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Reverse bifurcations in a quartic family
Computers & Graphics 24 (2000) 143}149
Chaos and graphics
Reverse bifurcations in a quartic family Michael Frame*, Shontel Meachem Mathematics Department, Union College, Bailey Hall, Schenectady, NY 12308-2311, USA
Abstract We show a family of real quartic maps exhibits reverse bifurcations. Also called cycle-annihilating transitions, reverse bifurcations have been known for the HeH non map for several years, but are not present in the logistic map. We use trapping squares to explain these quartic reverse bifurcations. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Period-doubling bifurcation; Reverse bifurcation; Trapping square
one-dimensional logistic map does not exhibit reverse bifurcations.
1. Introduction The familiar quadratic (logistic) map exhibits only two kinds of bifurcations: period doubling and tangent. The "rst are responsible for the `period doubling route to chaosa, the second for the formation of periodic windows the arise after the onset of chaos. In general, these processes are responsible for the stable (thus visible in the standard computer experiments) periodic behavior of the logistic map. For both types of bifurcations, once a cycle appears, it persists, even though eventually it becomes unstable. This persistence, established by Douady and Hubbard, Sullivan, and Milnor and Thurston [1] is called monotonicity. For the HeH non map
AB A
B
x j!x2#by H " , j y x Kan et al. [2] showed the x-coordinate bifurcations exhibit antimonotonicity. That is, as the j parameter increases some cycles disappear through reverse-bifurcation, the familiar bifurcations, but in reverse order. Through computer experiments and graphical analysis, we discovered a similar antimonoticity for a real quartic function q . Unlike the two-dimensional HeH non j map, our example is one-dimensional. We emphasize the
* Corresponding author. Tel.: #1-518-388-6246.
2. Quadratic bifurcations Many of the interesting phenomena that occur in dynamical systems are illustrated by the quadratic family, ¸ (x)"j ) x ) (1!x). For any point x 3[0,1], the j 0 forward orbit O`(x ) is the set Mx , x "¸ (x ), j 0 0 1 j 0 x "¸ (x ),2N of iterates of x . It is well known [3] 2 j 1 0 that O`(x ) remains bounded for j3[0,4], and for almost j 0 all x , O`(x ) evolves through a complicated collection 0 j 0 of sets as j increases from 0 to 4. For 0)j)1, the forward orbit converges to the "xed point 0. For j'1, the "xed point 0 becomes unstable, and for 1(j)3 most forward orbits converge to the "xed point 1!1/j. For j'3, the "xed point 1!1/j becomes unstable, and for 3(j)1#J6 most forward orbits converge to a 2-cycle. As j continues to increase, the 2-cycle becomes unstable and a stable 4-cycle appears. This 4-cycle becomes unstable and a stable 8-cycle appears. This sequence of period-doublings continues until the Myerberg point, j+3.56994562, where the "rst chaotic forward orbit appears. As j continues to increase to 4, an intricate interleaving of chaos and stable cycles arise. This information typically is assembled into a bifurcation diagram, plotting j on the horizontal axis and an approximation of the limit points of O`(1) on the vertical section j 2 above j (see Fig. 1). The combinatorial and many of the
0097-8493/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 9 9 ) 0 0 1 4 4 - 2
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M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
Fig. 1. Top: the bifurcation diagram of the logistic map, with "ve parameter values, j"2.5, 3.15, 3.48, 3.955, and 4.02 indicated. Bottom: the corresponding graphical iteration pictures.
metric properties of this diagram are well understood, but some fundamental questions remain unanswered. For example, although Jakobsen's theorem guarantees chaos occurs on a `largea set of j values (technically, on a set of positive one-dimensional Lebesgue measure), the `amounta of chaos (the measure of this set) remains unknown. The choice of O`(1) is dictated by Fatou's theorem, j 2 that a stable cycle must be the limit set of the forward orbit of some critical point. For each j, ¸ has exactly j one critical point, x"1, so O`(1) will reveal any stable 2 j 2 cycle. Fatou's theorem implies for each j. ¸ has at most j one stable cycle. Of course, ¸ need not possess any j stable cycle: for those j where O`(1) is chaotic, there can j 2 be no stable cycle. This is not the case for our quartic family. Fixed points x of ¸ are the points of intersection f j of the graphs of x "¸ (x ) and x "x , stable if n`1 j n n`1 n D(d/dx)¸ (x)D f D(1. Similarly, points of an m-cycle for j x ¸ (x) are "xed points of the composition ¸m(x). j j As mentioned earlier, chaos and stable cycles are interleaved for j beyond the Myerberg point. Fig. 2 shows a portion of the bifurcation diagram containing a stable 4-cycle. The right top shows the familiar observation that
a small copy of the whole bifurcation diagram grows from each branch of the cycle. The bottom of the "gure shows magni"cations of the graphical iteration of ¸4 in j a neighborhood of (1, 1). All these are in the square 22 [0.48, 0.52]][0.48, 0.52]. In addition, windows 1}5 indicate the trapping square determined by the "xed point, q, of ¸4 closest to 1. To construct this square, start from the j 2 "xed point (q, q) on the diagonal and draw a horizontal line to the closest intersection (r, q) with the graph of ¸4 . j Continue with the vertical line to (r, r) on the diagonal, then complete the square. Recall these are called trapping squares because, so long as the graph only crosses the square at corners, an iteration entering the square can never exit. Note that dynamics represented in the trapping squares of windows 1}5 in Fig. 2 correspond to those of windows 1}5 in Fig. 1. That is, in the j range of this 4-cycle, the restriction of the graph of ¸4 to the trapping j square is closely approximated by the graph of ¸ #ipped j vertically. Whereas ¸ has a maximum at 1 that increases j 2 as j increases, in this 4-cycle range ¸4 has a minimum j that decreases as j increases. Trivial as it may seem, this observation is the key to understanding the antimonotonicity of the quartic map.
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
145
Fig. 2. Top left: the part of the logistic bifurcation in the range 3.959)j)3.963 and 0)x )1; right: the range 3.96)j)3.9625 and n 0.464)x )0.548. Bottom: the graphical iteration pictures of ¸4 corresponding to the j values indicated on the top right. n j
3. A quartic family Motivated by the implications of Fatou's theorem, we sought a simple function with critical points having distinct future orbits, and possibly distinct limit sets. Symmetry considerations suggested a function with derivative a multiple of (x!1) ) (x!1) ) (x!3). To obtain 4 2 4 a family of functions q :[0,1]P[0,1] with all q (0)" j j q (1)"0, we took (Fig. 3) j
A
B
1 1 11 3 q (x)"!j ) x ) x3! x2# x! . j 4 2 32 32 Note q (1)" 9 , so to guarantee q :[0,1]P[0,1], we 14 1024 j restrict j to the interval 0)j)1024+113.778. Also, 9 note q (1)"q (3), so each q has two critical oribts, O`(1) j4 j4 j j 4 and O`(1). j 2 4. A quartic bifurcation diagram and reverse bifurcation windows Because it has two critical orbits, q (x) has two bifurcaj tion diagrams, one consisting of the limit points of O`(1), j 4 the other of the limit points of O`(1). These diagrams j 2
Fig. 3. The graph of x "q (x ) for j"100, together with n`1 j n the diagonal x "x . n`1 n
exhibit many similarities, but also some interesting di!erences. The left side of Fig. 4 shows a portion of the bifurcation diagram of O`(1). For each j value, iterations j 4 101}200 of the orbit of 1 are plotted in red. To emphasize 4
146
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
Fig. 4. Left: a portion of the bifurcation diagram of q (x) for the orbit of 1, together with the "rst few iterates of the critical point. Right: j 4 the same diagram for the orbit of 1; the small box is magni"ed below. 2
the role of the critical orbit, iterations 1}10 of 1 appear as 4 blue curves. Note the prominent 2-cycle window. The right side of Fig. 4 shows the same portion of the bifurcation diagram of O`(1) (in green); iterations 1}10 of j 2 1 appear as blue curves. Note that how much more 2 intricate these curves are in the range where the orbit of 1 4 converges to a 2-cycle. In particular, the small boxed region, magni"ed below, contains a complete quadratic bifurcation diagram. Consequently, there are j values where one critical point converges to a cycle, and the
other to chaos. While this is interesting, the magni"cation itself is a surprise. Note that the bifurcation diagram is oriented in the opposite way from the logistic diagram. In particular, this diagram appears to exhibit the same sort of reverese bifurcations seen in the HeH non map. Our goal here is to investigate this phenomenon in the bifurcation diagram of O`(1). j 2 We begin with the whole diagram (the left side of Fig. 5), then magnify to "nd the largest reverse bifurcation. The right side of Fig. 5 shows the portion 110)j)1024. 9
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
147
Fig. 5. Left: the bifurcation diagram of q (x) for the orbit of 1. Right: the portion of this diagram in the range 110)j)1024. j 2 9
Fig. 6. Left: a magni"cation of the right of Fig. 5. Right: a magni"cation, 0.45)x )0.55 of the bottom branch from the large window n on the left.
This diagram bears some resemblance to the general form of the quadratic bifurcation diagram, but the deeper magni"cation, 111.5)j)111.9, shown in Fig. 6 reveals a qualitatively di!erent structure. On the left, we see what appears to be a bifurcation pattern arising from a 4-cycle window. Magnifying the bottom branch of this picture reveals a copy of the quadratic bifurcation diagram, but oriented in the opposite direction. That is, the bifurcation diagram of q (x) j exhibits the antimonotonicity observed for the Henon map in [2]. In the next section we shall see how this reverse bifurcation diagram arises.
5. The source of antimonotonicity Because the antimonotonicity pictured arises in a 4-cycle window in the bifurcation diagram of 1, we shall 2 investigate the graphical iteration of the fourth composition q4(x) in a neighborhood of 1. In Fig. 7 the pictured graphs j 2 are for j"111.64, 111.65, 111.68, 111.70, 111.72, and 111.76. In all except the last, we show the trapping square determined by the "xed point nearest and less than 1. 2 The essence of the reverse bifurcation sequence is that as j increases in this range, q4(1) decreases, yet near 1 the j2 2 graph of q4(x) is concave down. In fact, the part of the j
148
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
graph of q4 (x) within this trapping square is well-apj proximated by the quadratic map in the unit square. We see parts of the familiar quadratic bifurcation sequence * "xed point to 2-cycle to 4-cycle to 2 to chaos to escape * repeated here, but as j decreases, not as j increases. Compare with the bottom of Fig. 2. This is the immediate reason for the reverse bifurcation sequence in this quartic family. Now, we turn to the question of why this family behaves this way, while the quadratic family does not. On the left of Fig. 8 we see the graph of (d2/dx2)q4(x)D for 110)j)113.778, showing the j x/1@2 graph of q4(x) is concave down at x"1 for j'105.4. On j 2 the right we see the graph of (d/dj)q4(1) in the same j2 j range. Combining these, we see indeed in the range of the 4-cycle window that q4(x) is concave down around j x"1 and q4(1) is a decreasing function of j. This combij2 2
nation of concavity and decreasing local maximum give rise to the reverse bifurcation sequence of q (x). j Fig. 9 shows the analogous graphs for ¸4(x). Note in j the 4-cycle window near x"1 the graph of ¸4 (x) is 2 j concave up and the local minimum ¸4 (1) is a decreasing j2 function of j. This accounts for the forward bifurcation in this 4-cycle window of ¸ (x). j While we have seen the quartic map has both forward and reverse bifurcations, perhaps a tedious inductive argument would show the logistic map always exhibits forward bifurcations. However, this was not the approach taken by Milnor and Thurston, leading us to suspect the straightforward argument would not work. The algebraic bookkeeping is daunting, absent a clever organizational scheme, elusive so far. This quartic family has other interesting features, including di!erent limit sets for O`(1) and O`(1). We j 4 j 2
Fig. 7. The graph of q4(x) in the neighborhood [0.46, 0.53]][0.46, 0.53] of (1, 1), at the j values indicated on the right side of Fig. 6. j 22
Fig. 8. Left: the second derivative plot of q4(x) at x"1. Right: the value of q4 (1), as a function of j. j 2 j2
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
149
Fig. 9. Left: the second derivative plot of ¸4(x) at x"1. Right: the value of ¸4 (1), as a function of j. j 2 j2
wonder if still more complicated functions will reveal qualitatively di!erent behavior, or if the catalog of onedimensional dynamics now is complete.
For j"1 to Iter Do x"f (x) j Plot(j, x).
PseudoCode References Suppose f (x) is a family of functions with critical point j p (and maybe other critical points). To study the bifur1 cation diagram of f (x) in the range j )j)j , use j min max For i"1 to j number Do j"j
#i/(j !j ) min max min x"p 1 For j"1 to Drop Do x"f (x) j
[1] Milnor J, Thurston W. On iterated maps of the interval. In: Alexander J, editor, Dynamical systems. New York: Springer, 1988. p. 465}563. [2] Kan I, Koc7 ak H, Yorke J. Antimonotonicity: concurrent creation and annihilation of periodic orbits. Annals of Mathematics 1992;136:219}52. [3] Devaney R. An introduction to chaotic dynamical systems, 2nd ed. Redwood City: Addison-Wesley, 1989.
Chaos and graphics
Reverse bifurcations in a quartic family Michael Frame*, Shontel Meachem Mathematics Department, Union College, Bailey Hall, Schenectady, NY 12308-2311, USA
Abstract We show a family of real quartic maps exhibits reverse bifurcations. Also called cycle-annihilating transitions, reverse bifurcations have been known for the HeH non map for several years, but are not present in the logistic map. We use trapping squares to explain these quartic reverse bifurcations. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Period-doubling bifurcation; Reverse bifurcation; Trapping square
one-dimensional logistic map does not exhibit reverse bifurcations.
1. Introduction The familiar quadratic (logistic) map exhibits only two kinds of bifurcations: period doubling and tangent. The "rst are responsible for the `period doubling route to chaosa, the second for the formation of periodic windows the arise after the onset of chaos. In general, these processes are responsible for the stable (thus visible in the standard computer experiments) periodic behavior of the logistic map. For both types of bifurcations, once a cycle appears, it persists, even though eventually it becomes unstable. This persistence, established by Douady and Hubbard, Sullivan, and Milnor and Thurston [1] is called monotonicity. For the HeH non map
AB A
B
x j!x2#by H " , j y x Kan et al. [2] showed the x-coordinate bifurcations exhibit antimonotonicity. That is, as the j parameter increases some cycles disappear through reverse-bifurcation, the familiar bifurcations, but in reverse order. Through computer experiments and graphical analysis, we discovered a similar antimonoticity for a real quartic function q . Unlike the two-dimensional HeH non j map, our example is one-dimensional. We emphasize the
* Corresponding author. Tel.: #1-518-388-6246.
2. Quadratic bifurcations Many of the interesting phenomena that occur in dynamical systems are illustrated by the quadratic family, ¸ (x)"j ) x ) (1!x). For any point x 3[0,1], the j 0 forward orbit O`(x ) is the set Mx , x "¸ (x ), j 0 0 1 j 0 x "¸ (x ),2N of iterates of x . It is well known [3] 2 j 1 0 that O`(x ) remains bounded for j3[0,4], and for almost j 0 all x , O`(x ) evolves through a complicated collection 0 j 0 of sets as j increases from 0 to 4. For 0)j)1, the forward orbit converges to the "xed point 0. For j'1, the "xed point 0 becomes unstable, and for 1(j)3 most forward orbits converge to the "xed point 1!1/j. For j'3, the "xed point 1!1/j becomes unstable, and for 3(j)1#J6 most forward orbits converge to a 2-cycle. As j continues to increase, the 2-cycle becomes unstable and a stable 4-cycle appears. This 4-cycle becomes unstable and a stable 8-cycle appears. This sequence of period-doublings continues until the Myerberg point, j+3.56994562, where the "rst chaotic forward orbit appears. As j continues to increase to 4, an intricate interleaving of chaos and stable cycles arise. This information typically is assembled into a bifurcation diagram, plotting j on the horizontal axis and an approximation of the limit points of O`(1) on the vertical section j 2 above j (see Fig. 1). The combinatorial and many of the
0097-8493/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 7 - 8 4 9 3 ( 9 9 ) 0 0 1 4 4 - 2
144
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
Fig. 1. Top: the bifurcation diagram of the logistic map, with "ve parameter values, j"2.5, 3.15, 3.48, 3.955, and 4.02 indicated. Bottom: the corresponding graphical iteration pictures.
metric properties of this diagram are well understood, but some fundamental questions remain unanswered. For example, although Jakobsen's theorem guarantees chaos occurs on a `largea set of j values (technically, on a set of positive one-dimensional Lebesgue measure), the `amounta of chaos (the measure of this set) remains unknown. The choice of O`(1) is dictated by Fatou's theorem, j 2 that a stable cycle must be the limit set of the forward orbit of some critical point. For each j, ¸ has exactly j one critical point, x"1, so O`(1) will reveal any stable 2 j 2 cycle. Fatou's theorem implies for each j. ¸ has at most j one stable cycle. Of course, ¸ need not possess any j stable cycle: for those j where O`(1) is chaotic, there can j 2 be no stable cycle. This is not the case for our quartic family. Fixed points x of ¸ are the points of intersection f j of the graphs of x "¸ (x ) and x "x , stable if n`1 j n n`1 n D(d/dx)¸ (x)D f D(1. Similarly, points of an m-cycle for j x ¸ (x) are "xed points of the composition ¸m(x). j j As mentioned earlier, chaos and stable cycles are interleaved for j beyond the Myerberg point. Fig. 2 shows a portion of the bifurcation diagram containing a stable 4-cycle. The right top shows the familiar observation that
a small copy of the whole bifurcation diagram grows from each branch of the cycle. The bottom of the "gure shows magni"cations of the graphical iteration of ¸4 in j a neighborhood of (1, 1). All these are in the square 22 [0.48, 0.52]][0.48, 0.52]. In addition, windows 1}5 indicate the trapping square determined by the "xed point, q, of ¸4 closest to 1. To construct this square, start from the j 2 "xed point (q, q) on the diagonal and draw a horizontal line to the closest intersection (r, q) with the graph of ¸4 . j Continue with the vertical line to (r, r) on the diagonal, then complete the square. Recall these are called trapping squares because, so long as the graph only crosses the square at corners, an iteration entering the square can never exit. Note that dynamics represented in the trapping squares of windows 1}5 in Fig. 2 correspond to those of windows 1}5 in Fig. 1. That is, in the j range of this 4-cycle, the restriction of the graph of ¸4 to the trapping j square is closely approximated by the graph of ¸ #ipped j vertically. Whereas ¸ has a maximum at 1 that increases j 2 as j increases, in this 4-cycle range ¸4 has a minimum j that decreases as j increases. Trivial as it may seem, this observation is the key to understanding the antimonotonicity of the quartic map.
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
145
Fig. 2. Top left: the part of the logistic bifurcation in the range 3.959)j)3.963 and 0)x )1; right: the range 3.96)j)3.9625 and n 0.464)x )0.548. Bottom: the graphical iteration pictures of ¸4 corresponding to the j values indicated on the top right. n j
3. A quartic family Motivated by the implications of Fatou's theorem, we sought a simple function with critical points having distinct future orbits, and possibly distinct limit sets. Symmetry considerations suggested a function with derivative a multiple of (x!1) ) (x!1) ) (x!3). To obtain 4 2 4 a family of functions q :[0,1]P[0,1] with all q (0)" j j q (1)"0, we took (Fig. 3) j
A
B
1 1 11 3 q (x)"!j ) x ) x3! x2# x! . j 4 2 32 32 Note q (1)" 9 , so to guarantee q :[0,1]P[0,1], we 14 1024 j restrict j to the interval 0)j)1024+113.778. Also, 9 note q (1)"q (3), so each q has two critical oribts, O`(1) j4 j4 j j 4 and O`(1). j 2 4. A quartic bifurcation diagram and reverse bifurcation windows Because it has two critical orbits, q (x) has two bifurcaj tion diagrams, one consisting of the limit points of O`(1), j 4 the other of the limit points of O`(1). These diagrams j 2
Fig. 3. The graph of x "q (x ) for j"100, together with n`1 j n the diagonal x "x . n`1 n
exhibit many similarities, but also some interesting di!erences. The left side of Fig. 4 shows a portion of the bifurcation diagram of O`(1). For each j value, iterations j 4 101}200 of the orbit of 1 are plotted in red. To emphasize 4
146
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
Fig. 4. Left: a portion of the bifurcation diagram of q (x) for the orbit of 1, together with the "rst few iterates of the critical point. Right: j 4 the same diagram for the orbit of 1; the small box is magni"ed below. 2
the role of the critical orbit, iterations 1}10 of 1 appear as 4 blue curves. Note the prominent 2-cycle window. The right side of Fig. 4 shows the same portion of the bifurcation diagram of O`(1) (in green); iterations 1}10 of j 2 1 appear as blue curves. Note that how much more 2 intricate these curves are in the range where the orbit of 1 4 converges to a 2-cycle. In particular, the small boxed region, magni"ed below, contains a complete quadratic bifurcation diagram. Consequently, there are j values where one critical point converges to a cycle, and the
other to chaos. While this is interesting, the magni"cation itself is a surprise. Note that the bifurcation diagram is oriented in the opposite way from the logistic diagram. In particular, this diagram appears to exhibit the same sort of reverese bifurcations seen in the HeH non map. Our goal here is to investigate this phenomenon in the bifurcation diagram of O`(1). j 2 We begin with the whole diagram (the left side of Fig. 5), then magnify to "nd the largest reverse bifurcation. The right side of Fig. 5 shows the portion 110)j)1024. 9
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
147
Fig. 5. Left: the bifurcation diagram of q (x) for the orbit of 1. Right: the portion of this diagram in the range 110)j)1024. j 2 9
Fig. 6. Left: a magni"cation of the right of Fig. 5. Right: a magni"cation, 0.45)x )0.55 of the bottom branch from the large window n on the left.
This diagram bears some resemblance to the general form of the quadratic bifurcation diagram, but the deeper magni"cation, 111.5)j)111.9, shown in Fig. 6 reveals a qualitatively di!erent structure. On the left, we see what appears to be a bifurcation pattern arising from a 4-cycle window. Magnifying the bottom branch of this picture reveals a copy of the quadratic bifurcation diagram, but oriented in the opposite direction. That is, the bifurcation diagram of q (x) j exhibits the antimonotonicity observed for the Henon map in [2]. In the next section we shall see how this reverse bifurcation diagram arises.
5. The source of antimonotonicity Because the antimonotonicity pictured arises in a 4-cycle window in the bifurcation diagram of 1, we shall 2 investigate the graphical iteration of the fourth composition q4(x) in a neighborhood of 1. In Fig. 7 the pictured graphs j 2 are for j"111.64, 111.65, 111.68, 111.70, 111.72, and 111.76. In all except the last, we show the trapping square determined by the "xed point nearest and less than 1. 2 The essence of the reverse bifurcation sequence is that as j increases in this range, q4(1) decreases, yet near 1 the j2 2 graph of q4(x) is concave down. In fact, the part of the j
148
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
graph of q4 (x) within this trapping square is well-apj proximated by the quadratic map in the unit square. We see parts of the familiar quadratic bifurcation sequence * "xed point to 2-cycle to 4-cycle to 2 to chaos to escape * repeated here, but as j decreases, not as j increases. Compare with the bottom of Fig. 2. This is the immediate reason for the reverse bifurcation sequence in this quartic family. Now, we turn to the question of why this family behaves this way, while the quadratic family does not. On the left of Fig. 8 we see the graph of (d2/dx2)q4(x)D for 110)j)113.778, showing the j x/1@2 graph of q4(x) is concave down at x"1 for j'105.4. On j 2 the right we see the graph of (d/dj)q4(1) in the same j2 j range. Combining these, we see indeed in the range of the 4-cycle window that q4(x) is concave down around j x"1 and q4(1) is a decreasing function of j. This combij2 2
nation of concavity and decreasing local maximum give rise to the reverse bifurcation sequence of q (x). j Fig. 9 shows the analogous graphs for ¸4(x). Note in j the 4-cycle window near x"1 the graph of ¸4 (x) is 2 j concave up and the local minimum ¸4 (1) is a decreasing j2 function of j. This accounts for the forward bifurcation in this 4-cycle window of ¸ (x). j While we have seen the quartic map has both forward and reverse bifurcations, perhaps a tedious inductive argument would show the logistic map always exhibits forward bifurcations. However, this was not the approach taken by Milnor and Thurston, leading us to suspect the straightforward argument would not work. The algebraic bookkeeping is daunting, absent a clever organizational scheme, elusive so far. This quartic family has other interesting features, including di!erent limit sets for O`(1) and O`(1). We j 4 j 2
Fig. 7. The graph of q4(x) in the neighborhood [0.46, 0.53]][0.46, 0.53] of (1, 1), at the j values indicated on the right side of Fig. 6. j 22
Fig. 8. Left: the second derivative plot of q4(x) at x"1. Right: the value of q4 (1), as a function of j. j 2 j2
M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149
149
Fig. 9. Left: the second derivative plot of ¸4(x) at x"1. Right: the value of ¸4 (1), as a function of j. j 2 j2
wonder if still more complicated functions will reveal qualitatively di!erent behavior, or if the catalog of onedimensional dynamics now is complete.
For j"1 to Iter Do x"f (x) j Plot(j, x).
PseudoCode References Suppose f (x) is a family of functions with critical point j p (and maybe other critical points). To study the bifur1 cation diagram of f (x) in the range j )j)j , use j min max For i"1 to j number Do j"j
#i/(j !j ) min max min x"p 1 For j"1 to Drop Do x"f (x) j
[1] Milnor J, Thurston W. On iterated maps of the interval. In: Alexander J, editor, Dynamical systems. New York: Springer, 1988. p. 465}563. [2] Kan I, Koc7 ak H, Yorke J. Antimonotonicity: concurrent creation and annihilation of periodic orbits. Annals of Mathematics 1992;136:219}52. [3] Devaney R. An introduction to chaotic dynamical systems, 2nd ed. Redwood City: Addison-Wesley, 1989.