Chaos, Solitons and Fractals 31 (2007) 1191–1202
www.elsevier.com/locate/chaos
Super-attracting cycles for the cosine-root family
D.A. Brown
a
a,*
, M.L. Halstead
b
Ithaca College, Department of Mathematics, Ithaca, NY 14850, USA
b
Havertown, PA 19083, USA
Accepted 19 October 2005
Communicated by Prof. M. S. El Naschie
Abstract
pffiffi
We investigate the dynamics of the cosine-root family, C k ðzÞ ¼ k cos z, where k 2 C. When k 2 R, we focus on the
distribution of super-attracting cycles associated to the two critical values. In particular, we locate the parameters leading to super-attracting three cycles for one of the critical values while the other critical value is attracted to a fixed point.
These results are used to verify observations made upon viewing Julia and parameter plane pictures.
2005 Elsevier Ltd. All rights reserved.
1. Introduction
pffiffi
In [4], the dynamics of the family of transcendental functions C k ðzÞ ¼ k cosð zÞ are discussed from the point of view
of computer generation of dynamical and parameter plane pictures. This family has also been mentioned by McMullen
[5] with regards to functions whose Julia sets have positive measure while not being the entire plane. Not much else has
appeared about this family. In this paper, we further explore the dynamics of this family, focusing on attracting cycles
for various values of k. In particular, we locate values of k for which the critical orbits of Ck(z) are attracted to distinct
attracting cycles, identifying these parameters among those that yield super-attracting cycles. Mentioned briefly in [4],
we develop the machinery to find those parameters leading to super-attracting three cycles for one of the critical values
while the other critical value is attracted to a fixed point. Several papers and books investigate the location of parameters exhibiting interesting attracting behavior for polynomial [1,6–9,13–15] and perturbed polynomial dynamics [10–
12]. We build on this work, extending to the transcendental case.
The interesting behavior of this family is highlighted by the existence of two critical orbits. The complex trigonometric functions k cos(z) and k sin(z) have infinitely many critical points, but only two critical values: ±k. However, for these
functions, the orbits of both critical values are determined by the orbit of k only. For the k cos(z) family, we have
k cos(k) = k cos(k) while k sin(k) = k sin(k). So, the orbit of k escapes to infinity if and only if the orbit of k
escapes to infinity for each of these families. Further, if k is in the basin of attraction of a cycle for k cos(z), then k
lies in the same basin. For the k sin(z) family, we have a symmetry between the orbit of k and k. That is, if k is attracted
*
Corresponding author.
E-mail addresses: [email protected] (D.A. Brown), [email protected] (M.L. Halstead).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2005.10.108
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D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
to a k-cycle, {f1, . . . , fk}, then k is attracted to the k-cycle {f1, . . . , fk}. Hence, for these families, we need only follow the orbit of one of the critical values in order to understand their dynamics.
pffiffi
In contrast, the critical values, ±k, for C k ðzÞ ¼ k cosð zÞ can have orbits that are independent of each other; that is,
they are attracted to distinct cycles. In this way, the dynamics of Ck contrasts with those of k sin z or k cos z in a fashion
similar to the way in which cubic polynomial dynamics contrasts with quadratic dynamics; see [7]. In fact, the orbits of
the critical values for the cosine-root family with fall into four categories related to attracting cycles:
(1)
(2)
(3)
(4)
both critical values escape to infinity;
one critical value escapes to infinity while the other is attracted to a cycle;
both critical values are attracted to the same cycle;
the critical values are attracted to distinct cycles.
Each of these behaviors will be illustrated in Section 2. We will focus on the convergence to distinct cycles as we
analyze the occurrence of super-attracting cycles along the real axis. In fact, we will show that this behavior occurs
at infinitely many places along the positive real axis. This is shown in Section 4. In Section 3, we look at the creation
of attracting two-cycles and the formation of saddle-node bifurcations.
Dynamical and parameter plane pictures in this paper were generated via a JAVA application written by the first
author; this application is available via email from the second author.
2. Escape criterion and dynamics
The Julia set of a given function f is defined as the set of points z 2 C on which the set of iterates, {fn}, fails to be a
normal family on each neighborhood U of z. The Julia set is also the closure of the set of repelling periodic points of f,
and in the case of entire transcendental functions, the Julia set is the closure of the points that escape to infinity [2]. In
pffiffi
pffiffiffiffi
the case of the function z7!k cosð zÞ, we follow [4] to say that z0 escapes if jIm zk j > 50 for some k < N, where N is the
k
maximum number of iterations of Ck and zk ¼ C k ðzÞ. One can easily check that
pffiffiffiffi
ðImðzk ÞÞ2
2500
jIm zk j > 50 () Reðzk Þ <
10; 000
giving a bounding parabola defining the escape locus. If for all points zk in the forward orbit of z0 we have zk inside our
bounding parabola, then we say that the orbit of z0 is bounded under iteration of Ck. See Fig. 1, which illustrates this
Fig. 1. Julia set for C1, with 400 6 Re(z) 6 2000 and jIm(z)j 6 1200.
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
1193
2
y
bounding parabola . That is, the points in color lie in the Julia set and clearly lie outside a parabola, x ¼ 10;000
2500,
defined by the points colored black.
We use the following algorithm to draw computer-generated Julia sets in order to observe the dynamics of a given
function Ck:
(1) Choose a rectangle in the complex plane.
(2) For each point z in the rectangle, examine the forward orbit of z, OðzÞ ¼ fz; C k ðzÞ; C 2k ðzÞ; . . . ; C Nk ðzÞg, where N is
some positive integer (normally we choose N between 50 and 1000).
(3) If every value in O(z) is inside the bounding parabola, color z black. Otherwise, color z some color according to
the smallest k such that Ckk(z) is not inside our parabola.
Once the picture is complete, we have a set of colored points that are in the Julia set; the black points represent those
points whose orbit is bounded under Ck. We must keep in mind that the calculations made to generate the image are
numerical and finite, and so can only give us an approximation of the Julia set.
Figs. 2–6 show Julia sets for various values of k. These values are chosen to highlight the fates of the orbits of the
pffiffi
critical values. In Fig. 2, the critical values ±5 both escape to infinity under iteration of z7! 5 cos z. When k = 3.5,
the critical values are both attracted to the two-cycle {11.33633,3.408340}. When k = 40 + i, we find that 40 + i is
attracted to the fixed point z = 40.60971 + .00003i, while the other critical value 40 i escapes. When k = 10,
the critical value k = 10 is attracted to the fixed point z = 9.99793, while k escapes. Finally, interesting behavior occurs
when k = 40.772, in which case k is attracted to the fixed point z = 40.60932 and k is attracted to the two-cycle
{40.77199, 12090.15069}. In Section 4, we describe how this last behavior occurs often in this family of functions.
Next, we begin to distinguish between these behaviors by examining parameter plane pictures.
2.1. Parameter plane
We generate parameter plane (the k plane) images by tracking the associated critical values in k-space using the following algorithm.
(1) Choose a rectangle in the k-plane.
(2) For each point in our rectangle, examine the forward orbits of ±k under Ck, using the same method as when we
generate Julia set images.
(3) Color the point k according to the fates of the critical orbits under Ck, as follows:
Fig. 2. Julia set for C5, with jRe(z)j 6 100 and jIm(z)j 6 100. Both ±k escape.
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D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
Fig. 3. Julia set for C40+i, with jRe(z)j 6 100 and jIm(z)j 6 100. k has a bounded orbit, while k escapes.
Fig. 4. Julia set for C10. k has a bounded orbit, while k escapes.
•
•
•
•
If
If
If
If
both k and k escape, then color k white.
k escapes while k has a bounded orbit, then color k red.
k has a bounded orbit while k escapes, then color k green.
both k and k have bounded orbits then color k black.
Fig. 7 shows a large view of the parameter plane. The quadratic-like behavior of the cosine-root family leads to the
appearance of green, red, and black Mandelbrot-like sets. A blow-up of the plane near the origin reveals a complicated
Mandelbrot-like structure, see Fig. 8. In this blow-up, the black points again represent those parameter values for which
the critical values are both bounded.
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
1195
Fig. 5. Julia set for C3.5. Both critical points are attracted to the same two-cycle.
Fig. 6. Julia set for C40.772. One critical value is attracted to a two-cycle and the other is attracted to a fixed point.
Fig. 7. Parameter plane of Ck, with jRe(k)j 6 100 and jIm(k)j 6 100.
Fig. 9 shows the parameter plane near k = 40, revealing the interesting phenomenon of the appearance of quadraticlike Julia sets, first observed in [4]. An extreme blow-up near k = 40 reveals a tiny Mandelbrot set nestled in amongst
these Julia-like sets; see Fig. 10. Parameters chosen from such black Mandelbrot sets appearing in these larger green
Mandelbrot sets leads to the distinct critical orbits.
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D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
Fig. 8. Parameter plane of Ck, with 2.2 6 Re(k) 6 7.8 and jIm(k)j 6 5.
Fig. 9. Parameter plane near k = 40.
2.2. Super-attracting fixed points
In this section, we prove results suggested by the parameter plane images and characterize fixed points of Ck when Ck
has a super-attracting fixed point.
When k 2 R; C k : R ! R, can be written as
(
C k ðxÞ ¼
pffiffiffi
k cos x;
pffiffiffiffiffi
k cosh jxj;
x P 0;
x < 0:
pffiffiffi
Proposition 1. If k P 0, then the orbit of every x 2 R under C k ðxÞ ¼ k cos x is bounded.
ð1Þ
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
1197
Fig. 10. Parameter Space of Ck, with 40.7675 6 Re(k) 6 40.7775 and jIm(k)j 6 .005.
pffiffiffi
pffiffiffi
Proof. If x P k, then k 6 C k ðxÞ 6 k cosh k. If x < k, then Ck(x) > 0 P k; so, k 6 C nk ðxÞ 6 k cosh k for all
n P 1. h
This confirms the observation that, in the parameter plane, the positive real k-axis is colored black. See Fig. 7.
Proposition 2. hIf k < 2, then theiorbit under Ck of every x < 0 is unbounded. Moreover, if for any k 2 N, x falls in the
2 4k1 2
closed interval 4k3
, then the orbit if x is unbounded.
2 p ;
2 p
Proof. First, suppose x < 0. Then
pffiffiffiffi
ffiffiffiffi
pffiffiffiffi pffiffiffiffi pffiffiffiffi
pffiffiffiffiffi k pjxj
e
þ e jxj < e jxj þ e jxj < e jxj < x 1.
C k ðxÞ ¼ k cosh jxj ¼
2
By induction, we find that C kk ðxÞ < x k. Hence, the sequence fx; C k ðxÞ; C 2k ðxÞ; C 3k ðxÞ; . . .g is decreasing and
unbounded.
h
2 4k1 2 i
The second case, with x 2 4k3
for some k 2 N, follows directly from the first: let y ¼ C k ðxÞ ¼
2 p ;
2 p
pffiffiffi
k cos x < 0. The orbit of y is unbounded, and therefore the orbit of x is unbounded. h
This also confirms an observation stemming from the parameter plane. When k < 2, the orbit of k under Ck is
unbounded. Therefore, we never see black or green points on the negative real k-axis (to the left of 2) (see Fig. 7).
The next result is useful in characterizing fixed points of Ck.
Lemma 3. Suppose p1, p2, and k are real numbers satisfying
(1) 0 < p1 < p2,
(2) Ck(p1) and Ck(p2) are positive, and
(3) jC 0k ðp1 Þj 6 jC 0k ðp2 Þj 6¼ 0.
Then Ck(p1) > Ck(p2).
Proof. We have
ffi
ffi
k sin pffiffiffiffi
0
k sin pffiffiffiffi
p1 0
p2 ¼ C k ðp1 Þ 6 C k ðp2 Þ ¼ pffiffiffiffiffi
ffi
2pffiffiffiffi
p1 2 p2 and since C 0k ðp2 Þ 6¼ 0,
1198
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
pffiffiffiffiffi
j 2k p1 j
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
j sin p1 j 6
pffiffiffiffiffi j sin p2 j < j sin p2 j.
j 2k p2 j
pffiffiffiffiffi
pffiffiffiffiffi
Now suppose j cos p1 j 6 j cos p2 j. This tells us that
1 ¼ sin2
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
pffiffiffiffiffi
p1 þ cos2 p1 < sin2 p2 þ cos2 p2 ¼ 1
pffiffiffiffiffi
pffiffiffiffiffi
thus j cos p1 j > j cos p2 j. Then
pffiffiffiffiffi
pffiffiffiffiffi
C k ðp1 Þ ¼ jC k ðp1 Þj ¼ jk cos p1 j > jk cos p2 j ¼ jC k ðp2 Þj ¼ C k ðp2 Þ:
Now, we characterize the fixed points of Ck when Ck has a super-attracting fixed point. The k values yielding these
super-attracting fixed points are found by solving the simultaneous transcendental equations
C k ðzÞ ¼ z;
C 0k ðzÞ ¼ 0.
Theorem 4. If k = (2np)2, with n 2 N, then Ck has exactly 2n + 1 fixed points {xij0 < x1 < x2 < x3 < < x2n + 1 = k},
where x2n+1 is super-attracting and the rest are repelling. Similarly, if k = ((2n 1)p)2, with n 2 N, then Ck has exactly
2
2n fixed points fxi j p4 < x1 < x2 < < x2n ¼ jkjg, where x2n is super-attracting and the rest are repelling.
Proof. Suppose k = (2np)2. It is easy to check that Ck(k) = k and C 0k ðkÞ ¼ 0, giving us a super-attracting fixed point at
x = k. Since Ck(x) > 0 when x 6 0 and Ck(x) 6 x when x P k, any other fixed points of Ck must be contained in the
interval I = (0, k).
For j = 1, 2, 3, . . . , 2n 1, let pj = (jp)2. So Ck(pj) = ±k for all j; thus pj cannot be a fixed point. Divide I into 2n
disjoint open intervals {(0, p1),(p1, p2), . . . , (p2n1, k)}, knowing that every fixed point in I has to fall into one of these. Let
Ij = (pj1, pj) be any of these intervals. We see that Ck is either strictly increasing or strictly decreasing, with a range of
(k, k), on Ij. Define g(x) = Ck(x) x on Ij. Then the range of g must contain the interval (k pj1, k pj); thus the
Intermediate Value Theorem tells us that there is some xj 2 Ij such that g(xj) = 0, and so xj is a fixed point of Ck.
Since C 0k ðx2nþ1 Þ ¼ 0, we know the graph of Ck lies above the line y = x for all x in the open interval (x2n, x2n+1);
therefore C 0k ðx2n Þ P 1. Now suppose some fixed point 0 < xi < x2n is non-repelling; that is, jC 0k ðxi Þj 6 1. Then Lemma 3
tells us that Ck(xi) > Ck(x2n) = x2n > xi, and so xi cannot be a fixed point. Hence, xj, j = 1, 2, . . . , 2n 1, are repelling
fixed points.
We need to show that Ck has no more than 2n fixed points on the open interval (0, jkj). Suppose there are more than
2n fixed points; that is, some open Ij, as defined above, must contain two fixed points t1 < t2. We know that Ck is either
increasing or decreasing on Ij. If it were monotonically decreasing, we would have no fixed points. So Ck is increasing on
Ij. The mean value theorem now provides us with a t3 2 (t1, t2) such that C 0k ðt3 Þ ¼ 1. Since t2 cannot be an attracting
fixed point, we know that jC 0k ðt2 Þj > 1. Then the points t2 and t3 fit the criteria for Lemma 3, which tells us that
Ck(t2) < Ck(t3); but t2 > t3 and Ck is increasing. Therefore Ck has no more than 2n + 1 fixed points.
Now suppose k = ((2n 1)p)2. It is easy to check that Ck(jkj) = jkjand C 0k ðjkjÞ ¼ 0, giving us a super-attracting
2
fixed point at x = jkj. Since Ck(x) < x when x 6 p4 (this follows from the proof of Proposition 2) and Ck(x) < x when
2
x > jkj, any other fixed points of fk must be contained in the interval I ¼ p4 ; jkj .
The remainder of this proof is identical to the first case; the only difference is that p2n1 = jkj, and so there are only
2n 1 intervals, resulting in exactly 2n fixed points. h
Corollary 5. Let k0 be of the form (2np)2 or ((2n 1)p)2, where n is an integer. Then there exists a neighborhood D
around jk0j such that when k 2 D, the orbit of at least one critical value of the function Ck is bounded.
Corollary 5 tells us that in the parameter space for C k0 , there should be a region around jk0jwithout any white; this is
evident in Figs. 7–10.
3. Saddle-node bifurcations and two-cycles
2
2
2
2
2
2
For ease of notation, define two sets Sþ
1 ¼ fð2pÞ ; ð4pÞ ; ð6pÞ ; . . .g and S1 ¼ fp ; ð3pÞ ; ð5pÞ ; . . .g, representing the positive and negative k values for which Ck has super-attracting fixed points.
Theorem 4 relies on the periodicity of the cosine function. Now, as we increase the amplitude jkj, we see more phases
reaching and crossing the line y = x. When k takes some value in Sþ
1 or S1 , we see that y = x crosses the peak of one of
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
1199
our cosine phases, representing a super-attracting fixed point of Ck. Looking at the graphs of Ck near x = jkj, we see
that the function is nearly tangent to the line y = x; there is a repelling fixed point very close to k. This leads to the
^
^
observation that for every k in Sþ
1 or S1 , there is a nearby k, with jkj < jkj, such that C ^k has a fixed point and a sad^
^
dle-node bifurcation at k. However, the value of k is not easy to find; we would need to solve the simultaneous equations
^k cos pffiffiffiffi
x0 ¼ x0 ;
p
ffiffiffiffi
k^ sin x0
pffiffiffiffi ¼ 1;
2 x0
where x0 is some value slightly less than jkj. This is not easy to do by hand, p
soffiffiffiffiffiffiffiffiffiffiffiffiffi
we use a computer to find a numerical
approximation for ^k. First, we combine the equations to find that x0 ¼ 2 4 þ ^
k2 ; we then replace this for x0 to see
^
that k must be a solution to both of the equations
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 4 þ ^k2 ¼ ^k cos 2 4 þ k^2 ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
^
^
2 2 4 þ k ¼ k sin 2 4 þ ^k2 .
In addition to saddle-node bifurcations, we conjecture that every k in Sþ
1 or S1 has a nearby k2, with jk2j > jkj such
that jk2jis in a super-attracting period two-cycle under C k2 . To find these values of k2, we use a computer to find numerical estimates of the solutions to the simultaneous equations
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
C 2k2 ðjk2 jÞ ¼ k2 cos k2 cos jk2 j ¼ jk2 j;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
2
k
ðsin
jk
j
Þ
sin k2 cos jk2 j
2
0
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C 2k2 ðjk2 jÞ ¼
¼ 0;
pffiffiffiffiffiffiffi
4 k2 jk2 j cos jk2 j
which yields
ðsin
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffi
jk2 jÞðsin k2 cos jk2 jÞ ¼ 0.
Tables 1 and 2 show the approximate values of ^k and k2 associated with the k values which give super-attracting fixed
points.
Table 1
Saddle-node bifurcations and two-cycles for k > 0
k
2
(2p) 39.47841760
(4p)2 157.91367042
(6p)2 355.30575844
(8p)2 631.65468167
(10p)2 986.96044011
^
k
k2
37.46100104
155.90941637
353.30387576
629.65362423
984.95976381
47.02763326
165.78516725
363.24708948
639.62131805
994.93902676
^k
k2
7.79271815
86.81883152
244.73739492
481.60923370
797.43712132
16.61749015
96.60621288
254.65643566
491.56721848
807.41154236
Table 2
Saddle-node bifurcations and two-cycles for k < 0
k
2
p 9.86960440
(3p)2 88.82643961
(5p)2 246.74011003
(7p)2 483.61061565
(9p)2 799.43795649
1200
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
It appears that the attracting two-cycle for C k2 is the sequence {jkj, jk2j, jkj, . . .}, where k is the element of Sþ
1 [ S1
which is closest to k2.
We see that as we move k along the real axis, Ck, near its attracting fixed points, behaves much like the quadratic
map Qc(z) = z2 + c, with c 2 R: we see a saddle-node bifurcation (at ^
k) and attracting two-cycles (near k2). So it is not
too surprising that we should see Mandelbrot-like sets in the parameter plane of our complex function Ck [3].
4. Super-attracting three-cycles
We know from Sarkovskii’s theorem that cycles of prime period 3 are the ‘‘strongest’’; the existence of a period-3
cycle implies the existence of cycles of any other period. We examine a special case of these: the set of super-attracting
period-3 cycles, specifically those that contain k. That is, we would like to find those values of k for which
C 3k ðkÞ ¼ k, but Ck(k) 5 k.
Let S3 denote the set of all positive k such
pffiffiffi that Ck has a super-attracting three-cycle containing k.
Notice that when k > 0, C k ðkÞ ¼ k cosh k > 0 > k and so, Ck (k) 5 k. Therefore k 2 S3 if and only if k > 0
and C 3k ðkÞ ¼ k. That is, we do not need to worry about k being a fixed point.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
Proposition 6. For positive values of k, k 2 S3 if and only if k cos k cosh k ¼ ð2n þ 1Þ2 p2 , where n is an integer.
Proof. ()) Suppose k > 0 and C 3k ðkÞ ¼ C k ðC k ðC k ðkÞÞÞ ¼ k. So
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi
pffiffiffi
pffiffiffi
k ¼ C k ðC k ðk cosh kÞÞ ¼ C k k cos k cosh k ¼ k cos k cos k cosh k
with p
the
last line coming from the fact that Ck(x) is always positive when x < 0. Dividing by k, we see that
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
k cos k cosh k ¼ ð2n þ 1Þ2 p2 , p
where
np
2 ffiffiN.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
(() Suppose k > 0 and k cos k cosh k ¼ ð2n þ 1Þ2 p2 for some integer n. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
C 3k ðkÞ ¼ C k k cos k cosh k ¼ C k ðð2n þ 1Þ2 p2 Þ ¼ k cos ð2n þ 1Þ2 p2 ¼ k cosðð2n þ 1ÞpÞ ¼ k:
We now define the continuous real-valued function
gn ðkÞ ¼ k cos
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
k cosh k ð2n þ 1Þ2 p2 .
Each gn is defined for all non-negative real numbers. Also, we can look at the infinite set of these functions,
G ¼ fgn jn ¼ 0; 1; 2; 3; . . .g. We see that k 2 S3 if and only if k is a root of some gn 2 G.
Proposition 7. The family of functions G ¼ fgn g has the following properties:
(1)
(2)
(3)
(4)
Each gn is bounded by the lines y1 = k (2n + 1)2p2 and y2 = k (2n + 1)2p2.
Each gn exhibits cosine-like ‘‘cycling’’, touching y1 and y2 on each cycle.
The gn functions ‘‘nest’’ inside each other, never intersecting.
gn has no roots in the interval (0, (2n + 1)2p2) (see Fig. 11).
10
20
30
40
-50
-100
-150
-200
-250
-300
Fig. 11. The first few gn functions: g1 (top), g2 (middle), g3 (bottom).
D.A. Brown, M.L. Halstead / Chaos, Solitons and Fractals 31 (2007) 1191–1202
Proof. Properties
1
and
3
1201
follow
directly from the definition of gn. Property 2 follows because
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
gn(k) = k cos(h(k))(2n + 1) p , where hðkÞ ¼ k cosh k is a continuous, strictly increasing function. The fourth
property is obvious since gn 6 k (2n + 1)2p2. h
2 2
We wish to describe the distribution of the roots of G, since each one corresponds to a super-attracting three-cycle.
In order to do so, we partition the real line into intervals, each one containing a certain finite number of roots (from G).
pffiffiffi
Definition 8. Let b be a positive integer. Define xb to be the (positive) solution to k cosh k ¼ b2 p2 ; that is,
pffiffiffiffiffi
xb cosh xb ¼ b2 p2 .
Proposition 9. If a and b are positive integers with a < b, then xa < xb.
pffiffiffiffi
pffiffiffiffi
Proof. We have xa cosh xa ¼ a2 p2 < b2 p2 ¼ xb cosh xb . Since the cosh and square root functions are increasing,
we have xa < xb. h
When b is even, we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffi
gn ðxb Þ ¼ xb cos xb cosh xb ð2n þ 1Þ2 p2 ¼ xb cosðb2 p2 Þ ¼ xb ð2n þ 1Þ2 p2 .
Therefore gn touches its upper-bounding line at xb. Similarly, if b is odd then gn(xb) = xb (2n + 1)2p2; so gn touches
its lower-bounding line. This observation leads us to the next result.
Theorem 10. Let n and b be positive integers, with xb > (2n + 1)2p2. Then gn has exactly one root on the interval (xb, xb+1).
Proof. Suppose b is even. Then gn(xb) = xb (2n + 1)2 p2 > 0. and gn(xb+1) = xb+1 (2n + 1)2p2 < 0. So there must
be at least one root on the open interval (xb, xb+1). Note that
get the same result if b is odd. Uniqueness follows from
pwe
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
the periodicity of cosine coupled with the fact that hðkÞ ¼ k cosh k is a strictly increasing concave up function. h
Corollary 11. Let n, a, b be positive integers such that a < b and xa > (2n + 1)2p2. Then gn has exactly b a roots on the
closed interval [xa, xb].
Corollary
12. Let a,
lpffiffiffi m
jpbffiffiffibekpositive integers with a < b. The number of roots of all gn 2 G on the interval (xa, xb) is between
xb p
xa p
ðb aÞ and
ðb aÞ, inclusive.
2p
2p
lpffiffiffi m
jpffiffiffi k
xb p
xa p
Proof. Let n1 ¼ 2p
and n2 ¼ 2p
. That is, for every n < n1, gn crosses the k axis to the left of xa and so, by
Corollary 11, has exactly b a roots on the interval. Similarly, for every n > n2, gn < 0 on (xa, xb) and so has no roots
on the interval.
h
If we look closely at the parameter plane, we can find evidence of these super-attracting three-cycles. They appear as
red or black Mandelbrot sets, symmetric across the real axis, pointing to the right. Since the three-cycles become more
concentrated the further we travel out along the real axis (follows from Corollary 12), we can see that some of the centers of these Mandelbrot sets, which represent super-attracting three-cycles containing the critical value k, will eventually land in the main cardioids of the green Mandelbrot sets, which are distributed along the elements of Sþ
1 . These
values of k are examples of ‘‘independent’’ behavior of critical values. At these points, one critical value is attracted to a
fixed point while the other is attracted to a three-cycle.
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