-9. s(b, w) with b ranging from to and w ranging from 0 to 3.
7T 7T





























Figure 3-24. gb,w(x) when b = 1.235 and w = 0.5 with three fixed point lines.

2 x + 0.8 sin(2 x)


Figure 3-25. 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 3-32 and Figure 3-33.









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 (2-6) 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 (2-6) 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 f-l + (f ))
2 i2k+1
j=1


k+1 _- ))

2j
j=1


(2-9)








(2-10)


(2-11)


Now to show that limn,,, Fn converges uniformly, consider 7 from (2-6). 7 is

degree-zero, 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:


(2-12)


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 n-2oo n

fn+1(.) f'(x)
lim = lim 2 f (X 2a-(x) = d(a(x))
n-oo 2n n-oO 2n

All of the maps in (2-8) project to the circle and so we also get

S1 f S1


S1 S1


(2-13)


(2-14)


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 3-11. 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 3-12. 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 3-7 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 3-10 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 3-10 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)) (3-2)

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'! i-l, -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 3-23).


2

1.75

1.5

1.25

1

0.75

0.5

0.25

0.2 0.4 0.6 0.8 1



Figure 3-23. 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 3-24). The area A should

also be familiar, as it di-,'1,v behavior like the area alpha in 3-10 for both critical points.

ab,w of this region is di- .1' '.1 in Figure 3-25.

Referring back to Figure 3-23, 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 3-10, 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 3-16, 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 3-16. Graph of gb,w, approximately at parameter point 3 in 3-10.

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 3-17 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 3-18.

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 3-19 is somewhat unenlightening.









3-26 gb,w(x) in region X, when b = 0.6 and w = 0.2. .................. .. 35

3-27 gb,w(x) in region Y, when b = 0.6 and w = 0.8. .................... .. 35

3-28 gb,w(x) in region Z, when b = 0.4 and w = 0. ................ 36

3-29 ab,w(x) in area nu when b = 0.85 and w = 0.3. .................... .. 36

3-30 ab,((x)in area lambda when b = 1.05 and w = 0.5. ................ 36

3-31 ab,w(x) in area theta when b= 1.05 and w= 0.1. ................ 37

3-32 ab,,w(x) in area U when b = 0.97 and w = 0.83. .................... .. 37

3-33 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 3-21. 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 3-14. Figure 3-22 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 3-21. 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 3-22. ab,w,(x ) when b = 0.6 and w = 0.6.


0.6 0.8 1









CHAPTER 2
THEORY

2.1 Degree-n 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 degree-n standard fi n,,:l; for parameters w, b E R:

g(x) = nx + w + bsin(27rx) (2-1)


3. A periodic function of period 1 (y(x + 1) = 7(x)) is the same as a degree-zero 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 3-13. Graph of gb,w at parameter point s in 3-10.

Finally, the graph of point a in 3-10 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 3-14 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 3-15 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 left-most 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 3-14 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, Springer-Verlag, Berlin, 1992.

[2] W. de Melo and S. van Strien, One-dimensional l;;,i/,,ii, vol. 25 of Ergebnisse der
Mathematik und ihrer G, .q, 1.: /..1 (3) [Results in Mathematics and Related Areas
(3)], Springer-V. 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 angle-doubling," Proc. Amer. Math. Soc., vol. 134,
no. 5, pp. 1299-1307 (electronic), 2006.

[5] W. Rudin, Principles of Mathematical A,...u.-.:- McGraw-Hill, 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. 260-267, 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
y--X


Mf(y) X| | f'(ci)| ly- Xy (2-15)
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 (2-17)
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 < R|y x < 6= (2-18)
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 3-17. Graph of gb,w in area 7 in the bifurcation diagram. w
create the plot.


-0.8 was used to


Figure 3-18. Graph of gb,w approximately at point 6 in the bifurcation diagram.



































Figure 3-19. 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 3-20.



1.2

1

0.8

0.6

0.4

0.2


0.2 0.4 0.6 .8 1

-0.2



Figure 3-20. 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

3-1 gb,w when w = 0.5 and b = 0.75. ............... ....... 19

3-2 gb,, when w = 0.2 and b = 0.2. ............... ...... 20

3-3 a when b = 0.87 and w = 0.3. ............... ...... 20

3-4 a when b 0.87 and w 0. .................. .. ...... 21

3-5 si(b,w). ...... ....... ................... .. ..22

3-6 2(b, w). ..... ......... ................. .. .. 22

3-7 Key for 3-5 and 3-6 ................ ........... .. 23

3-8 3D im age for 3-5. .. .. .. ... .. .. .. ... ... ... .. ... .. 23

3-9 sl(b,w) with b ranging from 1 to and w ranging from 0 to 3.. . . 23

3-10 si(b,w) with w ranging from -0.5 to 0.5. .............. ...... 25

3-11 Bifurcation diagram with the colors running through running seven times the
normal mod 1 .................. ................. .. 26

3-12 Graph of gb,w when b = 1 and the critical point is a fixed point. . ... 26

3-13 Graph of gb,w at parameter point s in 3-10. ............... .. .. 27

3-14 Graph of gb,w when b = 0.6 and w = -0.6 ................ .... 28

3-15 Graph of gb,w at parameter point a in 3-10. ................ .... 28

3-16 Graph of gb,w, approximately at parameter point 3 in 3-10. ........... .29

3-17 Graph of gb,w in area 7 in the bifurcation diagram. .............. 30

3-18 Graph of gb,w approximately at point 6 in the bifurcation diagram . .... 30

3-19 Graph of gb,w in the area c of the bifurcation diagram. ............ ..31

3-20 Graph of gb,w approximately at point Q in the bifurcation diagram. . 31

3-21 Double Bifurcation Diagram. .................. ...... 32

3-22 ab,w(X) when b = 0.6 and w = 0.6. .................... ...... 32

3-23 gb,w(x) when b = 0.726 and w = 0 with two fixed point lines. ......... ..33

3-24 gb,w(x) when b = 1.235 and w = 0.5 with three fixed point lines. . ... 34

3-25 ab,w(x) when b = 0.8 and w = 0. .................. ........ .. 34









In the situation shown in the diagram (2-14) (or (2-8)), 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 period-n point. The orbit o(x, f) of a periodic point x is called a periodic orbit. When x

is a period-n point, its orbit o(x, f) contains ,,. /l;, n elements. Note that for a period-n

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 one-sided 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 period-n point x is said to be an attractor, one-sided 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 DEGREE-TWO STANDARD FAMILY OF CIRCLE MAPS

By

William C!i i-I. 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 degree-2 standard family 2x + w + b sin(27rx) of circle maps iterated n times.

Beginning with an exploration of degree-n maps and their projections onto the circle, I

introduce some theory on lim.. g for continuous, degree-2 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 3-14. 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 3-15. Graph of gb,w at parameter point a in 3-10. 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 degree-zero 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 degree-zero:
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 degree-zero 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---


(2-2)


S1 -f S1
In the situation shown in diagram (2-2), 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 degree-n 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 (2-2), 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 3-4. 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 3-5 and 3-6. If x e S1, the color value for x in Figure

3-5 and Figure 3-6 can be found by the radius 0 = 27x on the circle in Figure 3-7.

To better visualize what the colors mean, I have also included the 3D version of the

bifurcation diagram in Figure 3-5 as Figure 3-8.

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 3-9

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 3-9. 3-9 also dispels the illusion that

there is noticeably more blue in si(b, w) than green. Similarly, the non-red 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 3-28. 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 3-29. ab,w(x) in area nu when b


0.85 and w


2 x+ 1.05 sin(2 T x)


0.5


Figure 3-30. ab,,(x)in area lambda when b


1.05 and w = 0.5.














Gb,,n (X)


2.5


1.5


2"


(3-1)


Figure 3-2. 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 3-3. 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 3-4. 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 3-10. 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 3-11).

In examining 3-11, 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 3-10. Point f is obvious and shown in Figure 3-12.

The graph in Figure 3-13 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 Degree-n Maps of the Real Line ........... ............ 9
2.2 Covering Space Point of View ......................... 10
2.3 The Degree-2 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 Degree-Two Maps ................ .. .. 15
2.6 Flat Spots in the Graph of a .......... . . .... 16
2.7 The (b, w) Diagram for the Degree-Two 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, 3-1 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 3-1. 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 y-axis while changing b has the effect of increasing or decreasing

the size of the bumps in the graph. To illustrate this effect, Figure 3-2 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 3-1 and Figure 3-2, 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

(3-1):









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 (f-2(x)) ... Df(x). (2-19)

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 (2-19) 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 (2-20)

An easy calculation shows that the degree-two standard family (2-4) 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

period-n 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 degree-two standard family can have at most two attracting periodic points

because it has exactly two critical points.

2.5 Periodic Orbits of Degree-Two 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




Full Text

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Iwishtothankmyadvisor,PhilipBoyland,forhisconstantsupportandpatienceduringtheresearchingandwritingofthisthesis.Thetimewhichhecommittedtothisprojecthasbeenacriticalpartofitssuccess,andIamextremelygratefulforallthathehastaughtme. 3

PAGE 4

page ACKNOWLEDGMENTS ................................. 3 LISTOFFIGURES .................................... 5 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2THEORY ....................................... 9 2.1Degree-nMapsoftheRealLine ........................ 9 2.2CoveringSpacePointofView ......................... 10 2.3TheDegree-2Case ............................... 11 2.4SomeDynamicalSystems ........................... 13 2.4.1PeriodicPointsandOrbits ....................... 13 2.4.2CriticalPoints .............................. 14 2.4.3SchwarzianDerivative .......................... 15 2.5PeriodicOrbitsofDegree-TwoMaps ..................... 15 2.6FlatSpotsintheGraphof 16 2.7The(b;w)DiagramfortheDegree-TwoStandardFamily .......... 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 3-1gb;wwhenw=0:5andb=0:75. ........................... 19 3-2gb;wwhenw=0:2andb=0:2. ........................... 20 3-3whenb=0:87andw=0:3. ............................ 20 3-4whenb=0:87andw=0. ............................. 21 3-5s1(b;w). ........................................ 22 3-6s2(b;w). ........................................ 22 3-7Keyfor 3-5 and 3-6 ................................. 23 3-83Dimagefor 3-5 ................................... 23 3-9s1(b;w)withbrangingfrom1 .......... 23 3-10s1(b;w)withwrangingfrom0:5to0:5. ...................... 25 3-11Bifurcationdiagramwiththecolorsrunningthroughrunningseventimesthenormalmod1. .................................... 26 3-12Graphofgb;wwhenb=1andthecriticalpointisaxedpoint. ......... 26 3-13Graphofgb;watparameterpointsin 3-10 ..................... 27 3-14Graphofgb;wwhenb=0:6andw=0:6 ..................... 28 3-15Graphofgb;watparameterpointin 3-10 .................... 28 3-16Graphofgb;w,approximatelyatparameterpointin 3-10 ............ 29 3-17Graphofgb;winareainthebifurcationdiagram. ................ 30 3-18Graphofgb;wapproximatelyatpointinthebifurcationdiagram. ........ 30 3-19Graphofgb;wintheareaofthebifurcationdiagram. .............. 31 3-20Graphofgb;wapproximatelyatpointinthebifurcationdiagram. ....... 31 3-21DoubleBifurcationDiagram. ............................ 32 3-22b;w(x)whenb=0:6andw=0:6. .......................... 32 3-23gb;w(x)whenb=0:726andw=0withtwoxedpointlines. ........... 33 3-24gb;w(x)whenb=1:235andw=0:5withthreexedpointlines. ......... 34 3-25b;w(x)whenb=0:8andw=0. ........................... 34 5

PAGE 6

................... 35 3-27gb;w(x)inregionY,whenb=0:6andw=0:8. ................... 35 3-28gb;w(x)inregionZ,whenb=0:4andw=0. .................... 36 3-29b;w(x)inareanuwhenb=0:85andw=0:3. ................... 36 3-30b;w(x)inarealambdawhenb=1:05andw=0:5. ................. 36 3-31b;w(x)inareathetawhenb=1:05andw=0:1. ................. 37 3-32b;w(x)inareaUwhenb=0:97andw=0:83. ................... 37 3-33b;w(x)inareaVwhenb=0:97andw=0:17. ................... 37 6

PAGE 7

7

PAGE 8

1 ],[ 2 ]and[ 3 ].ThisstrategyisexactlywhatisusedinthispaperinwhichIdescribethebifurcationdiagramforthedegree-2standardfamily.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

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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 ).isdegree-zero,so(x+1)=(x).Sinceisperiodicandcontinuous,thereexistsMwithj(x)j
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2{14 )(or( 2{8 )),themap 2.4.1PeriodicPointsandOrbits Denition3.

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.ThenforallnN,wehavethat: ="(2{18)Sowhenn!1,jfn(y)xj=0andthusfn(y)!x.2 14

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2{19 )thisimpliesthatDfj(y)=0forj>n. Denition6. 2f00 2{4 )isnotinjectiveifandonlyifb>1=,andwhenb>1=,S(gbw)(x)<0forallx. 6 ]: 15

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2{2 ),fn(x)=x+mforsomem2Z.Inthiscase,wesaythatxisatype(n;m)pointforf.NotethatifIchoseadierentliftofx,Iwouldgetadierentm.Infact,usingLemma 1 foranarbitraryk2Z, 1 ,thenusingdiagram( 2{8 ),

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2n
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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

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3-1 isaplotofgb;wwhenb=0:75andw=0:5withy=x+1included: Figure3-1. Thepositionofwandbingb;wgivesustheresultthatchangingwhastheeectofshiftingtheplotalongthey-axiswhilechangingbhastheeectofincreasingordecreasingthesizeofthebumpsinthegraph.Toillustratethiseect,Figure 3-2 isgb;wwhenbothbandwhavethevalueof0.2.Again,Ihaveplottedy=x+1withthegraph.Since 3-1 andFigure 3-2 ,wecanvisualizeallintersectionswithonlyy=x+1sincethegraphsonR2donotcrossy=xory=x+2.Comparingthesegraphs,wealsonoticethatthenumberofxedpointsin 3{1 ): 19

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2n(3{1) Figure3-2. Since Figure3-3. Whilethisplotisobviouslyquitecomplicated,ifwemerelyresetwto0,wegettheplotshowninFigure 3-4 .Nowthereareobviousatspots,andtheprocessusedtocreatetheplotisclear.Ifwearewillingtoholdeitherborwxed,wecandirectlyobservehow 20

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Figure3-4. Unfortunately,allthesewaysofviewingthe 3-5 and 3-6 .Ifx2S1,thecolorvalueforxinFigure 3-5 andFigure 3-6 canbefoundbytheradius=2xonthecircleinFigure 3-7 .Tobettervisualizewhatthecolorsmean,Ihavealsoincludedthe3DversionofthebifurcationdiagraminFigure 3-5 asFigure 3-8 .Ofcourse,eventhoughwemusttakeb1 3-9 isaplotof 3-9 3-9 alsodispelstheillusionthatthereisnoticeablymoreblueins1(b;w)thangreen.Similarly,thenon-redcolorsappearmoreevenwhenweexpandtherangeofbins2(b;w).Usingagraphicsprogramtocount 21

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Figure3-6. 22

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Keyfor 3-5 and 3-6 Figure3-8. 3Dimagefor 3-5 Figure3-9. 23

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3-7 andndthepositionofthemostprominentgreenandblueintheparameterspaceasanexample.Greenisfoundat2 3-10 showss1(b;w)again,withsomeofthemoreinterestingfeatureslabeled. 3-10 representsthewvalue(mod1)forwhichagivenbvaluemakesthecriticalpointaxedpointof.Thislinewasfoundnumericallybysolvingtheequation: 4 inChapter 24

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2,thexedpointisanattractoraslongasthederivativeofthepointislessthan1,sowecouldconjecturethattheparameterlineonwhichthederivativeofthexedpointequals1(makingthexedpointasaddlepoint)wouldformaboundaryfortheatspotofthecriticalpoint.Indeed,plottingthis"saddlepointline"onthebifurcationdiagramshowsthatitexactlyfollowsthelargestboundaryfortheredatregion(seeFigure 3-11 ).Inexamining 3-11 ,itisimportanttorecallthattheimagewasactuallyproducedusingGb;w;10insteadofb;wandthatthereisonlyalimitedamountofprecisionavailableindensityplotsofthisnature.Thelineappearsfurtherawayfromthecolorsasbdecreases,butthewidthofthecolorsincreasesinthisdirectionaswell.Onecaneasilyimaginethatifatrueplotofb;wwasavailable,thecolorswouldreachthesaddlepointlineforallvaluesofbgreaterthan1 3-10 .PointfisobviousandshowninFigure 3-12 .ThegraphinFigure 3-13 forpointsusesw=1:265. 25

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Bifurcationdiagramwiththecolorsrunningthroughrunningseventimesthenormalmod1.Onceagain,agraphicsprogramwasusedtocreatethepicture,andIhaveincludedcodewhichveriestheplacementoftheline. Figure3-12. Graphofgb;wwhenb=1andthecriticalpointisaxedpoint.Thexedpointlinemod1hasbeenincluded. 26

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Graphofgb;watparameterpointsin 3-10 Finally,thegraphofpointain 3-10 simplyshowsaxedpointthatisnotasaddlepointbutstillleftofthecriticalpoint.Ihaveomitteditforthesakeofbrevity. 3-14 isatypicalgraphofgb;wwithatrappingregiontakenfromthemiddleoftheredregionofthebifurcationdiagram.Notethatusingthexedpointline,wecanvisuallyseethatthecriticalpointnevermapsoutsidethebox.Figure 3-15 isthegraphofatypicalpointintheregionalphaofthebifurcationdiagram.Forthenextfewgraphs,Iwillalsoincludethehorizontallinethroughthexedpointwhichdenesthebaseofthetrappingregion.Sincethelineconnectingtheleft-mostxedpointintersectsgb;w(x),thecurveprovidesawaythatthecriticalpointmightescapeiftheboxwasextendedtotheright.NoticethatthiseectcouldnothappeninFigure 3-14 becausethehorizontallineconnectingtheleftmostxedpointneverintersectsthecurve.Sincethislineisdependentonthepositionofthesecondcriticalpoint,itisnotdirectlyvisibleonthebifurcation 27

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Graphofgb;wwhenb=0:6andw=0:6 Figure3-15. Graphofgb;watparameterpointin 3-10 .w=1:2wasusedtogeneratetheplot. 28

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3-10 ,thethresholdofthiseectdoesbecomevisibleifthebifurcationdiagramofthesecondcriticalpointwasshownontopofthediagramfortherstcriticalpoint.NowinFigure 3-16 ,weseethatbetaliesrightonthethresholdtherstcriticalpointescaping. Figure3-16. Graphofgb;w,approximatelyatparameterpointin 3-10 Abovepointbetaonthediagram,thecriticalpointescapesthetrappingregionontherstiterationofgb;w.Itisstillpossiblethatcriticalpointmayentertheregionunderalateriteration,butitnowhasanescapehatchwherepartsofthecurveriseabovethetopofthetrappingbox.ThegreenlinesinFigure 3-17 showhowthecriticalpointmapsbelowthetrappingregionintherstiteration. 3-18 .Forareaepsilon,iterationsofgb;wsometimestakethecriticalpointintothetrappingregionwhileothersallowittoescape.Becausewemustexamineb;wtodeterminetheresult,thegraphofgb;winFigure 3-19 issomewhatunenlightening. 29

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Graphofgb;winareainthebifurcationdiagram.w=0:8wasusedtocreatetheplot. Figure3-18. Graphofgb;wapproximatelyatpointinthebifurcationdiagram. 30

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Graphofgb;wintheareaofthebifurcationdiagram.w=0:65wasusedtomakethisparticularplot. Finally,phiissituatedonthelineinthebifurcationdiagramthatmarkswherethehorizontallinethroughtherightxedpointintersectstherstcriticalpoint.Asdiscussedearlierforthehorizontallinethroughtheleftxedpoint,thisthresholdhasparticularsignicanceforthesecondcriticalpoint,andIhavegrapheditinFigure 3-20 Figure3-20. Graphofgb;wapproximatelyatpointinthebifurcationdiagram.Ahorizontallinethroughtherightxedpointhasbeenincluded. 31

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3-21 .Obviously,someoftheseareascontainconceptswehavealreadyexaminedintheprevioussection.AreaF,forexample,hasalreadybeenshownundergb;winFigure 3-14 .Figure 3-22 isthe(x)ofthesamepointintheparameterspace(mod1forw). Figure3-21. DoubleBifurcationDiagram.Thebifurcationdiagramofthesecondcriticalpointontopofthebifurcationdiagramoftherstcriticalpoint. Figure3-22. 32

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3-23 ). Figure3-23. AnotherintersectionoccursatB.Atthispoint,thelargevalueofbingb;w(x)hasexpandedthegraphtojusttouchthreexedpointlines(Figure 3-24 ).TheareaAshouldalsobefamiliar,asitdisplaysbehaviorliketheareaalphain 3-10 forbothcriticalpoints.b;wofthisregionisdisplayedinFigure 3-25 .ReferringbacktoFigure 3-23 ,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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Figure3-25. Finally,bothareasUandVlieintherstiterationatspotofoneofthecriticalpointswhilesittinginalateriterationatspotoftheother.Therstiterationatspotseemstohavetheeectofswallowingtheother,sothattheresultisthegraphsshowninFigure 3-32 andFigure 3-33 34

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Figure3-27. 35

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Figure3-29. Figure3-30. 36

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Figure3-32. Figure3-33. 37

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[1] L.S.BlockandW.A.Coppel,Dynamicsinonedimension,vol.1513ofLectureNotesinMathematics,Springer-Verlag,Berlin,1992. [2] W.deMeloandS.vanStrien,One-dimensionaldynamics,vol.25ofErgebnissederMathematikundihrerGrenzgebiete(3)[ResultsinMathematicsandRelatedAreas(3)],Springer-Verlag,Berlin,1993. [3] L.Alseda,J.Llibre,andM.Misiurewicz,Combinatorialdynamicsandentropyindimensionone,vol.5ofAdvancedSeriesinNonlinearDynamics,WorldScienticPublishingCo.Inc.,RiverEdge,NJ,secondedition,2000. [4] P.Boyland,\Semiconjugaciestoangle-doubling,"Proc.Amer.Math.Soc.,vol.134,no.5,pp.1299{1307(electronic),2006. [5] W.Rudin,PrinciplesofMathematicalAnalysis,McGraw-Hill,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