## Abstract

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.

## 1. Introduction

An aperiodic and especially a chaotic attractor may consist of some number of bands (also denoted as connected components). Multi-band chaotic attractors (MBCAs), defined by , 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 period-doubling 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 *p*_{0}×2^{n} with *n* increasing from zero to infinity, the second one represents a sequence of MBCAs with *p*_{0}×2^{n} bands, whereby *n* decreases from infinity to zero. Hereby, at each bifurcation point, the bands of a (*p*_{0}×2^{n+1})-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 (*p*_{0}×2^{n}) 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 band-merging cascade described earlier, here the number of bands before the *n*th bifurcation is 2^{n+1}−1 and after the bifurcation it is 2^{n}−1, 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 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 shown in figure 1*a* consists of seven bands and the attractor of the Tinkerbell map shown in figure 1*b* 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 *Σ*/Δ modulators), was already investigated in many works. However, until now only bifurcation scenarios involving non-chaotic 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 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.

*Let* *be an MBCA of a dynamical system with a continuous system function. Let us assume that* *consists of* *bands* . *Then*, *each band* *has exactly one successor and one predecessor band*,

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 , there exists some number *N*_{1} (first return time) withfor an arbitrarily small *ϵ*. Let *ϵ* be smaller than the distance between adjacent bands, then . Hereby, due to theorem 2.1, the number *N*_{1} is a multiple of . Similarly, the second return time *N*_{2} for can be calculated, etc. Iterating the system for a sufficiently long time, we obtain the set of *M* return timeswhereby the number *M* can be arbitrarily high and is only limited by the computation time. The key point is that all return times *N*_{i} are multiples of the number of bands that means with some integer number *m*_{i}. Then, for a sufficiently large set , the number can be calculated as the greatest common divisor (GCD) of the return times

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 set of *M* subsequent points from the attractor: with . These points are denoted in the following as basis points. Within the next iteration steps, we find the first return times (for a given *ϵ*) for some of these points and obtain the set . Then, for a sufficiently large set the number can be calculated as the GCD of all detected return times, whereby the number of the required iteration steps 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 . Hereby, the necessary number of iteration steps remains acceptable if a sufficiently large set 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 is considered, which shows in the interval a band-merging cascade. From this sequence of bands, we detected numerically the bandcounts up to . As one can see, the results obtained using the GCD-based method (figure 2*c*) are in a good accordance with the usual bifurcation diagram (figure 2*b*) and with the boundaries of the chaotic attractors (figure 2*a*) 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 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 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 of clusters is counted. Note that this method requires a higher number of iterations than is the case for the GCD-based method. If 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(3.1)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 *x*=0 is given by the parameter *l*. Using a suitable scaling, it can be shown that system (3.1) can be reduced to three cases, *l*=−1, *l*=0 and *l*=1.

In the following, we use the parameter transformation(3.2)which maps the infinite parameter space onto the finite box 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 ±∞.

In the following, *O*_{σ} denotes the periodic orbit corresponding to the symbolic sequence *σ*, consisting of symbols (for a point *x*<0) and (for *x*>0). denotes the stability area of *O*_{σ}, which are bounded by the parameter subspaces , where the *i*th point of *O*_{σ} collides with the border from the left side or from the right side (*d*∈{*l*, *r*}), and by the parameter subspace *θ*_{σ}, where this orbit becomes unstable. Note that at least for simple symbolic sequences like and , these curves can be determined analytically for all *n* as presented in appendix B. Additionally, denotes the area in parameter space corresponding to *n*-periodic dynamics (obviously, for ) and , the area corresponding to a chaotic *n*-band attractor.

## 4. Continuous piecewise-linear map

Let us start with the case *l*=0. 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 piecewise-linear map(4.1)which is identical with system (3.1) for *l*=0 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 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 and a periodic (figure 3*a*,*b*) or a chaotic (figure 3*c*–*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 3*c*,*e*), whereas in other cases they are approximately uniformly distributed in the state space (figure 3*d*)? Researchers familiar with MBCAs may ask additionally, whether the attractors shown in figure 3*c* 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 3*f* have significant traces of a three-band attractor?

When dealing with these and similar questions, we state that system (4.1) can be further reduced to three cases, , and , 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 and . For , 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 and are equivalent up to change of the parameters and . For this reason, we restrict ourselves in the following to the case . Related to the original three-dimensional parameter space , it has to be kept in mind that for the fixed point shown in the left parts of figure 3 is stable in the area , i.e. .

A typical bifurcation scenario, which can be observed for a fixed 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 is followed by a 2*n*-band chaotic attractor emerging at the point , 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 , 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 . At this bifurcation, a one-band attractor emerges, which persists until the next stable periodic orbit (namely, the orbit with period *n*+1) emerges. We summarize this scenario in the following scheme:(4.2)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 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 are bounded by the curves and . Remarkably, for all *n* these curves originate from the same point 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 and 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 with the maximal distance from the point *B*^{1}, is the intersection point . Additionally, we state that for increasing *n* the sequence of points converges monotonously to the point with . Therefore, if one keeps *a* fixed to a value *a*<*a*^{*} and varies *b* towards −*π*/2, the areas are intersected for all *n* and the scenario (4.2) takes place *ad infinitum*. In contrast to this, for *a*>*a*^{*} only a finite number of areas can be intersected. Similarly, if *b* is fixed to a finite value and *a* is varied, it is not possible to intersect all areas , so that scenario (4.2) can be observed under variation of *a* in a truncated form only.

Note that the areas, and , shown in figure 5 play a special role for system (4.1), because these areas represent the beginning of the usual band-merging scenario(4.3)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 .

Next, let us go back to the three-dimensional parameter space of system (4.1). Intersecting the plane for some value of *a* and *b*, we leave the existence area of 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 or 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 a transition from the fixed point to which attractor. Using this road map, the questions mentioned at the beginning of this section can be easily explained. Indeed, in figure 3*a*,*b*, we observe a transition from the stable fixed point to the periodic attractors and , respectively. The corresponding parameter values are marked in figure 5*b* with letters (*a*), (*b*) and lie in and , respectively. Note that due to the used scaling of the parameter space, the area is very thin and therefore difficult to recognize. Owing to the fact that the location of the areas is ordered according to *n*, it can be easily guessed where this area lies. The chaotic attractors shown in figure 3*c*,*e* belong to areas and , whereby the parameter values lie close to the areas and , as marked in figure 5*a*,*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 3*c* are in fact four-band attractors and not two-band attractors, as one may assume. Similarly, the one-band attractor shown in figure 3*f* turns out to be located in the parameter space close to the bifurcation line between areas and , which explains the traces of a three-band attractor in the distribution of its invariant measure (figure 7).

Finally, note that due to the symmetry of the cases exact analogous results can be obtained for the other fixed point as well.

## 5. Discontinuous piecewise-linear map

Next, let us consider multi-band attractors of system (3.1) in the case *l*=−1. 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 piecewise-linear 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*×*μ* for *a*=−0.73. 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 −*π*/4<*a*<0, whereby for *a* close to −*π*/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*×*μ* for −*π*/4<*b*<0.

It can be shown that the only stable periodic orbits of system (3.1) in the parameter subspace we investigate are . Hereby, for each *n*, the areas and 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 dynamics by the line, consisting of pieces of curves and . Outside the parameter area , system (3.1) shows divergent behaviour. Now the question arises, how the area is organized and which MBCAs can be found within? Some examples of bifurcation scenarios observable under variation of parameters across the area are shown in figure 7. Unfortunately, it is a hard task to explain the interior structure of the area 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 , where the orbit becomes unstable, a 2(*n*+1)-band attractor emerges. As in the case of the piecewise-linear continuous map, this attractor undergoes a band-merging bifurcation, where it is replaced by an (*n*+1)-band attractor. Remarkably, both areas and are bounded by the curves and , where the unstable orbits and are destroyed by a border-collision bifurcation. This scenario can be summarized as follows:(5.1)This structure is repeated for all *n*>1, 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 and are interrupted by some regions with higher bandcounts .

In order to keep the presentation compact, we restrict our considerations in the following to the areas . Note that the structure of the areas represents an interesting topic as well and will be considered in future work. As an example, in figure 10, the structure of the area for *n*=3 is shown (i.e. the area , which begins at the line ). The bandcount of this area will be denoted as the basic bandcount . As shown in figure 10, the most expanded areas, which interrupt the area , form the sequence with bandcounts 22, 28, 34, etc. This sequence can be expressed in a closed form as(5.2)with *m*=1, 2, 3, … for *n*=3. Note that this equation holds for all and may be rewritten as with *m*=1, 2, 3, … and with *n*=1, 2, 3, ….

However, the sequence (5.2) still does not describe the complete structure of the area . Using a sufficiently high resolution in the parameter space, we state that between each two areas and there exists the area of attractors with(5.3)bands. Similarly, between and there is the area with bands, whereas between and the area with 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 , with *R*_{b}=0.001 and *R*_{μ}=0.002 around the codimension-2 bifurcation point , where all areas forming 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 of the state space containing the attractor was covered by 5×10^{6} partitions. This fine partitioning requires a corresponding high number of iteration steps (used value: 3×10^{8} steps). Note that performing the same calculation with lower accuracy (namely, 10^{6} partitions, what seems to be quite fine, and 6×10^{7} 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 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 we observe a symmetrical structure consisting of two sequences of areas , , similar to sequence (5.2). The structure within the area 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.

Finally, we remark that all areas involved in the scenario described earlier originate from the codimension-2 bifurcation point , 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 , , and intersect. In figure 9, two bifurcations belonging to this sequence are marked with *H*_{1–4–6–3} and *H*_{1–5–8–4}. As one can see, at these points four different chaotic attractors emerge. In order to characterize such a bifurcation point, we perform a one-dimensional parameter scan along the boundary of a sufficiently small convex neighbourhood. The results of this scans are shown in figure 14, where the parameter substitution , with *R*=0.05 for figure 14*a* and *R*=0.0075 for figure 14*b* is used. Hereby, we observe how the 2*n*-band attractor emerges as some kind of overlapping of an *n*-band and an (*n*+1)-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 non-robust chaotic attractors.

## Footnotes

- Received November 21, 2006.
- Accepted January 24, 2007.

- © 2007 The Royal Society