BIFURCATIONS OF THE DEGREE-TWO STANDARD FAMILY OF CIRCLE MAPS

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

c 2007 William Christopher Strickland 2

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. 3

TABLE OF CONTENTS page ACKNOWLEDGMENTS................................. 3 LIST OF FIGURES.................................... 5 ABSTRACT........................................ 7 CHAPTER INTRODUCTION.................................. 8 2 THEORY....................................... 9 2. Degree-n Maps of the Real Line........................ 9 2.2 Covering Space Point of View......................... 0 2.3 The Degree-2 Case............................... 2.4 Some Dynamical Systems........................... 3 2.4. Periodic Points and Orbits....................... 3 2.4.2 Critical Points.............................. 4 2.4.3 Schwarzian Derivative.......................... 5 2.5 Periodic Orbits of Degree-Two Maps..................... 5 2.6 Flat Spots in the Graph of α.......................... 6 2.7 The (b, w) Diagram for the Degree-Two Standard Family.......... 8 3 NUMERICAL RESULTS.............................. 9 3. Plots of the Standard Family......................... 9 3.2 The Bifurcation Diagram of the Standard Family.............. 2 3.3 Characterizing the Bifurcation Diagram.................... 24 3.3. Fixed Points and Saddle Points.................... 24 3.3.2 Trapping Regions and Other Effects.................. 27 3.4 Characterizing the Double Bifurcation Diagram............... 32 REFERENCES....................................... 38 BIOGRAPHICAL SKETCH................................ 39 4

Figure LIST OF FIGURES page 3- g b,w when w = and b = 0.75............................ 9 3-2 g b,w when w = 0.2 and b = 0.2............................ 20 3-3 α when b = 0.87 and w = 0.3............................. 20 3-4 α when b = 0.87 and w = 0.............................. 2 3-5 s (b, w)......................................... 22 3-6 s 2 (b, w)......................................... 22 3-7 Key for 3-5 and 3-6.................................. 23 3-8 3D image for 3-5.................................... 23 3-9 s (b, w) with b ranging from π to π and w ranging from 0 to 3........... 23 3-0 s (b, w) with w ranging from to....................... 25 3- Bifurcation diagram with the colors running through running seven times the normal mod..................................... 26 3-2 Graph of g b,w when b = and the critical point is a fixed point.......... 26 3-3 Graph of g b,w at parameter point s in 3-0...................... 27 3-4 Graph of g b,w when b = 0.6 and w = 0.6..................... 28 3-5 Graph of g b,w at parameter point α in 3-0..................... 28 3-6 Graph of g b,w, approximately at parameter point β in 3-0............. 29 3-7 Graph of g b,w in area γ in the bifurcation diagram................. 30 3-8 Graph of g b,w approximately at point δ in the bifurcation diagram......... 30 3-9 Graph of g b,w in the area ɛ of the bifurcation diagram............... 3 3-20 Graph of g b,w approximately at point φ in the bifurcation diagram........ 3 3-2 Double Bifurcation Diagram............................. 32 3-22 α b,w (x) when b = 0.6 and w = 0.6........................... 32 3-23 g b,w (x) when b = 0.726 and w = 0 with two fixed point lines............ 33 3-24 g b,w (x) when b =.235 and w = with three fixed point lines.......... 34 3-25 α b,w (x) when b = 0.8 and w = 0............................ 34 5

3-26 g b,w (x) in region X, when b = 0.6 and w = 0.2.................... 35 3-27 g b,w (x) in region Y, when b = 0.6 and w = 0.8.................... 35 3-28 g b,w (x) in region Z, when b = 0.4 and w = 0..................... 36 3-29 α b,w (x) in area nu when b = 0.85 and w = 0.3.................... 36 3-30 α b,w (x)in area lambda when b =.05 and w =.................. 36 3-3 α b,w (x) in area theta when b =.05 and w = 0................... 37 3-32 α b,w (x) in area U when b = 0.97 and w = 0.83.................... 37 3-33 α b,w (x) in area V when b = 0.97 and w = 0.7.................... 37 6

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 Chair: Philip Boyland Major: Mathematics By William Christopher Strickland August 2007 In this research I parameterize the fractals generated by lim n g n (x)/2 n where g(x) is the degree-2 standard family 2x + w + b sin(2πx) 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 n f n 2 n 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 g n (x)/2 n 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 g n (x)/2 n and their relation to flat regions in the bifurcation diagram. Finally, I conclude with a characterization of the bifurcation diagram of lim n g n (x)/2 n involving first one, and then both critical points. 7

CHAPTER 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 [], [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. 8

CHAPTER 2 THEORY 2. Degree-n Maps of the Real Line Definition. Let f : X X be a function. We define the k th iterate of f, or f k, as f f f... f(x) k times. Definition 2. If n Z, a function f : R R is said to be degree n when f(x + ) = f(x) + n for all x R. Lemma. If f : R R is degree n and m Z, then. f(x + m) = f(x) + nm. 2. f k is degree n k, i.e. f k (x + ) = f(x) + n k Proof: We can assume m 0. Suppose that m N, and proceed by induction on m. The base case is m =, and by definition we have our result. Now suppose the lemma holds for m m. Then f(x + m + ) = f((x + ) + m) = f(x + ) + nm = f(x) + nm + n = f(x) + n(m + ). We can quickly check that the lemma holds for m Z as well. Suppose m 0 and let k = m. Then f(x) = f((x k) + k) = f(x k) + kn, and so f(x) kn = f(x k). Thus f(x) + mn = f(x + m). Proof of 2: Proceed by induction. If k =, f(x + ) = f(x) + n = f(x) + n which proves the base case. Now assume the lemma is true for k = m. Then f m+ (x + ) = f(f m (x + )) = f(f(x) + n m ) by the induction hypothesis. Now f(f(x) + n m ) = f(x) + n n m = f(x) + n m+. Example. The following are some examples of degree n functions:. f(x) = nx is degree n. 2. The degree-n standard family for parameters w, b R: g(x) = nx + w + bsin(2πx) (2 ) 3. A periodic function of period (γ(x + ) = γ(x)) is the same as a degree-zero map. 4. If γ is degree zero, then f(x) = nx + γ(x) is degree n. 9

Lemma 2. The map f : R R is degree n if and only if there exits a degree-zero map γ with f(x) = nx + γ(x). Proof: Let f be degree n. Then f(x + ) = f(x) + n. We want to show that f(x) = nx + γ(x), so define γ(x) = f(x) nx. We just need to check that γ is degree-zero: γ(x + ) = f(x + ) n(x + ) = f(x) + n nx n = f(x) nx = γ(x). Now conversely, let γ(x) be a degree-zero map with f(x) = nx + γ(x). Then f(x + ) = n(x + ) + γ(x + ) = nx + n + γ(x) = f(x) + n. So f(x) is degree n. 2.2 Covering Space Point of View The circle is denoted S and treated as the real numbers mod, i.e. R/Z, or as the unit circle in the complex plane {z C : z = }. I will often use both of paradigms. The covering projection is the map p : R S defined alternately as p(x) = x x, i.e. the fractional part of x, or else p(x) = exp(2πix). The map f : R R is said to be a lift of a circle map if there is a map f : S S so that p f = f p, i.e. the following diagarm commutes: R p f R p (2 2) S f S 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) = x, is called a lift of x. Note that if x is a lift of x, then so is x + n for all n 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 Z. Proof: Suppose that f is the lift of a circle map. Note that for x, x R, p(x) = p(x ) if and only if x = x + n for some n Z. Now since p(x + ) = p(x), f(p(x + ) = f(p(x)) and by the commuting diagram (2 2), p(f(x + ) = p(f(x)). Thus (f(x + ) f(x)) Z and so, by continuity, there exists n such that f(x + ) f(x) = n. 0

Now conversely, let f be a degree-n function: f(x + ) = f(x) + n. For each y S, pick some r 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. 2.3 The Degree-2 Case From now on, we will assume that f : R R is degree 2 and that f : S S is its corresponding circle map. From Lemma 3, we may assume that f(x) = 2x + γ(x) (2 3) with γ periodic. The most important degree-2 map for us is the degree-2 standard family: g bw (x) = 2x + w + b sin(2πx) (2 4) The following theorem is essential in our discussion of g b,w (x) [4]. Theorem (Boyland). Let f : R R be a continuous, degree-2 map and define F n by: F n = f n 2 n (2 5) Then there exists a continuous, degree-one map α : R R with F n α uniformly. Furthermore, if γ is as in (2 3), then: and F n (x) = x + α(x) = x + n j= j= γ(f j (x)) 2 j (2 6) γ(f j (x)) 2 j (2 7) In addition, α f = d α where d(x) = 2x, i.e. the following diagram commutes: f R R α R α d R (2 8)

Proof: First, I will show that (2 6) holds by induction. Base case: If j =, then we have that F (x) = f(x) 2 = 2x + γ(x) 2 = x + γ 2. (2 9) Induction step: Suppose that (2 6) holds for n = k and consider k + : By the induction hypothesis, F k+ (x) = f k+ (x) = f(f k (x)) = 2 f k (x) + γ(f k (x)) 2 k+ 2 k+ 2 k+ F k+ (x) = x + = f k (x) 2 k + γ(f k (x)) 2 k+ = F k (x) + γ(f k (x)) 2 k+ (2 0) k j= γ(f j (x)) + γ(f k (x)) k+ = x + 2 j 2 k+ j= γ(f j (x)) 2 j (2 ) Now to show that lim n F n converges uniformly, consider γ from (2 6). γ is degree-zero, so γ(x + ) = γ(x). Since γ is periodic and continuous, there exists M with γ(x) < M for all x R. As a result, for each j N: γ(f j (x)) 2 j M 2 j (2 2) M 2 j converges, so by the Weierstrass M test [5], F n converges uniformly. All that remains is to check that α f = d α when d(x) = 2x: α(f(x)) = lim n F n (f(x)) = lim n f n (f(x)) 2 n = f n+ (x) lim = lim 2 f n (x) = 2α(x) = d(α(x)) (2 3) n 2 n n 2 n All of the maps in (2 8) project to the circle and so we also get α S S f S α d S (2 4) 2

In the situation shown in the diagram (2 4) (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 α is a semiconjugacy from (or of) f to d. 2.4 Some Dynamical Systems 2.4. Periodic Points and Orbits Definition 3. The orbit or trajectory of a point x is all iterates of x under f and is written as o(x, f). That is, o(x, f) = {f n (x) : n N}. In this section the space X is either the real line R, the interval [0, ] or the circle S, and f, g : X X are differentiable functions in the class C 3. 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 n (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 exactly n elements. Note that for a period-n point x, x Fix(f kn ) for all k N. Definition 5. The fixed point x for f is attracting if there is an open interval I containing x, and y I implies that f n (y) x as n. The largest 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 I implies that for some n > 0, f n (y) I. A period-n point x is said to be an attractor, one-sided attractor, or repeller if x meets the same definition for f n. Attractors and repellers are also sometimes called stable and unstable, respectively. Lemma 4. Assume that x is a fixed point for f. If f (x) <, then x is an attractor and if f (x) >, then x is a repeller. Proof: I will show that if f (x) <, then x is an attractor. The proof that if f (x) > then x is a repeller is similar. Let ε > 0. Since the derivative exists at x, f 3

is continuous at x. Thus, by continuity, there exists δ > 0 such that for all y x < δ, f (y) <. Now by the Mean Value Theorem, there exists c such that c [x, y] and f(y) f(x) = f (c y x ) <. Thus f(y) x = f(y) f(x) = f (c ) y x. We claim that f n (y) x = n f (c i ) y x (2 5) i= for some c i [x, y] with f (c i ) < and prove the claim by induction. Base case: There is nothing to prove when n =. Induction step: Assume that the claim is true for n = k. Then f k (y) x < δ by hypothesis, so (f k+ ) (y) < and by the Mean Value Theorem, f k+ (y) f k+ (x) = f (c f k (y) f k k+ ) < (2 6) (x) for some c k+ x < f k (y) x. Now we have that f k+ (y) f k+ (x) = f k+ (y) x = f (c k+ ) f k (y) f k (x) = f (c k+ ) f k (y) x. By the induction hypothesis, f k+ (y) x = f (c k+ ) which proves the claim. k k+ f (c i ) y x = f (c i ) y x (2 7) i= Now we claim that as lim i f (c i ) =. By way of contradiction, assume the contrary. Then by the continuity of f, there exists z such that z x y x < δ and f (z) =. But then z x < δ, which implies that f (z) <, a contradiction. As a result, we have that there is a number R < such that c i, f (c i ) R. Let N be such that R N < ε. Then for all n N, we have that: δ i= f n (y) x = n f (c i ) y x R n y x < ε δ δ = ε (2 8) i= So when n, f n (y) x = 0 and thus f n (y) x. 2.4.2 Critical Points A point c X is called a critical point for f if f (c) = 0. 4

It is convenient to avoid crowding the superscripts and write Df(x) for f (x). The chain rule implies that Df n (x) = Df(f n (x))df(f n 2 (x))... Df(x). (2 9) Thus the derivative of the n th iterate is the product of the derivatives along the orbit. A point y is called precritical if for some n, f n (y) = c, a critical point. By (2 9) this implies that Df j (y) = 0 for j > n. 2.4.3 Schwarzian Derivative Definition 6. For a function f as above, the Schwarzian derivative, S(f), is defined as S(f) = f 3 ( ) f 2 (2 20) f 2 f An easy calculation shows that the degree-two standard family (2 4) is not injective if and only if b > /π, and when b > /π, S(g bw )(x) < 0 for all x. Definition 7. For an iterated function f and a point x in the domain of f, we define the omega limit of x as all the points y such that for some subsequence n i, lim i f n i (x) = y. That is, ω(x) = {y : lim i f n i (x) = y for some n i } With this definition, I proceed with a fundamental result by Singer [6]: Theorem 2 (Singer). If f : X X satisfies S(f)(x) < 0, then for each attracting period-n point x 0 there is a critical point c in the basin of x 0 and hence ω(c) is equal to the orbit of x 0. 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 : S S. The main dynamic object of interest is the circle map, as the circle is 5

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 S has a period-n orbit under f and that x R is a lift of x. Now since f n (x) = x, then using (2 2), f n (x) = x + m for some m Z. In this case, we say that x is a type (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 for an arbitrary k Z, f n (x + k) = f n (x) + 2 n k = x + m + 2 n k = (x + k) + m + (2 n )k (2 2) and so x + k is of type (n, (2 n )k + m). This means that the type of a period-n point x S under f is only defined modulo 2 n. If x has type (n, m) for a degree-two f and α is as in Theorem, then using diagram (2 8), α(f n (x)) = α(x + m) d n (α(x)) = α(x) + m 2 n α(x) = α(x) + m which implies that α(x) = m. If I choose another lift (x + k) of x, I get that 2 n α(x + k) = (2n )k + m 2 n = k + m 2 n, and so α(x) α(x + k) mod. So on the circle we have that if x is a period-n point, then α(x) = m 2 n S for some 0 m < 2 n. 2.6 Flat Spots in the Graph of α Definition 9. A flat spot for the graph of α is a closed interval I so that α(x ) = α(x 2 ) for all x, x 2 I, and I is the maximal interval with this property, i.e. f is not constant on any interval J with J I and I J. 6

Lemma 5.. If I is an interval with f(i) I, then I is contained in a flat spot of α. 2. If I S is an interval with f n (I) I for some n > 0, then I is in a flat spot of α. 3. If I S is the basin of an attracting periodic point for f, then I is in a flat spot of α. Proof:. Let I be an interval, say I = [a, b]. Then by induction, f n (I) I [a, b]. So for all x I, a < f n (x) < b and thus a < f n (x) < b. Now by the squeeze theorem, 2 n 2 n 2 n f lim n (k) n = 0, so for all x I, α(x) = 0. As a result, I is contained in a flat spot 2 n of α. 2. The proof is the same as for, just add in the bars denoting that you are on the circle. 3. By definition, I is the maximal interval on which lim f n (x) = α(x) = x o, where x o is an attracting periodic point. By part two of this lemma, I is a flat spot of α. Remark. An example of a flat spot that does not come from the basin of a periodic point is a trapping region; a region in which the points cannot escape. The f-dynamics of a flat spot corresponding to α = 0 or α = is exactly that: a trapping region that does not go around the circle (see numerical results for specific examples and descriptions). Lemma 6.. If x is precritical and f n (x) = c, a critical point, then for all j > 0, F n+j(x) = 0, with F n as defined in Theorem. 2. If f has an attracting periodic point x of type (n, m) with 0 m < (2 n ), then for some critical point c, α(c) = and further, c is contained in a flat spot of α. Proof: m 2 n. Since f n (x) = c, F n = c and thus F 2 n n = 0. F n+ = f(c) and since f(x) is a real 2 n+ number, F n+ = 0. By induction, F n+j = 0. 2. We have that S(f) < 0, so by Singer s Theorem there exists a critical point c with ω(c) = o(x), the orbit of x. This fact means that c is in the basin of x and 7

by Lemma 5, c is in the flat spot of x. By the discussion in the previous section, α(x) = m m and since c is in the basin of x, we also have α(c) =. 2 n 2 n 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, ] and thus the circle map g b,w has two critical points. The critical points of g b,w are the solutions to g bw (x) = 0, and they depend on b but not w. I denote them as c (b) and c 2 (b) and solve for them as follows: arccos( bπ ) 2π 0 = g bw(x) = 2 + 2bπ cos(2πx) 2 2bπ = cos(2πx) 2π arccos( bπ = x = c (b), ) = x = c 2 (b) 2π The notes so far indicate the importance of studying the value of α at a critical point. Note that we get a different α for each (b, w). I indicate this by writing α b,w and for each (b, w), I define: s (b, w) = α b,w (c (b)) and s 2 (b, w) = α b,w (c 2 (b)). (2 22) Note that Lemma 6 says that if g b,w has an attracting periodic point x of type (n, m), then s j (b, w) = m 2 n for j = or 2, or both, but not necessarily the converse. Remark 2. As will become evident in the numerical results, we can say many things about the relationship between s and s 2. In particular, s (b, w) = s 2 (b, w), allowing us to study s and generalize to s 2. 8

CHAPTER 3 NUMERICAL RESULTS 3. Plots of the Standard Family We will now begin to look specifically at the family of maps α b,w = lim n (g n b,w /2n ) of the standard family g b,w = 2x + w + b sin(2πx) on the circle. To begin, 3- is a plot of g b,w when b = 0.75 and w = with y = x + included: 3 2.5 2.5 0.20.40.60.8 Figure 3-. g b,w when w = and b = 0.75. The position of w and b in g b,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 g b,w when both b and w have the value of 0.2. Again, I have plotted y = x + with the graph. Since g b,w is on the circle, the intersections of y = x + n with g b,w represent the fixed points of g b,w, n Z. In both Figure 3- and Figure 3-2, we can visualize all intersections with only y = x + since the graphs on R 2 do not cross y = x or y = x + 2. Comparing these graphs, we also notice that the number of fixed points in g b,w can change with the parameters. Now as in Theorem 3. of Chapter 2, define G b,w,n (x) as follows in equation (3 ): 9

G b,w,n (x) = gn b,w (x) 2 n (3 ) 3 2.5 2.5 0.20.40.60.8 Figure 3-2. g b,w when w = 0.2 and b = 0.2. Since G b,w,0 is a good approximation of α b,w under most of the graphical resolutions used in this paper, the reader may assume that G b,w,0 was used to produce results for α b,w unless stated otherwise. With this note in mind, here is an example of α: 2.75.5.25 0.75 0.25 g x 2x 0.87 sin 2Πx 0.3 0.2 0.4 0.6 0.8 Figure 3-3. α 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 20

α b,w changes with its other parameter by using animations or 3D plots. In addition, all of α b,w can be viewed by using an animation of 3D plots. 2.75.5.25 0.75 0.25 g x 2x 0.87 sin 2Πx 0.2 0.4 0.6 0.8 Figure 3-4. α when b = 0.87 and w = 0. Unfortunately, all these ways of viewing the α b,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 α b,w is to look at the behavior of the g b,w critical points as we vary b and w. 3.2 The Bifurcation Diagram of the Standard Family As stated in Chapter 2, g b,w has exactly two critical points whenever b, and so π there are two bifurcation diagrams, s (b, w) and s 2 (b, w). Graphed as a density plot, these bifurcation diagrams are shown in 3-5 and 3-6. If x S, the color value for x in Figure 3-5 and Figure 3-6 can be found by the radius θ = 2πx 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 π ourselves to b +, which is the upper bound for b in all the previous plots. Figure 3-9 π is a plot of s (b, w) in which the dimensions of the plot are 3x3. Recall that since s (b, w) is on the circle, 2x+(w+n)+b sin(2πx) 2x+w+b sin(2πx) mod which accounts for the vertical repetition in 3-9. 3-9 also dispels the illusion that there is noticeably more blue in s (b, w) than green. Similarly, the non-red colors appear more even when we expand the range of b in s 2 (b, w). Using a graphics program to count 2

0.8 0.6 0.4 0.2 0 0.4 0.6 0.8.2 Figure 3-5. s (b, w). 0.8 0.6 0.4 0.2 0 0.4 0.6 0.8.2 Figure 3-6. s 2 (b, w). 22

Figure 3-7. Key for 3-5 and 3-6. 0.75.25 2.5 0.25 0 0.75 Figure 3-8. 3D image for 3-5. 3 2.5 2.5 0 Figure 3-9. s (b, w) with b ranging from.5 π to 2 π 23 2.5 3 and w ranging from 0 to 3.

the pixels of the most prominent colors under different resolutions, I have arrived at the following statement: Conjecture. In the complete plot of s (b, w) and s 2 (b, w) (that is, when b ranges from π to infinity), the areas of corresponding colors is equivalent. This conjecture is further supported by the second part of Lemma 6 in Chapter 2. For different parameter values, c (c 2 ) could be attracted to different points in the same periodic orbit of g b,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 2π 3 and blue is at 4π, which means that green corresponds to 3 the value /3 and blue to 2/3 on S. These are exactly the values possible when g b,w has an attracting periodic point of type (2, m) for 0 m < (2 n ). 3.3 Characterizing the Bifurcation Diagram Now that we have generated these density plots for s (b, w) and s 2 (b, w), the obvious question to ask is what the features of the plot tell us. As mentioned at the end of Chapter 2, s (b, w) = s 2 (b, w) so in this section, I will only talk about s (b, w) since the behavior of s 2 (b, w) is similar. Figure 3-0 shows s (b, w) again, with some of the more interesting features labeled. 3.3. Fixed Points and Saddle Points The fixed point line in 3-0 represents the w value (mod ) for which a given b value makes the critical point a fixed point of α. This line was found numerically by solving the equation: c (b) = 2c (b) + w + b sin(2πc (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 α b,w (c ). The critical point still converges to the same place in the graph, the fixed point of g b,w. By Lemma 4 in Chapter 24

Figure 3-0. s (b, w) with w ranging from to. 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, so we could conjecture that the parameter line on which the derivative of the fixed point equals (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-). In examining 3-, it is important to recall that the image was actually produced using G b,w,0 instead of α b,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 α b,w was available, the colors would reach the saddle point line for all values of b greater than π. To further validate this claim, we will now examine g b,w at the points s, a, and f listed in 3-0. Point f is obvious and shown in Figure 3-2. The graph in Figure 3-3 for point s uses w =.265. 25

Figure 3-. Bifurcation diagram with the colors running through running seven times the normal mod. Once again, a graphics program was used to create the picture, and I have included code which verifies the placement of the line. 0.75 0.25-0.25 0.2 0.4 0.6 0.8 - -0.75 - Figure 3-2. Graph of g b,w when b = and the critical point is a fixed point. The fixed point line mod has been included. 26

0.75 0.25 0.2 0.4 0.6 0.8-0.25 - -0.75 - Figure 3-3. Graph of g b,w at parameter point s in 3-0. Finally, the graph of point a in 3-0 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 Chapter 2, the flat spots in α b,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 3-4 is a typical graph of g b,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 3-5 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 g b,w (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 3-4 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 27

Figure 3-4. Graph of g b,w when b = 0.6 and w = 0.6 Figure 3-5. Graph of g b,w at parameter point α in 3-0. w =.2 was used to generate the plot. 28

diagram. As we shall see when discussing the point phi in 3-0, 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 3-6, we see that beta lies right on the threshold the first critical point escaping. Figure 3-6. Graph of g b,w, approximately at parameter point β in 3-0. Above point beta on the diagram, the critical point escapes the trapping region on the first iteration of g b,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 3-7 show how the critical point maps below the trapping region in the first iteration. Conjecture 2. In any plot that falls within area gamma, the x values in the interval represented by the bottom of the trapping region form a cantor set of points that never escape the region. This statement should be easy to prove using a code space. At point delta, g b,w is on a threshold of no longer using the curve to fall below the trapping region, as seen in Figure 3-8. For area epsilon, iterations of g b,w sometimes take the critical point into the trapping region while others allow it to escape. Because we must examine α b,w to determine the result, the graph of g b,w in Figure 3-9 is somewhat unenlightening. 29

Figure 3-7. Graph of g b,w in area γ in the bifurcation diagram. w = 0.8 was used to create the plot. Figure 3-8. Graph of g b,w approximately at point δ in the bifurcation diagram. 30

Figure 3-9. Graph of g b,w in the area ɛ 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..2 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8-0.2 Figure 3-20. Graph of g b,w approximately at point φ in the bifurcation diagram. A horizontal line through the right fixed point has been included. 3

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 3-2. Obviously, some of these areas contain concepts we have already examined in the previous section. Area F, for example, has already been shown under g b,w in Figure 3-4. Figure 3-22 is the α(x) of the same point in the parameter space (mod for w). Figure 3-2. Double Bifurcation Diagram. The bifurcation diagram of the second critical point on top of the bifurcation diagram of the first critical point. 2.75.5.25 0.75 0.25 2x 0.6 sin 2Πx 0.6 0.2 0.4 0.6 0.8 Figure 3-22. α b,w (x) when b = 0.6 and w = 0.6. 32

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 g b,w. It is the threshold for both critical points having an escape hatch. Point S represents the place where g b,w has a double saddle point, since it is the intersection between both saddle point lines (Figure 3-23). 2.75.5.25 0.75 0.25 0.2 0.4 0.6 0.8 Figure 3-23. g b,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 g b,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 displays behavior like the area alpha in 3-0 for both critical points. α b,w of this region is displayed 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 g b,w, and the other fixed point line is too low to play a role. Area Y is just the opposite, and area Z places both critical points between the fixed point lines. The plots of α b,w (x) will vary depending on the choice of b and w in these regions, since later iterations of g b,w (x) have a pronounced effect on the limit. This is also the case for areas nu, mu, lambda, and theta. Below are the α b,w plots for nu, lambda, and theta respectively. mu is simply a 80 degree rotation of nu. 33

3 2.5 2.5 0.20.40.60.8 Figure 3-24. g b,w (x) when b =.235 and w = with three fixed point lines..2 2x 0.8 sin 2Πx 0.8 0.6 0.4 0.2-0.2 0.2 0.4 0.6 0.8 Figure 3-25. α b,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. 34

2.75.5.25 0.75 0.25 0.2 0.4 0.6 0.8 Figure 3-26. g b,w (x) in region X, when b = 0.6 and w = 0.2. 3 2.75 2.5 2.25 2.75.5.25 0.2 0.4 0.6 0.8 Figure 3-27. g b,w (x) in region Y, when b = 0.6 and w = 0.8. 35

2.75.5.25 0.75 0.25 0.2 0.4 0.6 0.8 Figure 3-28. g b,w (x) in region Z, when b = 0.4 and w = 0. 2.75.5.25 0.75 0.25 2x 0.85 sin 2Πx 0.3 0.2 0.4 0.6 0.8 Figure 3-29. α b,w (x) in area nu when b = 0.85 and w = 0.3. 2.75.5.25 0.75 0.25 2x.05 sin 2Πx 0.2 0.4 0.6 0.8 Figure 3-30. α b,w (x)in area lambda when b =.05 and w =. 36

.5.25 0.75 0.25-0.25-2x.05 sin 2Πx 0. 0.2 0.4 0.6 0.8 Figure 3-3. α b,w (x) in area theta when b =.05 and w = 0.. 2.5 2.25 2.75.5.25 0.75 2x 0.97 sin 2Πx 0.83 0.2 0.4 0.6 0.8 Figure 3-32. α b,w (x) in area U when b = 0.97 and w = 0.83..5.25 0.75 0.25-0.25-2x 0.97 sin 2Πx 0.7 0.2 0.4 0.6 0.8 Figure 3-33. α b,w (x) in area V when b = 0.97 and w = 0.7. 37

REFERENCES [] L. S. Block and W. A. Coppel, Dynamics in one dimension, vol. 53 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 992. [2] W. de Melo and S. van Strien, One-dimensional dynamics, vol. 25 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 993. [3] L. Alsedà, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, vol. 5 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co. Inc., River Edge, NJ, second edition, 2000. [4] P. Boyland, Semiconjugacies to angle-doubling, Proc. Amer. Math. Soc., vol. 34, no. 5, pp. 299 307 (electronic), 2006. [5] W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, Inc., New York, 976. [6] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., vol. 35, no. 2, pp. 260 267, 978. 38

BIOGRAPHICAL SKETCH William Christopher Strickland was born on April 6, 983 in Houston, Texas. He grew up in Oxford, Mississippi and graduated from Oxford High School in 200. 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. Christopher then entered graduate school at the University of Florida in order to continue his studies in mathematics. He completed his M.S. in 2007. 39