*Corresponding author. Tel.: #1-518-388-6246.

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, reversebifurcations have been known for the HeH non map for several years, but are not present in the logistic map. We usetrapping squares to explain these quartic reverse bifurcations. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Period-doubling bifurcation; Reverse bifurcation; Trapping square

1. Introduction

The familiar quadratic (logistic) map exhibits only twokinds of bifurcations: period doubling and tangent. The"rst are responsible for the `period doubling route tochaosa, the second for the formation of periodic windowsthe arise after the onset of chaos. In general, these pro-cesses are responsible for the stable (thus visible in thestandard computer experiments) periodic behavior of thelogistic map. For both types of bifurcations, once a cycleappears, it persists, even though eventually it becomesunstable. This persistence, established by Douady andHubbard, Sullivan, and Milnor and Thurston [1] iscalled monotonicity. For the HeH non map

HjAx

yB"Aj!x2#by

x B,Kan et al. [2] showed the x-coordinate bifurcationsexhibit antimonotonicity. That is, as the j parameterincreases some cycles disappear through reverse-bifurca-tion, the familiar bifurcations, but in reverse order.

Through computer experiments and graphical analy-sis, we discovered a similar antimonoticity for a realquartic function qj . Unlike the two-dimensional HeH nonmap, our example is one-dimensional. We emphasize the

one-dimensional logistic map does not exhibit reversebifurcations.

2. Quadratic bifurcations

Many of the interesting phenomena that occur indynamical systems are illustrated by the quadraticfamily, ¸j (x)"j )x ) (1!x). For any point x

03[0,1], the

forward orbit O`j (x0) is the set Mx

0,x

1"¸j (x0

),x2"¸j (x1

),2N of iterates of x0. It is well known [3]

that O`j (x0) remains bounded for j3[0,4], and for almost

all x0, O`j (x

0) evolves through a complicated collection

of sets as j increases from 0 to 4. For 0)j)1, theforward orbit converges to the "xed point 0. For j'1,the "xed point 0 becomes unstable, and for 1(j)3most 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 toa 2-cycle. As j continues to increase, the 2-cycle becomesunstable and a stable 4-cycle appears. This 4-cycle be-comes unstable and a stable 8-cycle appears. This se-quence of period-doublings continues until the Myerbergpoint, j+3.56994562, where the "rst chaotic forwardorbit appears. As j continues to increase to 4, an intricateinterleaving of chaos and stable cycles arise. This in-formation typically is assembled into a bifurcation dia-gram, plotting j on the horizontal axis and an approxi-mation of the limit points of O`j (1

2) on the vertical section

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

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 guaranteeschaos occurs on a `largea set of j values (technically, ona set of positive one-dimensional Lebesgue measure), the`amounta of chaos (the measure of this set) remainsunknown.

The choice of O`j (12) is dictated by Fatou's theorem,

that a stable cycle must be the limit set of the forwardorbit of some critical point. For each j, ¸j has exactlyone critical point, x"1

2, so O`j (1

2) will reveal any stable

cycle. Fatou's theorem implies for each j. ¸j has at mostone stable cycle. Of course, ¸j need not possess anystable cycle: for those j where O`j (1

2) is chaotic, there can

be no stable cycle. This is not the case for our quarticfamily.

Fixed points xf

of ¸j are the points of intersectionof the graphs of x

n`1"¸j(xn

) and xn`1

"xn, stable if

D(d/dx)¸j(x)Dxf

D(1. Similarly, points of an m-cycle for¸j(x) are "xed points of the composition ¸mj (x).

As mentioned earlier, chaos and stable cycles are inter-leaved for j beyond the Myerberg point. Fig. 2 showsa portion of the bifurcation diagram containing a stable4-cycle. The right top shows the familiar observation that

a small copy of the whole bifurcation diagram growsfrom each branch of the cycle. The bottom of the "gureshows magni"cations of the graphical iteration of ¸4j ina neighborhood of (1

2, 12). All these are in the square

[0.48, 0.52]][0.48, 0.52]. In addition, windows 1}5 indi-cate the trapping square determined by the "xed point, q,of ¸4j closest to 1

2. To construct this square, start from the

"xed point (q, q) on the diagonal and draw a horizontalline to the closest intersection (r, q) with the graph of ¸4j .Continue with the vertical line to (r, r) on the diagonal,then complete the square. Recall these are called trappingsquares because, so long as the graph only crosses thesquare at corners, an iteration entering the square cannever exit.

Note that dynamics represented in the trappingsquares of windows 1}5 in Fig. 2 correspond to those ofwindows 1}5 in Fig. 1. That is, in the j range of this4-cycle, the restriction of the graph of ¸4j to the trappingsquare is closely approximated by the graph of ¸j #ippedvertically. Whereas ¸j has a maximum at 1

2that increases

as j increases, in this 4-cycle range ¸4j has a minimumthat decreases as j increases. Trivial as it may seem, thisobservation is the key to understanding the anti-monotonicity of the quartic map.

144 M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149

Fig. 2. Top left: the part of the logistic bifurcation in the range 3.959)j)3.963 and 0)xn)1; right: the range 3.96)j)3.9625 and

0.464)xn)0.548. Bottom: the graphical iteration pictures of ¸4j corresponding to the j values indicated on the top right.

Fig. 3. The graph of xn`1

"qj (xn) for j"100, together with

the diagonal xn`1

"xn.

3. A quartic family

Motivated by the implications of Fatou's theorem, wesought a simple function with critical points havingdistinct future orbits, and possibly distinct limit sets.Symmetry considerations suggested a function with de-rivative a multiple of (x!1

4) ) (x!1

2) ) (x!3

4). To obtain

a family of functions qj :[0,1]P[0,1] with all qj(0)"qj(1)"0, we took (Fig. 3)

qj(x)"!j )x )A1

4x3!

1

2x2#

11

32x!

3

32B.Note q

1(14)" 9

1024, so to guarantee qj :[0,1]P[0,1], we

restrict j to the interval 0)j)10249

+113.778. Also,note qj(14)"qj (34), so each qj has two critical oribts,O`j (1

4)

and O`j (12).

4. A quartic bifurcation diagram and reversebifurcation windows

Because it has two critical orbits, qj(x) has two bifurca-tion diagrams, one consisting of the limit points of O`j (1

4),

the other of the limit points of O`j (12). These diagrams

exhibit many similarities, but also some interesting di!er-ences. The left side of Fig. 4 shows a portion of thebifurcation diagram of O`j (1

4). For each j value, iterations

101}200 of the orbit of 14are plotted in red. To emphasize

M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149 145

Fig. 4. Left: a portion of the bifurcation diagram of qj (x) for the orbit of 14, together with the "rst few iterates of the critical point. Right:

the same diagram for the orbit of 12; the small box is magni"ed below.

the role of the critical orbit, iterations 1}10 of 14appear as

blue curves. Note the prominent 2-cycle window.The right side of Fig. 4 shows the same portion of the

bifurcation diagram of O`j (12) (in green); iterations 1}10 of

12

appear as blue curves. Note that how much moreintricate these curves are in the range where the orbit of 1

4converges to a 2-cycle. In particular, the small boxedregion, magni"ed below, contains a complete quadraticbifurcation diagram. Consequently, there are j valueswhere one critical point converges to a cycle, and the

other to chaos. While this is interesting, the magni"ca-tion itself is a surprise. Note that the bifurcation diagramis oriented in the opposite way from the logistic diagram.In particular, this diagram appears to exhibit the samesort of reverese bifurcations seen in the HeH non map. Ourgoal here is to investigate this phenomenon in the bifur-cation diagram of O`j (1

2).

We begin with the whole diagram (the left side of Fig. 5),then magnify to "nd the largest reverse bifurcation. Theright side of Fig. 5 shows the portion 110)j)1024

9.

146 M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149

Fig. 5. Left: the bifurcation diagram of qj (x) for the orbit of 12. Right: the portion of this diagram in the range 110)j)1024

9.

Fig. 6. Left: a magni"cation of the right of Fig. 5. Right: a magni"cation, 0.45)xn)0.55 of the bottom branch from the large window

on the left.

This diagram bears some resemblance to the generalform of the quadratic bifurcation diagram, but the deepermagni"cation, 111.5)j)111.9, shown in Fig. 6 revealsa qualitatively di!erent structure.

On the left, we see what appears to be a bifurcationpattern arising from a 4-cycle window. Magnifying thebottom branch of this picture reveals a copy of thequadratic bifurcation diagram, but oriented in the oppo-site direction. That is, the bifurcation diagram of qj (x)exhibits the antimonotonicity observed for the Henonmap in [2]. In the next section we shall see how thisreverse bifurcation diagram arises.

5. The source of antimonotonicity

Because the antimonotonicity pictured arises in a4-cycle window in the bifurcation diagram of 1

2, we shall

investigate the graphical iteration of the fourth compositionq4j(x) in a neighborhood of 1

2. In Fig. 7 the pictured graphs

are for j"111.64, 111.65, 111.68, 111.70, 111.72, and111.76. In all except the last, we show the trapping squaredetermined by the "xed point nearest and less than 1

2.

The essence of the reverse bifurcation sequence is thatas j increases in this range, q4j(12) decreases, yet near 1

2the

graph of q4j(x) is concave down. In fact, the part of the

M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149 147

Fig. 8. Left: the second derivative plot of q4j(x) at x"12. Right: the value of q4j (12), as a function of j.

Fig. 7. The graph of q4j (x) in the neighborhood [0.46, 0.53]][0.46, 0.53] of (12, 12), at the j values indicated on the right side of Fig. 6.

graph of q4j (x) within this trapping square is well-ap-proximated by the quadratic map in the unit square. Wesee parts of the familiar quadratic bifurcation sequence* "xed point to 2-cycle to 4-cycle to 2 to chaos toescape * repeated here, but as j decreases, not as j in-creases. Compare with the bottom of Fig. 2. This is theimmediate reason for the reverse bifurcation sequence inthis quartic family. Now, we turn to the question of whythis family behaves this way, while the quadratic familydoes not.

On the left of Fig. 8 we see the graph of(d2/dx2)q4j (x)D

x/1@2for 110)j)113.778, showing the

graph of q4j(x) is concave down at x"12for j'105.4. On

the right we see the graph of (d/dj)q4j (12) in the samej range. Combining these, we see indeed in the range ofthe 4-cycle window that q4j(x) is concave down aroundx"1

2and q4j(12) is a decreasing function of j. This combi-

nation of concavity and decreasing local maximum giverise to the reverse bifurcation sequence of qj(x).

Fig. 9 shows the analogous graphs for ¸4j(x). Note inthe 4-cycle window near x"1

2the graph of ¸4j (x) is

concave up and the local minimum ¸4j (12) is a decreasingfunction of j. This accounts for the forward bifurcation inthis 4-cycle window of ¸j (x).

While we have seen the quartic map has both forwardand reverse bifurcations, perhaps a tedious inductiveargument would show the logistic map always exhibitsforward bifurcations. However, this was not the ap-proach taken by Milnor and Thurston, leading us tosuspect the straightforward argument would not work.The algebraic bookkeeping is daunting, absent a cleverorganizational scheme, elusive so far.

This quartic family has other interesting features, in-cluding di!erent limit sets for O`j (1

4) and O`j (1

2). We

148 M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149

Fig. 9. Left: the second derivative plot of ¸4j(x) at x"12. Right: the value of ¸4j (12), as a function of j.

wonder if still more complicated functions will revealqualitatively di!erent behavior, or if the catalog of one-dimensional dynamics now is complete.

PseudoCode

Suppose fj (x) is a family of functions with critical pointp1

(and maybe other critical points). To study the bifur-cation diagram of fj(x) in the range j

min)j)j

max, use

For i"1 to j number Do

j"jmin

#i/(jmax

!jmin

)

x"p1

For j"1 to Drop Do

x"fj (x)

For j"1 to Iter Do

x"fj (x)

Plot(j,x).

References

[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: concurrentcreation and annihilation of periodic orbits. Annals ofMathematics 1992;136:219}52.

[3] Devaney R. An introduction to chaotic dynamical systems,2nd ed. Redwood City: Addison-Wesley, 1989.

M. Frame, S. Meachem / Computers & Graphics 24 (2000) 143}149 149