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Proc. R. Soc. A (2007) 463, 1339–1358
doi:10.1098/rspa.2007.1826
Published online 6 March 2007
On detection of multi-band chaotic attractors
B Y V IKTOR A VRUTIN , B ERND E CKSTEIN
AND
M ICHAEL S CHANZ *
Institute of Parallel and Distributed Systems (IPVS), University of Stuttgart,
Universitätstrasse 38, 70569 Stuttgart, Germany
In this work, we present two numerical methods for the detection of the number of bands
of a multi-band chaotic attractor. The first method is more efficient but can be applied
only for dynamical systems with a continuous system function, whereas the second one is
applicable for dynamical systems with a discontinuous system function as well. Using the
developed methods, we investigate a one-dimensional piecewise-linear map and report for
both cases of a continuous and a discontinuous system functions some new bifurcation
scenarios involving multi-band chaotic attractors.
Keywords: multi-band chaotic attractors; piecewise-linear maps;
discontinuous maps; band merging; bandcount adding
1. Introduction
An aperiodic and especially a chaotic attractor may consist of some number
KR 1 of bands (also denoted as connected components). Multi-band chaotic
attractors (MBCAs), defined by KO 1, represent a well-known phenomenon in
the field of nonlinear dynamics and are often involved in several bifurcations,
such as, interior crisis (Grebogi et al. 1982). It is also well known that the perioddoubling cascade occurring in many dynamical systems is typically followed by
an inverse band-merging cascade (Collet & Eckmann 1980; Romeiras et al. 1988).
Whereas the first one is formed by a sequence of periodic attractors with periods
p0!2n with n increasing from zero to infinity, the second one represents a
sequence of MBCAs with p0!2n bands, whereby n decreases from infinity to
zero. Hereby, at each bifurcation point, the bands of a ( p0!2nC1)-band chaotic
attractor collide pairwise with each other and with a limit cycle, which become
unstable at the n th period-doubling bifurcation. The MBCA emerging at this
bifurcation has ( p0!2n) bands. A similar scenario is also known in the case of the
border-collision period-doubling scenario (Avrutin & Schanz 2004, 2005), which
is followed by an inverse band-merging cascade as well. In contrast to the bandmerging cascade described earlier, here the number of bands before the nth
bifurcation is 2nC1K1 and after the bifurcation it is 2nK1, with n decreasing
from infinity to zero. These bifurcation scenarios are well known and investigated
in detail. However, the question which other types of bifurcation scenarios
involving MBCAs exist, is still insufficiently investigated, whereby efficient
algorithms for investigation of MBCAs are missing. Considering a dynamical
* Author for correspondence ([email protected]).
Received 21 November 2006
Accepted 24 January 2007
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1340
V. Avrutin et al.
(a)
(b)
0.4
0.4
0
y
0
–0.4
–0.4
–1.5
–0.8
0
x
1.5
–0.4
–0.2
0
0.2
x
Figure 1. Examples of MBCAs. (a) The seven-band attractor of the Hénon map at aZ1.2668, bZ
0.3 and (b) the 22-band attractor of the Tinkerbell map at aZ0.9, bZK0.5169, cZ2, dZ0.5.
system with fixed parameters, we might be able to count the bands by simply
looking at a graphical representation of the attractor.
For instance, we would
assume that the attractor of the Hénon map xnC1 Z 1Kax 2n C yn ; ynC1 Z bxn
shown in figure 1a consists of seven bands and the attractor of the Tinkerbell
map ðxnC1 Z x 2n Ky 2n C axn C byn ; ynC1 Z 2xn yn C cxn C dyn Þ shown in figure 1b
consists of 22 bands. Obviously, such an assumption can be erroneous, especially
in the case when attractors close to a band-merging bifurcation are considered.
However, when dealing with chaotic attractors of a dynamical system under
variation of some parameters, their number of bands has to be detected
automatically. This seemingly simple question of how to determine the number
of bands of an MBCA automatically, represents in fact a hard task from the
numerical point of view.
The rest of the paper is organized as follows. Firstly, in §2, we present two
methods for numerical detection of the number of bands of an MBCA. The first
method is developed for dynamical systems with a continuous system function.
In this case, we are able to prove that the bands of an MBCA are visited by an
orbit in the same order for all times and therefore we are able to solve the given
task efficiently, with low requirements with respect to computation time and
memory. The second developed method does not use any assumptions related to
the properties of the system function and hence can be applied for dynamical
systems with a discontinuous system function as well. The price for this is the
slower convergence rate, when compared with the first method. Some
implementation issues and some typical problems, which may occur by
application of the developed methods, are discussed as well, but in order to
keep the presentation compact, these are moved to appendices C and D.
In the following sections, we discuss some application examples for the
developed methods. We consider a one-dimensional piecewise-linear map, whose
system function can be continuous or discontinuous depending on a parameter
(§3). This map, closely related to some models of electronic circuits of practical
interest (DC/DC converters and S/D modulators), was already investigated in
many works. However, until now only bifurcation scenarios involving nonchaotic attractors are considered. In this work, we report for the first time some
bifurcation scenarios occurring in the area of chaotic behaviour. In the case of the
continuous piecewise-linear map (§4), we explain the overall structure of this
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On detection of multi-band chaotic attractors
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area and describe where in the parameter space which chaotic attractors can be
found. In the more complex case of the discontinuous piecewise-linear map (§5),
we restrict ourselves mainly to one interesting bifurcation scenario, which we call
the bandcount-adding scenario. This scenario turns out to be similar to the usual
period-adding scenario, but in contrast to this one, involves no periodic but only
chaotic attractors. The presented results lead to numerous open problems, some
of them are briefly presented in §6.
Note that this paper does not pretend to explain the bifurcations leading to
occurrence of MBCAs. Instead, the goals of this paper are the first to present a
numerical framework for the detection of MBCAs and secondly to report some
typical scenarios where these attractors are involved in.
2. Numerical determination of the bandcounts
For the numerical determination of bandcounts (the number of bands of
MBCAs), we developed two methods that are described below. The first method
is based on the following theorem.
Theorem 2.1. Let A be an MBCA of a dynamical system with a continuous
system function. Let us assume that A consists of K bands B0 ; .; BKK1 . Then,
each band Bi ðiZ 0; .; KK1Þ has exactly one successor and one predecessor band,
c n 2N ;
ci Z 0; .; KK1;
x nK1 2A;
x n 2Bi 0 x nC1 2BðiC1Þmod K
and x nK1 2BðiK1Þmod K :
A sketch of the proof is presented in appendix A. Note that for dynamical
systems with a discontinuous system function such a theorem does not exist.
Using the same notation as for theorem 2.1, we state the following. For an orbit
started at a point x 0 2B0 , there exists some number N1 (first return time) with
jx 0 Kf ½N1 ðx 0 Þj! 3;
for an arbitrarily small 3. Let 3 be smaller than the distance between adjacent bands,
then f ½N1 ðx 0 Þ 2B0 . Hereby, due to theorem 2.1, the number N1 is a multiple of K.
Similarly, the second return time N2 for x 0 can be calculated, etc. Iterating the
system for a sufficiently long time, we obtain the set of M return times
N Z fNi ; ji Z 1; .; M g
with jx i Kf ½Ni ðx i Þj! 3;
whereby the number M can be arbitrarily high and is only limited by the
computation time. The key point is that all return times Ni are multiples of the
number of bands that meansciZ 1; .; M : Ni Z mi K with some integer number mi.
Then, for a sufficiently large set N , the number K can be calculated as the greatest
common divisor (GCD) of the return times N1 ; .; NM
K Z lim GCDðmi K; .; mM KÞ:
M/N
However, this kind of calculation is not practicable because it requires a very
large number of iteration steps. In order to make the calculation possible using a
realistic number of iteration steps, we can use the following idea. Let us consider a
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1342
V. Avrutin et al.
(a)
0.8
(b)
(c)
8
128
6
64
log2
32
x
0.5
4
16
8
2
4
2
0.3
0.5
a
0.8 0.5
a
0.8
0
0.5
1
a
0.8
Figure 2. Band-merging cascade of the tent map: (a) analytically calculated boundaries of chaotic
attractors, (b) numerically calculated bifurcation diagram and (c) bandcount diagram. Detected
are bandcounts K up to 128.
set S of M subsequent points from the attractor: SZ fx 0 ; .; x MK1 g 2A with
x iC1 Z f ðx i Þ. These points are denoted in the following as basis points. Within the
next I iteration steps, we find the first return times (for a given 3) for some of these
points and obtain the set N . Then, for a sufficiently large set S the number K can be
calculated as the GCD of all detected return times, whereby the number of the
required iteration steps I C M remains acceptable. Note that in this case, it is not
guaranteed that the number of determined return times is equal to M. For some of
the basis points more than one return time may be found, whereas for some other
point not a single one. In order to avoid this problem, it is possible to iterate until
exact M return times are found, instead of the fixed number of iteration steps I .
Hereby, the necessary number of iteration steps remains acceptable if a sufficiently
large set S is used. Technical details related to the implementation of this method
(denoted in the following as the GCD-based method) are discussed in appendix C.
An application example for the GCD-based method is shown in figure 2. Here, the
well-known tent map xnC1 Z að1K2jxn K1=2jÞ is considered, which shows in
the interval a 2½1=2; 1 a band-merging cascade. From this sequence Km Z 2m of
bands, we detected numerically the bandcounts up to K7 Z 128. As one can see, the
results obtained using the GCD-based method (figure 2c) are in a good accordance
with the usual bifurcation diagram (figure 2b) and with the boundaries of the
chaotic attractors (figure 2a) calculated analytically using the standard technique
based on kneading orbits (Collet & Eckmann 1980; Jensen & Myers 1985; Milnor &
Thurston 1987).
As follows from the proof of theorem 2.1, for dynamical systems with a
discontinuous system function the GCD-based method cannot be applied.
Therefore, we use a different idea which does not require any assumptions related
to the properties of the system function. The technique is similar to the boxcounting
techniques, often applied to calculate numerically the invariant measure of
attractors and for other purposes. The area in the state space where the attractor is
located in is subdivided in small partitions (boxes). The bands of the attractor
correspond to clusters of adjacent boxes separated from each other by empty boxes.
Then, the number K is calculated as the number of these clusters. Therefore, the
area of the state space where the attractor is located in is covered by a uniform grid
with p partitions in each direction. The size of a single partition in this grid
represents a parameter of the method and has to be sufficiently small (smaller than
the distance between neighbouring bands of the attractor). Then, we calculate I
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On detection of multi-band chaotic attractors
1343
points of the attractor and mark the partitions, where these points are located in.
After that the clusters consisting of marked partitions are determined, whereby two
partitions are assumed to belong to the same cluster, if they have at least a common
corner (so-called ‘Moore neighbourhood’). Finally, the number K of clusters is
counted. Note that this method requires a higher number of iterations I than is the
case for the GCD-based method. If I is set too small, a band may split apart and
hence will be counted more than once. Other technical details related to the
implementation of this method (denoted as the boxcounting-based method) are
discussed in appendix C.
3. Piecewise-linear map
A dynamical system showing several bifurcation scenarios, where MBCAs are
involved in, is the one-dimensional
( piecewise-linear map given by
if xn ! 0;
a~xn C m~
xnC1 Z
ð3:1Þ
~ n C m~ C l if xn O 0:
bx
This map is studied in many works (Jain & Banerjee 2003; Avrutin et al. 2006;
Hogan et al. 2006) and actually considered by many authors as some kind of normal
form of the discrete time representation of many non-smooth systems of practical
interest in the neighbourhood of the point of discontinuity. In general, the system
function (3.1) is discontinuous, and the gap at the discontinuity point xZ0 is given
by the parameter l. Using a suitable scaling, it can be shown that system (3.1) can be
reduced to three cases, lZK1, lZ0 and lZ1.
In the following, we use the parameter transformation
~
a Z arctanð~
a Þ; b Z arctanðbÞ;
m Z arctanðm~Þ;
ð3:2Þ
3
which maps the infinite parameter space R onto the finite box ½Kp=2; p=23
preserving the topological structure of the parameter space. Although this
parameter mapping is not significant from the mathematical point of view, it is
preferable to use this mapping for a better graphical representation of the
parameter space especially for parameter values tending to GN.
In the following, Os denotes the periodic orbit corresponding to the symbolic
sequence s, consisting of symbols L (for a point x!0) and R (for xO0).
P s denotes the stability area of Os, which are bounded by the parameter
subspaces xi;d
s , where the ith point of Os collides with the border from the left
side or from the right side (d2{l, r}), and by the parameter subspace qs, where
this orbit becomes unstable. Note that at least for simple symbolic sequences like
LRn and Ln R, these curves can be determined analytically for all n as presented
in appendix B. Additionally, P n denotes the area in parameter space
corresponding to n-periodic dynamics (obviously, P s 3P n for nZ jsj) and Qn ,
the area corresponding to a chaotic n-band attractor.
4. Continuous piecewise-linear map
Let us start with the case lZ0. Since the system function (3.1) in this case is
continuous, we can use the GCD-based method to determine the number of
bands of MCBAs. Since the pioneer works (Nusse & Yorke 1992), the continuous
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V. Avrutin et al.
(a)
(c)
(b)
0
0
1
0
x
–25
–2
–3000
(d)
(f)
1
(e)
0
0
0
x
–14
–1
0
∼
m
1
–35
–1
0
∼
m
1
–3.5
–1
0
∼
m
1
Figure 3. System (4.1): transitions from a fixed point to a (a) five-periodic, (b) 12-periodic, (c) fourband chaotic, (d ) five-band chaotic, (e) 12-band chaotic and ( f ) one-band chaotic attractors.
The insets show blow-ups of the rectangles marked in the bifurcation diagrams. Parameter
~
~
~
settings: (a) a~Z 0:42; bZK26:42,
(b) a~Z 0:48; bZK3000,
(c) a~Z 0:445; bZK2:432,
(d)
~
~
~
a~Z 0:51789; bZK14:259, (e) a~Z 0:502; bZK31:351 and ( f ) a~Z 0:52; bZK4:33.
piecewise-linear map
(
xnC1 Z
a~xn C m~ if
xn ! 0;
~ n C m~
bx
xn O 0;
if
ð4:1Þ
which is identical with system (3.1) for lZ0 was studied by many authors
(Maistrenko et al. 1993, 1995; Nusse & Yorke 1995; di Bernardo et al. 1999;
Dutta et al. 1999; Zhusubaliyev & Mosekilde 2003). Especially, it is well known
that under variation of m~ this map demonstrates transitions from a fixed point to
several periodic dynamics and to one-band and MBCAs. Figure 3 shows some
typical examples for these bifurcations. Straightforward calculations show
hereby that the stable fixed point is destroyed at the bifurcation point via a
border collision. However, the seemingly obvious question, for which values of a~
and b~ a periodic (figure 3a,b) or a chaotic (figure 3c –f ) attractor emerges, is
according to our knowledge not investigated systematically until now. Related to
the chaotic attractors some further questions arise, like for instance, why in some
cases bands lie pairwise close to each other (figure 3c,e), whereas in other cases
they are approximately uniformly distributed in the state space (figure 3d )?
Researchers familiar with MBCAs may ask additionally, whether the attractors
shown in figure 3c are in fact four-band attractors. The reason for this question is
that they could also represent two-band attractors calculated with an insufficient
numerical accuracy. In this case, the points of the bands close to the unstable
two-periodic solution are detected only after a very large number of iterations.
Another question may be related with the invariant measure of the chaotic
attractors: why does the one-band attractor shown in figure 3f have significant
traces of a three-band attractor?
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On detection of multi-band chaotic attractors
(a)
x
3,l
3
q
2 x
2,l
2
(b)
q
0
12
o
10
2
–5
p,
x
–10
o
6
6
4
4
4
4
3
2
2
–15
–p/2
8
8
3
2
1
1
2
1
0
b
–1
b
Figure 4. System (4.1): (a) bifurcation diagram for aZ0.45, mZp/4. The inset shows a blow-up of
the marked rectangle. (b) Period p and bandcount K. Marked are some of the areas of periodic
attractors and MBCAs.
When dealing with these and similar questions, we state that system (4.1) can
be further reduced to three cases, m~ ZK1, m~ Z 0 and m~ Z 1, using a simple linear
transformation of the state variable and the parameters. This explains the
linearity of the bifurcation diagrams shown in figure 3 for both cases m~ ! 0 and
m~ O 0. For m~ Z 0, system (4.1) shows neither stable periodic nor chaotic dynamics
and therefore, this case is not relevant for the aims of our current work. Further,
~
the cases m~ ZK1 and m~ Z 1 are equivalent up to change of the parameters a~ and b.
For this reason, we restrict ourselves in the following to the case m~ Z 1. Related
to the original three-dimensional parameter space a~ ! b~ ! m~, it has to be kept in
mind that for m~ ! 0 the fixed point xL Z m~=ð1K a~Þ shown in the left parts of
figure 3 is stable in the area j~
a j! 1, i.e. jaj! p=4.
A typical bifurcation scenario, which can be observed for a fixed m~ O 0 under
variation of one of the remaining parameters a or b is shown in figure 4 (recall that we
use hereby and in the following parameter transformation (3.2)). It represents a
sequence of periodic dynamics with chaotic windows sandwiched in-between. Each
n-periodic attractor ORLnK1 is followed by a 2n-band chaotic attractor emerging at
the point qRLnK1 , where this n-periodic orbit becomes unstable. Next, we observe a
usual collision of the bands pairwise with each other and with the n-periodic orbit
ORLnK1 , which became unstable at the previous bifurcation. Hereby, an n-band
chaotic attractor emerges. This attractor exists until the next bifurcation, where it
is destroyed by the collision with another unstable n-periodic orbit OLRnK1 . At this
bifurcation, a one-band attractor emerges, which persists until the next stable
periodic orbit (namely, the orbit ORLn with period nC1) emerges. We summarize
this scenario in the following scheme:
P n / Q2n / Qn / Q1 / P nC1 //:
ð4:2Þ
This sequence represents an embedding of MBCAs in the period-increment scenario
with increasing periods and appears in figure 4 for decreasing values of the
parameter b which means from right to left. As already stated in Nusse & Yorke
(1995), depending on the parameters the scenario described earlier can be observed
either ad infinitum or in a truncated form. In the last case, for some n there are no
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V. Avrutin et al.
(a)
(b)
4
–1.1
2
6
8, 4
–1.49
3
10,
(d)
(c)
12,
b
(f )
14,
–π / 2
0
a
6
(a)
(e)
8
4
5
(d)
(a)
(e) –π / 2
0.7
0
0.5
7
(b)
a
Figure 5. System (4.1): bifurcation scenario (4.2). Horizontal arrows show the parameter settings
corresponding to figure 3. The rectangle marked in (a) is shown enlarged in (b). The dashed line
corresponds to the scenario shown in figure 4.
periodic windows with periods greater than or equal to n. In order to clarify the
question, under which conditions the scenario described earlier becomes truncated,
it is necessary to consider the bifurcation structure of the two-dimensional
parameter plane a!b.
Straightforward calculations show that in the parameter space a!b the areas
n
P RLn are bounded by the curves x0;r
RLn and qRL . Remarkably, for all n these
curves originate from the same point B 1 Z ð0;Kp=2Þ as shown in figure 5.
According to the notation introduced, for instance in Avrutin & Schanz (2006)
and Avrutin et al. (2006), this point represents a codimension-2 big bang
bifurcation point (defined as a bifurcation point, where an infinite number of
codimension-1 bifurcation curves intersect). Big bang bifurcations act typically
as organizing centres for stable periodic dynamics. For system (4.1), this
bifurcation organizes not only the areas of stable periodic dynamics but also the
areas of MBCAs. As shown in figure 5, for each n the areas Q2n and Qn originate
from this point as well.
Now, the truncation of the one-dimensional scenario presented earlier (figure 3)
can be explained. In order to do this, it is sufficient to note that the point of
the area P RLn with the maximal distance from the point B1, is the intersection
0;r
point cRLn Z xRL
n h qRLn . Additionally, we state that for increasing n the
sequence of points cRLn converges monotonously to the point ða ;Kp=2Þ with
a Z arctanð1=2Þ. Therefore, if one keeps a fixed to a value a!a and varies b
towards Kp/2, the areas P RLn are intersected for all n and the scenario (4.2)
takes place ad infinitum. In contrast to this, for aOa only a finite number of
areas P RLn can be intersected. Similarly, if b is fixed to a finite value and a is
varied, it is not possible to intersect all areas P RLn , so that scenario (4.2) can be
observed under variation of a in a truncated form only.
Note that the areas, Q2 and Q4 , shown in figure 5 play a special role
for system (4.1), because these areas represent the beginning of the usual
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On detection of multi-band chaotic attractors
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16 8
–π/4
2
4
b
div
1
3
6
–1.3
0.3
a
p /2
Figure 6. System (4.1): bifurcation scenario (4.3). The dashed line corresponds to the band-merging
scenario of the tent map (figure 2).
band-merging scenario
Q1 / Q2 / Q4 / Q8 / Q16 / Q32 //;
ð4:3Þ
as shown in figure 2 for the tent map. The occurrence of this scenario in system
(4.1) (figure 6) is not surprising, since the tent map is equivalent to system (4.1)
~
in the case a~ZKb.
Next, let us go back to the three-dimensional parameter space a~ ! b~ ! m~ of
system (4.1). Intersecting the plane m~ Z 0 for some value of a and b, we leave the
existence area of xL and reach the area of periodic and chaotic dynamics shown
in figure 5. Depending on the particular values of a and b, we may reach one of
the areas P RLn or QK from the scenarios (4.2) and (4.3) described earlier. In
some sense, figure 5 serves us as a road map, which describes, for which values of
a and b system (4.1) undergoes under variation of m~ a transition from the fixed
point xL to which attractor. Using this road map, the questions mentioned at the
beginning of this section can be easily explained. Indeed, in figure 3a,b, we
observe a transition from the stable fixed point xL to the periodic attractors
ORL4 and ORL11 , respectively. The corresponding parameter values are marked
in figure 5b with letters (a), (b) and lie in P RL4 and P RL11 , respectively. Note
that due to the used scaling of the parameter space, the area P RL11 is very thin
and therefore difficult to recognize. Owing to the fact that the location of the
areas P RLn is ordered according to n, it can be easily guessed where this area
lies. The chaotic attractors shown in figure 3c,e belong to areas Q4 and Q12 ,
whereby the parameter values lie close to the areas Q2 and Q6 , as marked in
figure 5a,b with letters (c) and (e). For this reason, the bands of these attractors
lie pairwise close to each other. However, the attractors shown in figure 3c are in
fact four-band attractors and not two-band attractors, as one may assume.
Similarly, the one-band attractor shown in figure 3f turns out to be located in
the parameter space close to the bifurcation line between areas Q3 and Q1 ,
which explains the traces of a three-band attractor in the distribution of its
invariant measure (figure 7).
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V. Avrutin et al.
(a)
1.0
(b)
(c)
1
0.5
x
0
0
– 0.5
0.83
1 0.4
b
m
0.75
–1
0
m
0.3
Figure 7. System (3.1): some bifurcation scenarios observed under variation of one parameter
across the area Q of chaotic dynamics. (a) aZK0.73, mZ0.36; (b) aZK0.73, bZ0.83; and
(c) aZK0.73, bZ0.86.
Finally, note that due to the symmetry of the cases m~ ZG1 exact analogous
~ as well.
results can be obtained for the other fixed point xR Z m~=ð1K bÞ
5. Discontinuous piecewise-linear map
Next, let us consider multi-band attractors of system (3.1) in the case lZK1.
Since the system function is discontinuous in this case, we have to use the
boxcounting-based method for the determination of bandcounts. Numerical
experiments show that the bifurcation structure of the area of chaotic dynamics
is in this case much more complex than in the case of the continuous piecewiselinear map discussed in §4. A detailed investigation of this structure is far beyond
the scope of this paper, so that we restrict ourselves in the following to one
example case, namely the investigation of the plane b!m for aZK0.73. For this
example, we report some specific bifurcations and bifurcation scenarios, which
require a more elaborate investigation in the future. Note that the phenomena we
present in the following can be observed at least for all values Kp/4!a!0,
whereby for a close to Kp/4 their investigating is more simple from the
numerical point of view. Owing to the symmetry of system (3.1), the same
scenarios take place in the plane a!m for Kp/4!b!0.
It can be shown that the only stable periodic orbits of system (3.1) in the
parameter subspace we investigate are OLRn . Hereby, for each n, the areas P LRn and
P LRnC1 overlap pairwise as shown in figure 8, so that the corresponding attractors
coexist. The overall area of periodic dynamics is separated from the area of chaotic
n
dynamics Q by the line, consisting of pieces of curves xl;0
LRn and qL R . Outside the
parameter area gN
nZ1 P LRn g Q, system (3.1) shows divergent behaviour. Now the
question arises, how the area Q is organized and which MBCAs can be found within?
Some examples of bifurcation scenarios observable under variation of parameters
across the area Q are shown in figure 7. Unfortunately, it is a hard task to explain the
interior structure of the area Q based only on the one-dimensional parameter scans
like the ones presented in this figure.
Considering the structure of the two-dimensional parameter space, we are able to
explain some of the observed phenomena (figure 9). Especially, we state that along
the line qLRnK1 , where the orbit OLRnK1 becomes unstable, a 2(nC1)-band attractor
emerges. As in the case of the piecewise-linear continuous map, this attractor
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div
0.6
0.4
0.2
0
0.8
1.0
b
Figure 8. System (3.1): areas of periodic P RLn , chaotic Q and divergent P div behaviour in the plane
b!m P LRn is bounded from the right side by the stability boundary qLRn . From above and below,
r;n
P LRn is bounded by the existence boundaries xl;0
LRn and xLRn , respectively.
undergoes a band-merging bifurcation, where it is replaced by an (nC1)-band
attractor. Remarkably, both areas Q2ðnC1Þ and QnC1 are bounded by the curves
xnC1;r
and xnC2;r
, where the unstable orbits OLRnC1 and OLRnC2 are destroyed by a
LRnC1
LRnC2
border-collision bifurcation. This scenario can be summarized as follows:
P n / Q2nC2 / QnC1 / Q1 :
ð5:1Þ
This structure is repeated for all nO1, resulting in the self-similarity of the
parameter space. However, the sequence (5.1) represents only a rough
approximation of a more complex phenomenon. As shown in figure 9, for each n,
both areas Q2nC2 and QnC1 are interrupted by some regions QK with higher
bandcounts K.
In order to keep the presentation compact, we restrict our considerations in
the following to the areas Q2nC2 . Note that the structure of the areas QnC1
represents an interesting topic as well and will be considered in future work. As
an example, in figure 10, the structure of the area Q2nC2 for nZ3 is shown (i.e.
the area Q8 , which begins at the line qLR2 ). The bandcount of this area will be
denoted as the basic bandcount K3b Z 8. As shown in figure 10, the most expanded
areas, which interrupt the area Q8 , form the sequence with bandcounts 22, 28,
34, etc. This sequence can be expressed in a closed form as
Knm Z 2Knb C mDKn Z 2Knb C mðKnb K2Þ;
ð5:2Þ
with mZ1, 2, 3, . for nZ3. Note that this equation holds for all Knb Z 2nC 2 and
may be rewritten as Knm Z 2nðmC 2ÞC 4 with mZ1, 2, 3, . and with nZ1, 2, 3, ..
However, the sequence (5.2) still does not describe the complete structure of
the area Q2nC2 . Using a sufficiently high resolution in the parameter space, we
state that between each two areas QKnm and QKnmC1 there exists the area QK0 of
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(b)
div
12
q
7
4
0.6
10
q
6
3
8
5
H1–5–8–4
m 0.4
(a)
(c)
0.2
4
6
q
2
H1–4–6–3
q
0
1
3
1
0.8
b
Figure 9. System (3.1): bifurcation structure of the plane b!m between the area of periodic
div
behaviour gN
. The dashed lines marked with (a), (b) and (c)
nZ1 P LRn and divergent behaviour P
correspond to the one-dimensional scans shown in figure 7. The rectangle marked bold is shown
enlarged in figure 10.
attractors with
ð5:3Þ
K 0 Z Knm C KnmC1 KKnb ;
00
bands. Similarly, between QK 0 and QKnm there is the area QK00 with K Z 2Knm C
KnmC1 K2Knb bands, whereas between QK0 and QKnmC1 the area QK00 with K 00 Z
Knm C 2KnmC1 K2Knb bands is located. This scenario continues further, but in
contrast to equations (4.2), (5.1) and (5.2) it cannot be expressed in closed form
by an equation. Instead, it is governed by an infinite-adding structure like the
period-adding scheme (Avrutin & Schanz 2006) and similar to the well-known
Farey trees (Lagarias & Tresser 1995; figure 11). Therefore, we denote it as the
bandcount-adding scenario. As shown in several works, period-adding structures
occur in many applications. Now, we report for the first time that a very similar
scenario formed by MBCAs exists as well. The only difference between the
period- and bandcount-adding schemes is that in the first case the periods of a
child sequence is the sum of the periods of two parent sequences, whereas in the
second case the bandcounts of the parents will be added and additionally the
basic bandcount will be subtracted.
A typical result of the numerical investigation of the bandcount scenario using
the boxcounting-based method is shown in figure 12. Here, the parameters are
varied along the curve bZ Rb cosð4Þ, mZ Rm sinð4Þ with R bZ0.001 and R mZ0.002
around the codimension-2 bifurcation point qLR2 h x0;l
LR , where all areas forming
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(a)
(b)
0.543
58
28
52
82
8
46
62
40
96
42
34
90
66
56
28
Q54
70
22
42
22
34
62
120
56
94
154
30
44
68
160
38
100
132
0.452
0.813
40
0.829
b
Figure 10. System (3.1): (a) bandcount-adding bifurcation scenario. Some areas from the first three
layers of the bandcount-adding scheme are marked. The bold curve around the codimension-2
bifurcation point qLRn h x0;l
LR shows the parameter values corresponding to the one-dimensional
parameter scan shown in figure 12. (b) Two examples for the first layers of the bandcount-adding scheme.
Figure 11. First layers of the infinite-adding scheme generating bandcounts of the MBCAs within
the bandcount-adding scenario.
the bandcount-adding scenario originate. The bandcounts corresponding to the
first five layers of the bandcount-adding scheme are detected well up to the
maximal considered value, which in this case is 512. Remarkably, in order to
calculate this figure with the presented quality, for each parameter value the area
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V. Avrutin et al.
500
400
5
4
300
3
2
200
1
(b)
(a)
100
0
p/8
p/4
j
3p/8
Figure 12. System (3.1): bandcount-adding scenario along the curve marked in figure 10. The grey
lines mark bandcounts which correspond to the first five layers of the bandcount-adding scheme
(the layers are marked with boxed numbers). The rectangles marked with (a) and (b) are shown
enlarged in figure 13.
of the state space containing the attractor was covered by 5!106 partitions. This
fine partitioning requires a corresponding high number of iteration steps (used
value: 3!108 steps). Note that performing the same calculation with lower
accuracy (namely, 106 partitions, what seems to be quite fine, and 6!107
iteration steps), we observe that the fifth layer bandcounts are detected correctly
only up to some critical parameter value. Beyond this value, the bandcounts
detected numerically decrease instead of increasing further. This represents an
error, caused by the fact that some bands of attractors corresponding to fifth
layer bandcounts are separated from each other by gaps, which are smaller than
the used partition size (see appendix D for more details).
It may be surprising, but the bifurcation structure of the area Q2nC2 is still not
completely described by the bandcount-adding scenario. Using a sufficiently high
resolution, we state that within areas forming this scenario some further areas
with higher bandcounts exist. An example of this structure is shown in figure 13.
As one can see, within the area Q22 we observe a symmetrical structure
consisting of two sequences of areas Q64 , Q78 , Q92 similar to sequence (5.2). The
structure within the area Q28 is identical to this structure up to the scaling in the
parameter space and the bandcount values. Hereby, there is some numerical
evidence that the parameter space between these areas is also organized by the
bandcount-adding scenario. However, this question has to be investigated in
more detail in future work.
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(b)
(a)
150
100
75
100
50
50
25
0
0.4206
j
0.4503
0
0.7156
0.733
j
Figure 13. System (3.1): identical interior structures of area (a) Q22 and (b) Q28 .
(a)
(b)
1
1
x 0
0
–1
–0.5
0
p
j
2p
0
p
j
2p
Figure 14. System (3.1): bifurcation scenarios around the codimension-2 points H1–4–6–3 (a) and
H1–5–8–4 (b) marked in figure 9. (a) Q1 / Q4 / Q6 / Q3 and (b) Q1 / Q5 / Q8 / Q4 .
Finally, we remark that all areas involved in the scenario described earlier
originate from the codimension-2 bifurcation point qLR2 h x0;l
LR , which therefore
turns out to be a codimension-2 big bang bifurcation point. In contrast to the big
bang bifurcation discussed in §4, it organizes a structure formed by MBCAs
without any periodic inclusions.
Note that the phenomenon presented above explains bandcounts shown in
figure 12 above the ‘curve’ of the bandcounts corresponding to the fifth layer of
the bandcount-adding scheme. As one can see, these bandcounts occur in the
middle of the parameter intervals leading to first- and second-layer bandcounts.
Of course, using a higher resolution in the parameter space, more of these
structures can be detected.
Note that in the investigated parameter plane some further codimension-2
bifurcations occur. Especially interesting is the sequence of bifurcations,
where the boundaries of the areas Q2n , Qn , QnC1 and Q1 intersect. In
figure 9, two bifurcations belonging to this sequence are marked with H1–4–6–3
and H1–5–8–4. As one can see, at these points four different chaotic attractors
emerge. In order to characterize such a bifurcation point, we perform a onedimensional parameter scan along the boundary of a sufficiently small convex
neighbourhood. The results of this scans are shown in figure 14, where the
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parameter substitution bZ R cosð4Þ, mZ R sinð4Þ with RZ0.05 for figure 14a
and RZ0.0075 for figure 14b is used. Hereby, we observe how the 2n-band
attractor emerges as some kind of overlapping of an n-band and an (nC1)band attractor.
6. Summary and open questions
In this work, we report two methods for numerical calculation of the number of
bands (bandcounts) for MBCAs. For both methods, we presented the basic
algorithm as well as some practical hints related to an efficient implementation
and practical usage. Typical problems which may occur if parameters of the
methods are chosen inappropriately are also discussed. Both methods
are implemented within the ANT.4669 software package, which is a free
simulation and analysis tool for dynamical systems and can be downloaded at
www.AnT4669.de.
As an application example for both methods, we considered a one-dimensional
piecewise-linear map (3.1). Determining the bandcounts in extended areas of
parameter space, we explain several complex bifurcation structures, where
MBCAs are involved in. Especially, the bandcount-adding scenario is reported
for the first time.
Related to the bifurcation scenarios reported in this work, numerous questions
remain for future work. First of all, several codimension-1 bifurcation involved in
these scenarios have to be investigated in more detail. Especially, the
relationship between the reported scenarios and existence areas of unstable
periodic orbits has to be investigated. It is remarkable that the geometrical
structure of an attractor and consequently its bandcount may change if an
unstable orbit is destroyed. When dealing with these questions, system (3.1) may
be useful, since the existence areas of many unstable periodic orbits can be
determined analytically.
Another series of questions arises related to the bandcount-adding scenario.
So far, we assume that this scenario continues ad infinitum. Since this
hypothesis cannot be verified numerically, some analytical ways are necessary.
If this hypothesis will be confirmed, the next challenging question will be
given by the accumulation points of the bandcount-adding scenario. As in the
case of the period adding in a final parameter interval, this scenario has an
infinite number of accumulation points. At these points, we have to deal with
chaotic attractors consisting of an infinite number of bands. Remarkably,
because the overall attractors remain bounded, each of the bands has to be
infinitely small.
Finally, in this context, the concept of robust attractors should be
reconsidered. Many authors refer to robust chaotic attractors, if one can show
that in some parameter interval no windows with stable periodic dynamics exist.
However, at the bifurcation points where the bandcount changes, the attractor
cannot be denoted as robust, because its geometrical structure changes.
Therefore, the bandcount-adding scenario contains an infinite number of nonrobust chaotic attractors.
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On detection of multi-band chaotic attractors
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Appendix A. Proof of theorem 2.1 (sketch)
We have to demonstrate two facts, namely that if the system function f is
continuous, then a band of the attractor has exactly (i) one successor band and
(ii) one predecessor band (Eckstein 2006).
(i) Let us consider a band B0 and assume that it has two successor bands B1
and B2. Then, ce dx 1;2 2B0 with jx 1 Kx 2 j! e, f ðx 1 Þ 2B1 and f ðx 2 Þ 2B2 .
From the definition of the continuity of f follows that for e/ 0, the distance
jf ðx 1 ÞKf ðx 2 Þj becomes infinitely small and therefore the distance between
B1 and B2 as well. Hence, B1 and B2 represent one band.
(ii) Let us consider a band B0 and assume that it has two predecessor bands B1
and B2. Then, there are two paths in the attractor, B1 1 B0 1 .1 B1 and
B2 1 B0 1 .1 B2 . In this case, one of the direct or indirect successor
bands of B0 must have two successors, and as shown earlier, this is
not possible.
Appendix B. Analytical results related to system (3.1)
n;r
In order to calculate the parameter subspaces x0;l
LRn and xLRn , where the orbits
OLRn undergo border-collision bifurcations, we have to solve the equations x0Z0
½n
and xnZ0. Hereby, the point x0 is defined as the solution of x 0 Z fr ðfl ðx 0 ÞÞ and
½nK1
the point xn is determined by xn Z fr ðfl ðx 0 ÞÞ. Then, we obtain
n
o
~ m~; lÞjlð1Kb~n Þ Z m~ðb~nC1 K1Þ ;
x0;l
Z
ð~
a
;
b;
n
LR
n
o
n
nC1
nC1
n;r
~ m~; lÞj~
~ C b~n l :
xLR
ð~
a ; b;
a ðb~ Kb~ Þð~
m C lÞ Z b~ m~ K b~m~ K bl
n Z
Similarly, we obtain for the border-collision bifurcations where the orbits ORLn
are involved in
~ m~; lÞjlð~
x0;r
a; b;
an K~
anC1 Þ Z mð~
a nC1 K1Þ ;
RLn Z ð~
n;l
~ m~; lÞjb~
~mð1K a~Þ Z m~a~Kl C a~l K~
a; b;
a1Kn m~ :
xRL
n Z ð~
Then parameter subspaces qLRn and qRLn are defined by the stability conditions
~ 1, respectively.
j~
ab~ jZ 1 and j~
a n bjZ
Appendix C. Implementation issues
The GCD-based algorithm can be implemented straightforwardly. The only
critical point is a suitable data structure for the set S of basis points. If an
unsorted array is used here, the run-time of the algorithm becomes OðI $M Þ.
Owing to the large values of I and M which are required for the correct
estimation of K, this is not feasible. The run-time can be reduced using any kind
of data structure, which supports a fast search. For instance, using a sorted list
we obtain the run-time OðM C I $log2 M Þ.
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Note that after the calculation is completed, it is still possible that the
investigated attractor is periodic and the detected value represents in fact
the period instead of the bandcount. Therefore, it has to be checked whether the
attractor is chaotic and the detected value is indeed the bandcount K.
An optimization of the method can be achieved by saving only such points in
the set S, which do not lie in the 3-neighbourhood of each other. In doing so,
some unnecessary calculations can be avoided.
For the implementation of the boxcounting-based algorithm, a suitable data
structure for the partitions is also important. This data structure has to support
not only a fast search like in the GCD-based method, but also a fast access to the
adjacent boxes. For low-dimensional dynamical systems, an array-based solution
is preferable. Depending on the used number of partitions p, this solution may
lead to a memory consumption which is too high. In this case, it is more suitable
to use a hash table-based solution.
Another important aspect of this method is the clusterization algorithm. The
most straightforward solution of this task is a recursive approach. However, due to
its high memory consumption, this algorithm can be used efficiently mainly for onedimensional dynamical systems. Therefore, an iterative solution is preferable.
Appendix D. Parameter adjustment and typical problems
For the practical application of both methods presented in this work, some
parameters have to be adjusted. Hereby, the parameter setting, which is optimal
in the sense of computation time and precision, depends on the specific
dynamical system under investigation. However, there are some general rules,
which can help to reduce the inaccuracy of the results.
Firstly, the setting of the parameters is less critical for attractors with
approximative uniform distribution of the invariant measure within each band.
In the case the distribution becomes strongly non-uniform, the parameters have
to be adjusted more carefully.
Next, we remark that the GCD- and the boxcounting-based methods show
different behaviour in the case that the first points of the investigated orbit still
belong to the transient phase of the dynamics with respect to the chosen
accuracy. For the GCD-based method, this is of comparatively low importance.
The only effect may be that for the corresponding points no return times will be
found. However, if the set S is sufficiently large and the return times will not be
detected only for a few points, the final result may still be correct. In contrast to
this, in the case of the boxcounting-based method, the points from the transient
phase of the dynamics may lead to totally incorrect results. If such a point lies far
away from the attractor, it will be interpreted as a singular partition, which will
be counted as an additional cluster (band). In order to avoid this problem, it can
be assumed that singular partitions containing only a few points represent a
numerical error and have to be eliminated. However, this technique should be
applied very carefully, because for attractors with a high non-uniform
distribution of the invariant measure, some partitions corresponding to bands
with low density may become eliminated in this way. Instead, the transient time
has to be increased.
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The computation accuracy of the developed methods is determined by the
compare precision 3 (GCD-based method) and the grid cell size (boxcountingbased method). Theoretically, these parameters can be set arbitrarily small.
Hereby, one has to keep in mind that the gaps between the bands become
typically very narrow with increasing bandcounts, so that it becomes in fact
necessary to perform the calculations with high accuracy. For instance, in the
case of the band-merging scenario of the tent map shown in figure 2, the compare
precision is set to 3Z 10K14 . However, one has to be careful because a high
accuracy requires a higher number of iteration steps to be performed. If the used
number of iterations is insufficient, the bands may split apart (boxcountingbased method) or the number of return times found is insufficient (GCD-based
method). In both cases, the detected bandcount will be incorrect.
Two typical problems are related with the numerical determination of the
bifurcation points, where attractors before and after the bifurcation are chaotic. In
the case of the band-merging bifurcations, it is obvious that separate bands before
the bifurcations become arbitrarily close to each other. Obviously, when dealing
with these bifurcations, the parameter value detected numerically corresponds not
to the correct bifurcation point, but to the point, where the gap between the bands
can no longer be resolved with respect to the used accuracy. Hence, this numericcaused shift of the bifurcation points can be reduced using a higher accuracy, but
cannot be completely avoided. Additionally, if attractors before the bifurcation
have 2K bands and after the bifurcation K bands, then the numerical calculation
may lead to some incorrect values monotonously decreasing from 2K to K. This
phenomenon (‘waterfall error’) is caused by the fact that the gaps between
attractors reach the calculation accuracy at different parameter values.
Another problem occurs when dealing with interior crises. In this case, the
invariant measure of attractors after the bifurcation is distributed strongly nonuniform. In order to guarantee that the orbit covers the attractor dense (up to
the given accuracy), the number of iterations has to be increased significantly.
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