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Proc. R. Soc. A (2008) 464, 2247–2263
doi:10.1098/rspa.2007.0299
Published online 15 April 2008
The bandcount increment scenario.
II. Interior structures
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
Bifurcation structures in the two-dimensional parameter spaces formed by chaotic
attractors alone are still far away from being understood completely. In a series of three
papers, we investigate the chaotic domain without periodic inclusions for a map, which is
considered by many authors as some kind of one-dimensional canonical form for
discontinuous maps. In this second part, we investigate fine substructures nested into the
basic structures reported and explained in part I. It is demonstrated that the overall
structure of the chaotic domain is caused by a complex interaction of bandcount increment,
bandcount adding and bandcount doubling structures, whereby some of them are nested
into each other ad infinitum leading to self-similar structures in the parameter space.
Keywords: bandcount increment; bandcount adding; bandcount doubling;
interior crises; merging crises; piecewise-linear discontinuous maps
1. Introduction
The investigation of the chaotic domain in a multi-dimensional parameter space
represents a challenging task from the theoretical point of view. Of course, this
investigation is also important for applications that use dynamical systems
operating in the chaotic domain. Hereby, it is especially important to know how
the variation of parameters influences the dynamics. For example, one is
interested in knowing whether the chaotic domain in the parameter space is
interrupted by periodic inclusions (‘windows’) or whether the attractors are oneband or multi-band attractors, and so on. Consequently, bifurcations or crises
occurring in the chaotic domain are of great importance. When dealing with
piecewise-smooth models that originate from a broad spectrum of applications
such as mechanical oscillators with impacts and/or stick–slip effects, switching
electronic circuits, power converters and others, the situation of the so-called
robust chaos is well known (see references in Banerjee & Verghese 2001,
Zhusubaliyev & Mosekilde 2003 and Bernardo et al. 2007). Typically, this
notation refers to the fact that the chaotic domain does not contain periodic
inclusions (Banerjee et al. 1998). However, it was shown in previous publications
and especially in part I of this work (Avrutin et al. 2008) that this domain can,
nevertheless, posses a complex structure formed by crises bifurcations. At these
* Author for correspondence ([email protected]).
Received 2 November 2007
Accepted 18 March 2008
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V. Avrutin et al.
bifurcations, the chaotic nature of the attractors persists, but their geometrical
and topological structure changes. Consequently, these attractors are robust in
the sense of Banerjee et al. (1998), but not robust in the sense of Milnor (1985).
Note that the change in the geometrical and topological structure is typically
associated with a change in the number of bands of the attractor. In this work,
we avoid discussion about the exact mathematical definition of a band and
consider it as a strongly connected (dense) component. For the techniques used
for numerical detection of the number of bands (bandcount), we refer to Avrutin
et al. (2007a). Since the dynamical system we investigate here has a
discontinuous system function, we have to use a box-counting-based algorithm
for the detection of the bandcounts.
In part I of this work, we investigated the piecewise-linear discontinuous map
given by
(
f[ ðxn Þ Z axn C m;
for xn ! 0;
xnC1 Z
ð1:1Þ
fr ðxn Þ Z bxn C m C l; for xn O 0:
This map represents a special case of a well-known two-dimensional piecewiselinear normal form investigated by many authors (for references, see Bernardo
et al. 2007). Concerning the periodic solutions, we considered the characteristic
case of negative discontinuity (l!0, whereby the investigated system can always
be reduced to the case lZK1 by suitable scaling) and the case a!0, where the
periodic domain is organized by the period increment scenario (Avrutin et al.
2007b) with coexisting attractors (sometimes denoted as ‘multi-stability’). This
scenario is formed by a sequence of periodic orbits OsLRk with kZ1, 2, ., whose
stability regions (in the parameter space) P LRk overlap pairwise.1 The question
we investigated in part I of this work was how these orbits influence the
dynamics after the transition to chaos.
The first step towards understanding the structure of the chaotic domain was
carried out by investigation of the special case aZK1. It was shown that the main
structure-forming component is given by a sequence of regions of multi-band chaotic
attractors organized by the bandcount increment scenario, as shown in figure 1.
As already mentioned, the numerical results are obtained by a box-countingbased algorithm, whereas the analytical results are from part I of this work. Note
that for a better graphical representation, in all figures in this paper we use the
same topology preserving scaling of parameters as in part I of this work, namely
p p
SðnÞ : ðKN;NÞ1 K ; ; with SðnÞ Z arctanðnÞ; for n 2 fa; b; mg: ð1:2Þ
2 2
The boundaries between the regions involved in this scenario represent curves
of merging crises2 caused by the orbits OuLRk . In this sense, both the periodic and the
chaotic domain are related, and the bandcount increment scenario in the chaotic
domain reflects the period increment scenario with coexisting attractors in the
periodic domain. Owing to the pairwise overlapping of the regions P uLRk , the
bandcount increment scenario includes regions of two types, namely triangle-like
1
For details related to the notation used here we refer to part I of this work.
To avoid confusion we emphasize that the term ‘merging crisis’ in this work refers to bifurcations,
where some of the bands of a multi-band chaotic attractor collide pairwise and not to the
bifurcation where two coexisting chaotic attractors collide (Ott 2002).
2
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Bandcount increment: interior structures
(a)
1
4
(b)
div
10
3
10
0.6
4
8
2
8
3
5
5
3
4
0.4
2
4
0.2
6
2
6
1
1
3
3
0
1
4
0.9
1.0
1
4
0.9
1.0
Figure 1. Bandcount increment structures calculated (a) numerically and (b) analytically for the
case aZK1.
regions Q 2kC4
and trapezoidal regions Q kC2
. As indicated by the subscripts
LRk hLRkC1
LRk
and superscripts, each region of the first type is influenced by two unstable periodic
orbits OuLRk and OuLRkC1 and contains (2kC4)-band attractors. By contrast, each
region of the second type is influenced by only one orbit OuLRk so that the
corresponding attractors have kC2 bands.
Comparing the numerical and analytical results shown in figure 1, we conclude
that the overall structure of the chaotic domain is already explained (Avrutin
et al. 2008). However, our numerical results indicate that each of the regions
forming the bandcount increment scenario has a complex interior substructure.
More precisely, there are two different substructures we need to explain: one
is within the regions Q 2kC4
and the other within the regions Q kC3
. Again,
LRk hLRkC1
LRkC1
the question arises of how the bandcounts in these substructures are organized
and which unstable orbits are responsible for the corresponding crises. This
question represents the main topic of the current work.
2. Bandcount adding within the region Q 6LRhLR2
Let us start with the structure of the regions Q 2kC4
and consider as a first
LRk hLRkC1
step the case kZ1. The numerically calculated structure of the region Q 6LRhLR2
is shown in figure 2a. Comparing this structure with the bandcount adding
structures reported in Avrutin & Schanz (in press), we recognize an
unambiguous similarity. As shown in figure 2a, the region located in the middle
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V. Avrutin et al.
(a)
(b)
0.4
2
(
32
28
)2 (
2 2
)
24
20
)2 (
(
0.2
2 2
)
6
6
16
2
2
(
22
2 2
) (
)
28
34
40
0.1
6
(
)2 (
2 2
)
2
0
1
4
0.9
1
4
0.9
(c) 100
60
40
52
40
34
44
28
20
32
22
46
38
30
16
20
24
2832
6
0
0.1
0.2
0.4
Figure 2. Substructures within the region Q 6LRhLR2 calculated (a) numerically and (b) analytically.
the families of regions Q 16C4m
and Q 16C6m
with
In (b), above and below Q 16
ðLRÞ2 ðLR2 Þ2
ðLRÞ2mC2 ðLR2 Þ2
ðLRÞ2 ðLR2 Þ2mC2
mZ1–10 are shown. The bandcount adding scenario along the dashed line in (a) is shown in (c).
of Q 6LRhLR2 has bandcount 16, and above this region we observe a sequence of
regions with bandcounts KmZ20, 24, 28, 32, .Z16C4m, m2N, whereas below
we detect a sequence of regions with KmZ22, 28, 34, 40, .Z16C6m, m2N.
Between each two subsequent regions in these sequences, further regions with
higher bandcounts are located, organized exactly in the same way as described in
Avrutin & Schanz (in press). These bandcounts are shown in figure 2c, where the
parameter m is varied along the dashed line S(b)Z0.7895 in figure 2a. As one can
see, between the regions with bandcounts 20 and 24 a region with bandcount 38
is located, between the regions with bandcounts 24 and 28 a region with
bandcount 46 is located, and so on. In contrast to the bandcount increment
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Bandcount increment: interior structures
2251
scenario described above, all involved regions within this structure are bounded
by interior crises and not by merging crises. Again, the question arises of which
unstable periodic orbits cause this structure to emerge.
In order to explain the observed bifurcation structure, we first have to recall
that all involved regions are nested into the region Q 6LRhLR2 . As a consequence,
each multi-band attractor within this region has two gaps occupied by the
unstable orbit OuLR and three gaps occupied by the unstable orbit OuLR2 . In
particular, for the region of 16-band attractors located in the middle of Q 6LRhLR2 ,
we state that from its 15 gaps, 5 are already occupied by these orbits. This leads
us to the assumption that at the interior crises where these attractors emerge,
an unstable orbit with period 10 is involved. Searching numerically for such an
orbit within the 16-band region, we can confirm this assumption and find the
unstable 10-periodic orbit OuðLRÞ2 ðLR2 Þ2 . The border-collision curves bounding
the region of existence P ðLRÞ2 ðLR2 Þ2 of this orbit (figure 2b) can easily be calculated,
b4 Kb 2 K 1
6;[
xðLRÞ2 ðLR2 Þ2 Z ðb; mÞ m ZK
ð2:1Þ
ðb C 1Þðb 2 C 1Þ
and
b4 Kb2 C 1
2;r
xðLRÞ2 ðLR2 Þ2 Z ðb; mÞ m Z
:
ð2:2Þ
ðb C 1Þðb4 C 1Þ
Note that the region P ðLRÞ2 ðLR2 Þ2 originates from the point bZ1, mZ1/4. As a
consequence, this orbit does not exist for b!1 and is unstable in its complete
region of existence. In the following, we denote such orbits as completely unstable.
Now we are able to calculate the boundaries of the region Q 16
.
ðLRÞ2 ðLR2 Þ2
u
The curves of interior crises caused by the orbit OðLRÞ2 ðLR2 Þ2 result from the
intersections of specific points of this orbit with suitable points of the kneading
orbit. Solving the equations
ðLRÞ2 ðLR2 Þ2
x6
Z x ko
[ rð[ r2 Þ2 [ r
we obtain the two curves
h[ðLRÞ2 ðLR2 Þ2
and
hrðLRÞ2 ðLR2 Þ2
ðLRÞ2 ðLR2 Þ2
and x 2
Z x ko
r[ r2 ð[ rÞ3 ;
b8 Kb6 C b4 K2b 2 C 2
Z ðb; mÞ m Z 8
ðb C b4 Kb 2 C 1Þðb C 1Þ
b8 Kb6 Kb4 K2b 2 C 2
Z ðb; mÞ m ZK 6
;
ðb C b4 C b 2 K1Þðb C 1Þ
ð2:3Þ
ð2:4Þ
ð2:5Þ
. As one can see from figure 2b, this region
bounding the region Q 16
ðLRÞ2 ðLR2 Þ2
originates from the same point bZ1, mZ1/4 as the region P uðLRÞ2 ðLR2 Þ2 . Hence, the
codimension-2 bifurcation occurring at this point is more complex than initially
assumed because not only two border-collision curves are involved, but also the
curves of interior crises.
The complete families of regions above and below the middle region
Q 16
with bandcounts
ðLRÞ2 ðLR2 Þ2
Km Z 16 C 4m
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V. Avrutin et al.
shown in figure 2b, can be explained in a similar way. It turns out that the first
family is caused by interior crisis bifurcations of the sequence of completely
unstable orbits OuðLRÞ2mC2 ðLR2 Þ2 and the second family by interior crisis
bifurcations of the sequence of completely unstable orbits OuðLRÞ2 ðLR2 Þ2mC2 . The
bandcounts of the corresponding attractors result from 4mC6 and 6mC4 gaps,
respectively, occupied by the points of these orbits and five further gaps where
the orbits OLR and OLR2 are located. The boundaries of the corresponding
regions Q 16C4m
and Q 16C6m
can be calculated analytically for an
ðLRÞ2mC2 ðLR2 Þ2
ðLRÞ2 ðLR2 Þ2mC2
arbitrary m. By solving the equations
ðLRÞ2mC2 ðLR2 Þ
Z x ko
[ rð[ r2 Þ2 ð[ rÞ2mC1
ð2:7Þ
ðLRÞ2mC2 ðLR2 Þ
Z x ko
r[ r2 ð[ rÞ2mC3 ;
ð2:8Þ
x 4mC6
and
x 4mC2
[ =r
we obtain the curves hðLRÞ2mC2 ðLR2 Þ bounding the family of regions Q 16C4m
.
ðLRÞ2mC2 ðLR2 Þ2
16C6m
The boundaries of the regions Q ðLRÞ
can be calculated analogously.
2
ðLR2 Þ2mC2
Comparing the numerical and analytical results shown in figure 2, we state
that the substructures within the region Q 6LRhLR2 include many more regions
and Q 16C6m
. In fact, these families
than the two families Q 16C4m
ðLRÞ2mC2 ðLR2 Þ2
ðLRÞ2 ðLR2 Þ2mC2
represent only the first generation of the fully developed bandcount adding
scenario. Owing to the complete analogy of this scenario with the fully developed
bandcount adding scenario reported in Avrutin & Schanz (in press), the
identification of further regions forming this scenario is straightforward.
Therefore, let us define the sequences dðmÞZ ðLRÞ2mC2 ðLR2 Þ2 . Then, between
16C4ðmC1Þ
and Q dðmC1Þ
of the first generation, we
each two subsequent regions Q 16C4m
dðmÞ
30C8m
observe the region Q dðmÞdðmC1Þ of the second generation. As one can see for mZ
0 and 1, this results in the bandcounts 30 and 38 marked in figure 2c. Between
40C12m
and Q 30C8m
the regions Q 16C4m
dðmÞ
dðmÞdðmC1Þ , there exists the region Q dðmÞdðmÞdðmC1Þ , which
belongs to the third generation (e.g. the bandcount 52 shown in figure 2c). This
process continues ad infinitum and explains the bandcounts shown in figure 2c on
the r.h.s. of the region Q 16
. The bandcounts shown in figure 2c on the
ðLRÞ2 ðLR2 Þ2
can
be explained in exactly the same way using
l.h.s. of the region Q 16
ðLRÞ2 ðLR2 Þ2
the sequences dðmÞZ ðLRÞ2 ðLR2 Þ2mC2 . In both cases, each orbit Ous , involved in
this bandcount adding scenario, is a completely unstable orbit responsible for the
emergence of the region Q K
s with KZ jsjC 6. Furthermore, the relative location of
is
the
same as shown in figure 2b for the orbit OðLRÞ2 ðLR2 Þ2 .
the regions P s and Q K
s
Namely, both regions originate from a point at the line bZ1 and the curves
[ =r
hs are tangent to the boundaries of P s at this point.
The only difference between the bandcount adding structure reported in
Avrutin & Schanz (in press) and the structure we observe here concerns the type
of the unstable periodic orbits responsible for the interior crises. In the cited
work, these orbits originate from the domain of periodic dynamics, where they
are stable. By contrast, the orbits forming the structure described above are
completely unstable.
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Bandcount increment: interior structures
350 316
200
100
156
76
36
0
1
4
0.80
16
0.82
6
0.85
Figure 3. Bandcount doubling cascade in the middle part of the region Q 16
.
ðLRÞ2 ðLR2 Þ2
3. Bandcount doubling
In figure 2a, it is clearly visible that the regions forming the bandcount adding
structure within the region Q 6LRhLR2 still have some further substructures. In
fact, along the middle line of each region of the bandcount adding scenario, we
observe a nested sequence of regions organized by a bandcount doubling
scenario.3 For instance, within the region Q 16
this cascade is formed
ðLRÞ2 ðLR2 Þ2
by attractors with bandcounts 16, 36, 76, ., as shown in figure 3. Applying the
same techniques as in Avrutin & Schanz (in press), we state that the unstable
periodic orbits responsible for this cascade have periods 10, 20, 40, ..
In particular, the bandcount 36 is explained by 2C3C10C20 gaps occupied
by the orbits OLR , OLR2 , OðLRÞ2 ðLR2 Þ2 and OðLRÞ4 ðLR2 Þ4 , respectively. The next
bandcount 76 is explained by 2C3C10C20C40 gaps, and in general we obtain
Kn Z 1 C 2 C 3 C 10
n
X
2i Z 6 C 10ð2nC1 K1Þ;
n Z 0; 1; 2; . :
ð3:1Þ
i Z0
The symbolic sequences sn for nZ0–3 corresponding to the unstable periodic
orbits Osn leading to this cascade are presented in table 1. These sequences and
all further sequences sn for nO3 can be generated using the technique presented
in Avrutin & Schanz (in press). Based on these sequences, the regions
6C10ð2nC1 K1Þ
Q sn
forming the bandcount doubling cascade in the middle of Q 16
s0 can
[ =r
be calculated. In figure 4, the boundaries of these regions (the curves hsi
u
of interior crises caused by the orbits Osi ) are shown for the sequences siZs0–s3
3
Related to the notation ‘bandcount doubling’, one has to keep in mind that it is slightly different
from the usual notation ‘period doubling’. In the case of a period doubling cascade, the periods of
the attractors are, in fact, doubled at each bifurcation. By contrast, in a bandcount doubling
cascade the bandcounts are not doubled at each bifurcation, but the periods of the unstable orbits
leading to these bandcounts are doubled. In the most simple case, the bandcounts may be doubled
as well, as in the case of logistic
maps. Here, the unstable orbits with periods piZ2i lead to
P and tent
n
the bandcounts Kn Z 1 C nK1
p
Z
2
.
In this case, bandcounts and periods of the responsible
iZ0 i
orbits are doubled. By contrast, the scenario described by equation (3.1) represents an example
where the periods of the responsible orbits, but not the bandcounts, are doubled.
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V. Avrutin et al.
Table 1. Symbolic sequences sn for nZ0 – 3, corresponding to the unstable periodic orbits Osn ,
leading to the bandcount doubling cascade within the region Q 16
. (The third and
ðLRÞ2 ðLR2 Þ2
6C10ð2nC1 K1Þ
fourth columns show the period of the orbit Osn and the region Q sn
attractors caused by the interior crises where the orbit Osn is involved.)
of multi-band
n
sn
jsnj
region
0
1
2
3
ðLRÞ2 ðLR2 Þ2
ðLRÞ4 ðLR2 Þ4
ðLRÞ2 ðLR2 Þ4 ðLRÞ2 ðLR2 Þ2 ðLRÞ4 ðLR2 Þ2
ðLRÞ4 ðLR2 Þ4 ðLRÞ2 ðLR2 Þ2 ðLRÞ4 ðLR2 Þ4 ðLRÞ4
ðLR2 Þ2 ðLRÞ2 ðLR2 Þ4
10
20
40
80
Q 16
s0
Q 36
s1
Q 76
s2
Q 156
s3
36
given in table 1. In figure 4a, the regions Q 16
s0 and Q s1 are presented as they
appear in the parameter space b!m. However, a better graphical representation
of the results is possible by introducing a sequence of local coordinate systems
6C10ð2nC1 K1Þ
with respect to the middle lines msn Z ðh[sn C hrsn Þ=2 of the regions Q sn
.
6C10ð2nC1 K1Þ
6C10ð2nC2 K1Þ
, Q snC1
are shown
Hence, in figure 4b –d the pairs of regions Q sn
using these local coordinate system. It is clearly visible that in each pair the next
6C10ð2nC2 K1Þ
region Q snC1
is slightly displaced with respect to the middle curve msn of
6C10ð2nC1 K1Þ
the previous region Q sn
. Furthermore, it can be clearly seen that the
6C10ð2nC1 K1Þ
Q sn
become very narrow for increasing n. This behaviour is
areas
explained in Avrutin & Schanz (in press), where the scaling properties of the
bandcount doubling scenario are investigated. Since this scenario takes place in a
two-dimensional parameter space, it has two scaling constants in the parameter
space (Lyubimov et al. 1989). Both these scaling constants are defined similarly
to the well-known Feigenbaum constant d of the period doubling cascade
(Feigenbaum 1979). Hereby, one of the scaling constants has a finite value,
whereasnC1the other one has the ‘value’ N. Therefore, the width of the regions
6C10ð2
K1Þ
Q sn
decreases faster with increasing n so that it becomes a hard task to
detect them numerically. Owing to both reasons (displacement and rapidly
decreasing width), we must use the middle curve ms4 of the region Q 316
s4 for the
calculation of figure 3.
The same bandcount doubling scenario takes place not only within the region
Q 16
, but also within each of the regions involved in the bandcount adding
ðLRÞ2 ðLR2 Þ2
scenario described above. For instance, within the region Q 46
,
ðLRÞ6 ðLR2 Þ2 ðLRÞ8 ðLR2 Þ2
24
28
located between the regions Q ðLRÞ6 ðLR2 Þ2 and Q ðLRÞ8 ðLR2 Þ2 ; this cascade leads to
the bandcounts given by
Kn Z 1 C 2 C 3 C 40
n
X
2i Z 6 C 40ð2nC1 K1Þ;
n Z 0; 1; 2; . :
ð3:2Þ
i Z0
Again, the unstable periodic orbits responsible for this cascade have doubled
periods, namely 40, 80, 160 and so on.
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Bandcount increment: interior structures
(a) 0.245
(b) 1.1
× 10 –2
1
0.220
0
0
0
1
0
1
0
0
1
0
–1.1
0.190
1
4
(d)
1
0
2
1
2
2
–1.0
1
4
1
4
0.84
1.4
× 10 –5
1.0
× 10 – 4
(c)
0.84
3
2
3
2
0
1
–1.4
0.81
1
4
0.798
Figure 4. Bandcount doubling in Q 16
. (a) The actual appearance in the parameter space.
ðLRÞ2 ðLRÞ2Zs
0
(b –d ) Local coordinate systems with respect to the middle lines msn are used. In (b), the same area
as in (a) is presented, whereas (c) shows the first doubling area Q 36
s1 and (d ) shows the second
doubling area Q 76
s2 . The sequences sn are shown in table 1.
4. Further bandcount adding
It is confirmed numerically and can be shown analytically too that a complete
bandcount doubling cascade occurs in each region forming the bandcount adding
scenario inside the region Q 6LRhLR2 . However, the interior structure of these
regions turns out to be even more complex. In order to demonstrate this, let us
consider figure 5, where the parameters are varied across the region Q 16
ðLRÞ2 ðLR2 Þ2
along the arc marked in figure 4a. As expected, in the middle of this figure we
observe the bandcounts 36 and 76, which belong to the bandcount doubling
scenario described above. However, around these regions we observe further
bandcounts, organized by the bandcount adding scenario. The first generation of
this scenario is given by two sequences of regions with bandcounts 46, 56, 66,
76, . converging from inside the region Q 16
to the outside, which means
ðLRÞ2 ðLR2 Þ2
from the middle line of this region towards the bounding interior crisis curves.
The results shown in figure 5 can be explained by two families of completely
unstable orbits Ousðj Þ and Ourðj Þ with
9
sð j Þ Z ðLRÞ4 ðLR2 Þ4 ððLRÞ2 ðLR2 Þ2 Þ j
>
=
and
ð4:1Þ
>
;
4
2 j
2 4
2 2
rð j Þ Z ðLR Þ ðLRÞ ððLR Þ ðLRÞ Þ ; j Z 1; 2; 3; . :
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V. Avrutin et al.
100
86
86
76
66
56
60
86
86
86
66
66
66
56
46
46
36
40
16
20
0
86
76
76
– 60
–50
–40
–30
Figure 5. Bandcount adding scenario within the region Q 16
along the arc marked in
ðLRÞ2 ðLR2 Þ2
figure 4a.
Note that these sequences are derived from the symbolic sequence ðLRÞ2 ðLR2 Þ2 h
and the symbolic sequence
ðLR2 Þ2 ðLRÞ2 of the ‘host region’ Q 16
ðLRÞ2 ðLR2 Þ2
4
4
2 4
2 4
36
ðLRÞ ðLR Þ h ðLR Þ ðLRÞ of the ‘first region’ Q ðLRÞ4 ðLR2 Þ4 from the bandcount
doubling cascade, which corresponds to the case jZ0 in both families. The periods
of the orbits are the same in both the cases, namely 20C10jZ30, 40, 50, .
Consequently, the bandcount 46 shown on the l.h.s. of figure 5 is structured by 2
gaps occupied by OuLR , 3 gaps occupied by OuLR2 , 10 gaps occupied by OuðLRÞ2 ðLR2 Þ2
and finally by 30 gaps occupied by Ousð2Þ . By contrast, bandcount 46 shown on the
r.h.s. of figure 5 is structured by 15 gaps occupied by the same orbits OuLR , OuLR2 ,
OuðLRÞ2 ðLR2 Þ2 and by a further 30 gaps where the other 30-periodic orbit, namely
Ourð2Þ , is located.
36C10j
36C10j
and Q rðj
, influenced by the families of orbits Ous ðj Þ and
The regions Q sðj
Þ
Þ
u
Or ðj Þ, form the first generation of the bandcount adding scenario similar to the
one described in §2. As in the previous case, for each j between two subsequent
36C10ð jC1Þ
66C20j
regions Q 36C10j
and Q sð jC1Þ
, we find the region Q sð
sðj Þ
j Þsð jC1Þ , which belongs to
the second generation of the bandcount adding. Examples of these regions shown
in figure 5 are the bandcounts 66, which correspond to the regions Q 66
sð0Þsð1Þ and
,
as
well
as
the
bandcounts
86
(located
between
the
bandcounts
46
and 56),
Q 66
rð0Þrð1Þ
86
86
corresponding to the regions Q sð1Þsð2Þ and Q rð1Þrð2Þ . Note that there are four further
regions with the bandcount 86, which are explained differently. Two of them,
86
namely Q 86
sð5Þ and Q rð5Þ belong to the first generation and the other two
86
86
(Q sð0Þsð0Þsð1Þ and Q rð0Þrð0Þrð1Þ ) belong to the third generation.
As one can see, the bandcount scenario within the region Q 16
involves an
ðLRÞ2 ðLR2 Þ2
jdjC16
infinite number of regions Q d
. Recall that each of these regions is embedded into
the region of existence of the corresponding completely unstable periodic orbit Oud .
Note that all these regions P ud originate from that point at the bZ1 line, where the
originates from. Consequently, we state that the codimension-2
region Q 16
ðLRÞ2 ðLR2 Þ2
bifurcation occurring at this point is quite complex. At this point, two former
stable periodic orbits OLR and OLR2 become unstable and an infinite number of
different multi-band chaotic attractors emerge, organized by an infinite number
of different unstable periodic orbits, which emerge at the same point.
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Bandcount increment: interior structures
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Remarkably, the results we have obtained so far are valid not only for the
region Q 16
, but also for any region involved in the bandcount adding
ðLRÞ2 ðLR2 Þ2
scenario within the region Q 6LRhLR2 , as described in §2. This means, within each
of these regions a nested self-similar bandcount adding structure exists,
originating from the same point as the surrounding region. As a consequence,
we state that at the boundary of the chaotic domain P ch, which means the line
bZ1, an infinite number of bifurcations occurs where, from each bifurcation
point, an infinite number of bifurcation curves originates forming the nested
doubling and adding structures.
5. Bandcount adding within Q 2kD4
regions
LRkhLRkD1
So far the structure of the region Q 6LRhLR2 is explained. It can be verified both
numerically and analytically that all regions Q 2kC4
for kO1 have the same
LRkhLRkC1
structure. In particular, in the middle of each of these regions there exists a
region Q 6kC10
of (6kC10)-band attractors. Of course, this number of bands
ðLRk Þ2 ðLRkC1 Þ2
is explained by (kC1)C(kC2)C(4kC6) gaps, occupied by the orbits OuLRk ,
OuLRkC1 and OuðLRk Þ2 ðLRkC1 Þ2 . Above this region, we observe the family of regions
10C6kC2mC2km
10C6kC4mC2km
and below the family of regions Q ðLR
Q ðLR
k Þ2mC2 ðLRkC1 Þ2
k Þ2 ðLRkC1 Þ2mC2 , which
together form the first generation of the bandcount adding scenario within
this specific region Q 2kC4
. Remarkably, the boundaries of these regions can
LRkhLRkC1
still be calculated analytically, even for large values of k and m. Examples of
these structures for kZ2 and 3 are shown in figures 6 and 7. For increasing k
values, the area occupied by these structures decreases (figure 1), but the
structures remain topologically equivalent. This is not surprising because the
‘host regions’ Q 2kC4
are organized in the same way by the orbits of the OLRk
LRkhLRkC1
family for all k values. However, recall that the substructures within these
regions are formed by orbits that are completely unstable. Similar structures
within the regions Q 2kC4
reflect the fact that these completely unstable
LRkhLRkC1
orbits are forming a self-similar structure like the OLRk family.
6. Overlapping structures within Q kD2
LRk
As already mentioned, the fine substructures within the regions Q 2kC4
and
LRkhLRkC1
kC2
Q LRk are different. After the first type of these substructures is explained, let us
consider the second one by focusing first on the case kZ2. Figure 8a shows the
structure of the region Q 4LR2 calculated numerically. As can be seen, the main
component of this structure is formed by the regions with bandcounts 6, 8 and 10
in-between. Consequently, we have to explain five, seven and nine gaps of the
corresponding attractors, whereby three of these gaps are already occupied by
the OuLR2 orbit. Hence, it seems natural to assume that the remaining two, four
and six gaps can be explained simply as follows: two gaps are occupied by a
period-2 orbit; four gaps by a period-4 orbit; and in the region in-between, both
orbits exist and occupy six gaps of the 10-band attractor. It is also not difficult to
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V. Avrutin et al.
(a)
(b)
0.58
34
(
26
3 2
) (
)
28
(
24
3 2
) (
)
2
34
28
2 2
(
0.52
3 2
) (
)
22
2 2
(
30
38
3 2
) (
8
)
8
2
0.48
3
3
0.800
0.845
0.800
0.845
Figure 6. Substructures within the region Q 8LR2hLR3 calculated (a) numerically and (b) analytically.
In (b), the regions Q 22C6m
and Q 22C8m
for mZ0 –10 are presented.
ðLR2 Þð2mC2Þ ðLR3 Þ2
ðLR2 Þ2 ðLR3 Þð2mC2Þ
(a)
(b)
0.64
44
(
36
4 2
36
(
34
4 2
) (
)
) (
)
44
3 2
(
36
4 2
) (
)
0.61
3
28
48
38
10
0.59
3 2
(
10
4 2
) (
3
)
4
4
0.800
0.825
0.800
0.825
Figure 7. Substructures within the region Q 10
calculated (a) numerically and (b) analytically. In
LR3hLR4
28C8m
28C10m
(b), the regions Q ðLR3 Þð2mC2Þ ðLR4 Þ2 and Q ðLR3 Þ2 ðLR4 Þð2mC2Þ for mZ0–10 are presented.
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Bandcount increment: interior structures
(a) 0.50
(b)
5
8
3
10
5
8
0.35
3
6
0.25
4
2
6
4
6
2
1
0.15
1
4
0.85
0.95
1
4
0.85
0.95
Figure 8. Substructures within the regions Q 4LR2 calculated (a) numerically and (b) analytically.
confirm that the regions of 6- and 10-band attractors also lie within the region
P uLR of the unstable period-2 orbit OuLR , and the regions of 8- and 10-band
attractors within the region P uLR3 of the unstable period-4 orbit OuLR3 .
Consequently, these orbits could be the orbits we are looking for. The only
difficulty with this assumption is that in all cases we have observed so far, each
unstable periodic orbit causing a region of a specific bandcount to emerge was
involved in exactly two interior or merging crises. Now, we have found an
example where this is not the case. In fact, it turns out that for any kO1 the orbit
OuLRk is involved in six different merging crises, as shown in figure 9a for the
special case kZ3. Two of the merging crises were already calculated and
[ =r
presented in part I of this work. Their corresponding curves gLRk in the
parameter space form the triangle-like regions in the centre part of figure 9a,
[ =r
shown also in figures 1, 2 and 7. The remaining four crises bifurcation curves g^LRk
[ =r
and g~LRk confine the ‘rabbit ear’-like regions above and below this triangle-like
[ =r
centre region. The calculation of these curves is similar to the calculation of gLRk ,
[ =r
namely the curves g^LRk can be obtained using the conditions
k
x LR
Z x ko
0
[ rk [ rk C1 [ rkC1
and the curves
[ =r
g~LRk
k
k
and x LR
Z x ko
0
r[ rk C2 [ rk C1 ;
ð6:1Þ
using the conditions
Z x ko
x LR
0
[ rkK2 [ rkK1 [ rk
k
and x LR
Z x ko
0
r[ rk [ rkK1 [ rkK1 :
ð6:2Þ
Hereby, the upper curve g^rLRk is at the bZ1 tangent to the upper existence
[
, where this unstable orbit is destroyed
boundary of the orbit OuLRk (the curve x0;
LR3
by a border-collision bifurcation). Similarly, the lower merging crisis curve g~[LRk
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V. Avrutin et al.
1
(a) 4
(b)
0
3
div
0.7
div
3
3
4
3
3
4
2
0.5
3
3
3
3
3
3
0.3
0.75 1
4
0.90
0.95
0.75 1
4
0.90
0.95
Figure 9. (a) Six merging crises bifurcation curves caused by the orbit OuLR3 . The border-collision
[
bifurcation curves x0;
and x3;r
bounding the region of existence of this orbit are shown as solid
LR3
LR3
lines when the stable orbit OsLR3 is destroyed via a border-collision bifurcation and as dashed lines
when the unstable orbit undergoes this bifurcation. (b) Overlapping of the ‘rabbit ear’-like regions
induced by the orbits OuLRn . The merging crises curves caused by the orbit OuLR are shown as thin
grey lines, the ones caused by OuLR2 as solid black lines, the ones caused by OuLR3 as thick grey lines
and the ones caused by OuLR4 as dashed lines. The vertical dashed line corresponds to the onedimensional bifurcation scenario presented in figure 10.
is at the bZ1 tangent to the other border-collision curve xk;r
(figure 9a). In the
LRk
following, let us denote the upper rabbit ear-like region (confined by the curves
[ =r
^ LRk and the lower rabbit ear-like region (confined by the curves g~[ =rk )
g^LRk ) by U
LR
~ LRk .
by U
The most important fact now is that this complete structure (triangle-like
centre region together with both upper and lower rabbit ear-like regions) occurs
for all kO1 and that the complete structures corresponding to adjacent values of
k overlap. This is a direct consequence of the period increment scenario with the
coexistence of attractors, where the solutions in the periodic regime overlap
pairwise for adjacent values of k. As a consequence of the overlapping structures,
^ LRkK1 overlaps with the U
~ LRkC1 within a part ðQ kC2k Þ of the trianglethe region U
LR
like centre region ðULRk Þ influenced by OuLRk . As an example, in figure 9b the
~ LR4 (dashed curves) and the region U
^ LR2 (thick grey
overlapping of the region U
curves) is shown. As one can see, this overlapping is located within the region
~ LR3 (black curves) and
Q 5LR3 . Similarly, within the region Q 4LR2 , the regions U
^ LR (thin grey curves) overlap. This explains the bandcounts 6, 8 and 10
U
mentioned at the beginning of this section. As one can see, the results obtained
[ =r
[ =r
numerically are explained by the analytically calculated curves g^LR and g~LR3 , as
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Bandcount increment: interior structures
3
6
8
2
2
4
3
2
1.0
0
– 0.8
0.31
0.40
0.51
3
3
3
Figure 10. Bifurcation scenario along the vertical line S(b)Z0.83 marked in figure 9b. The scenario
starts in the region Q 6LRhLR2 , transits the region Q 4LR2 and ends in the region Q 8LR2hLR3 . Within
^ LR and U
~ LR3 are shown, which lead to the
the region Q 4LR2 , both the overlapping interior regions U
bandcounts 6, 10 and 8.
^ LR and U
~ LR3 , as well as their
shown in figure 8b. Remarkably, the regions U
^ LR h U
~ LR3 , possess some further interior substructures. The investigaoverlap U
tion of these substructures remains for future work.
As a last step, let us illustrate the described behaviour under the variation of
one parameter. Figure 10 demonstrates the bifurcation diagram along the
vertical dashed line in figure 9b. The presented transitions between several multiband chaotic attractors (except the fine substructures mentioned above) can be
explained easily using the knowledge about the bifurcation structures in the twodimensional parameter space. The transition between the six- and four-band
attractors at grLR is caused by the unstable orbit OuLR . Between g rLR and g^ [LR ,
this orbit is located within the four-band attractor until it causes the next crisis
[
at g^[LR , where a four-band to a six-band transition occurs. The next crisis at g~LR
3
u
is induced by the unstable orbit OLR3 , which emerges at the border-collision
bifurcation x3;r
closely before g~[LR3 (figure 9a). This crisis increases the number
LR3
of bands from 6 to 10. The next crisis occurs at g^rLR and decreases the number of
bands from 10 to 8. It is the last crisis, which is caused by the unstable orbit OuLR ,
that will be destroyed closely after that by the border-collision bifurcation
[
~rLR3 and g[LR3 are induced by the orbit OuLR3 . As one
x0;
LR . The last two crises at g
^ LR and U
~ LR3 in the
can see, the beginning and end of the regions Q 4LR2 , U
bifurcation diagram presented in figure 10 can be explained by the six crisis
curves mentioned above.
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V. Avrutin et al.
7. Summary
In this work, we have investigated the behaviour of a specific piecewise-linear
discontinuous map in the chaotic domain. This map represents an approximation
of general piecewise-smooth maps in the neighbourhood of the point of
discontinuity and serves, therefore, as a normal form for these maps. Typically,
the dynamics in this domain are characterized by the term ‘robust chaos’, which
refers to the absence of periodic inclusions (windows) within. Nevertheless, an
infinite number of crisis bifurcations occur in this domain. These bifurcations
represent several interior and merging crises, which are organized in a complex
self-similar structure. The first level of this structure is given by the overall
bandcount increment scenario reported in part I of this work. The orbits OLRn ,
causing this scenario to emerge, originate from the periodic domain where they
are responsible for the formation of the period increment scenario with the
coexistence of attractors. At the boundary between the periodic and the chaotic
domain, these orbits become unstable and thereby responsible for the formation
of the overall bandcount increment scenario. A similar relation between the
bifurcation structures in the periodic and chaotic domain was already reported in
Avrutin & Schanz (in press) for the period adding and overall bandcount adding
scenarios. It was also reported in the cited work that further interior substructures denoted as interior bandcount adding and interior bandcount doubling
scenarios are nested into the regions induced by the overall bandcount adding
scenario. Regarding the present system, we found analogously interior bandcount
adding and interior bandcount doubling scenarios nested into the regions induced
by the overall bandcount increment scenario. Remarkably, the organizing
principles, as well as the scaling properties of these scenarios, are the same in the
case of the overall bandcount adding and the overall bandcount increment
scenarios. This leads us to the assumption that they are of some universality.
Note that the overall bandcount adding scenario, as well as the overall
bandcount increment scenario, are caused by an infinite number of crises
bifurcations induced by former stable periodic orbits. By contrast, the interior
scenarios (bandcount adding and bandcount doubling) are caused, in both cases,
by an infinite number of crises bifurcations induced by completely unstable
(nowhere stable) orbits.
Additionally, we have shown that the unstable periodic orbits OuLRn are
responsible not only for the formation of the overall bandcount increment
scenario. In fact, each of them leads to at least six merging crisis bifurcations.
Two of these crises represent a part of the overall bandcount increment scenario,
whereas the remaining four determine some regions with higher bandcounts
located within. This is explained by the fact that the existence regions of the
responsible unstable periodic orbits overlap pairwise (as a direct consequence of
the period increment scenario with the coexisting attractors, which occurs in the
periodic domain). This leads to an interaction of three subsequent unstable orbits
with periods nK1, n and nC1, which cause the bandcounts 2n, 2nC2 and 3nC1
to emerge.
So far, we have described almost all components that form the bifurcation
structure in the plane b!m. However, we have to keep in mind that all the results
presented so far are obtained for a specific, and not generic, value of the third
parameter, namely for aZK1. As a next step, in part III of our work we will
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Bandcount increment: interior structures
2263
present that the results obtained for this specific parameter value are helpful for
the understanding of the much more complex behaviour at arbitrary values of a.
This is required in order to transfer our results to applications, since it can not be
expected that a practical application operates exactly at those parameter
settings that correspond to the singular case aZK1. This will be demonstrated
in part III by investigating the chaotic domain of a two-dimensional map, which
is typically considered as the piecewise-linear normal form for many practical
systems in the neighbourhood of the point of discontinuity.
References
Avrutin, V. & Schanz, M. In press. On the fully developed bandcount adding scenario.
Nonlinearity.
Avrutin, V., Eckstein, B. & Schanz, M. 2007a On detection of multi-band chaotic attractors. Proc.
R. Soc. A 463, 1339–1358. (doi:10.1098/rspa.2007.1826)
Avrutin, V., Schanz, M. & Banerjee, S. 2007b Codimension-3 bifurcations: explanation of the
complex 1-, 2- and 3 -D bifurcation structures in nonsmooth maps. Phys. Rev. E 75, 066 205.
(doi:10.1103/PhysRevE.75.066205)
Avrutin, V., Eckstein, B. & Schanz, M. 2008 The bandcount increment scenario. I. Basic
structures. Proc. R. Soc. A 464, 1867–1883. (doi:10.1098/rspa.2007.0226)
Banerjee, S. & Verghese, G. (eds) 2001. Nonlinear phenomena in power electronics—attractors,
bifurcations, chaos, and nonlinear control. New York, NY: IEEE Press.
Banerjee, S., Yorke, J. & Grebogi, C. 1998 Robust chaos. Phys. Rev. Lett. 80, 3049–3052. (doi:10.
1103/PhysRevLett.80.3049)
Bernardo, M. d., Budd, C. J., Champneys, A. R. & Kowalczyk, P. 2007 Piecewise smooth
dynamical systems: theory and applications, vol. 163. Berlin, Germany: Springer.
Feigenbaum, M. J. 1979 The universal metric properties of nonlinear transformations. J. Stat.
Phys. 21, 669–707. (doi:10.1007/BF01107909)
Lyubimov, D., Pikovsky, A. & Zaks, M. 1989 Universal scenarios of transitions to chaos via
homoclinic bifurcations, vol. 8. London, UK: Harwood Academic Publishers.
Milnor, J. 1985 On the concept of attractor. Commun. Math. Phys. 99, 177–195. (doi:10.1007/
BF01212280)
Ott, E. 2002 Chaos in dynamical systems. Cambridge, UK: Cambridge University Press.
Zhusubaliyev, Z. T. & Mosekilde, E. 2003 Bifurcations and chaos in piecewise smooth dynamical
systems, vol. 44. Singapore: World Scientific.
Proc. R. Soc. A (2008)