## Abstract

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated 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 Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

## 1. Introduction

Investigating the chaotic domain in a multidimensional parameter space represents a challenging task important both from the theoretical point of view and with respect to applications. In the case that a technical system operates in a chaotic domain, it is important to know how the variation in parameters influences the dynamics. For example, one is interested to know whether the chaotic domain in the parameter space is interrupted by periodic inclusions (‘windows’), or whether the attractors are one-band or multi-band attractors and so on. Consequently, bifurcations and especially crisis bifurcations occurring in the chaotic domain are of great importance. When dealing with piecewise-smooth models that originate from a broad spectrum of applications, e.g. mechanical oscillators with impacts and/or stick–slip effects, switching electronic circuits, power converters and others, the situation that so-called robust chaos occurs is well known (see references in Banerjee & Verghese (2001), Zhusubaliyev & Mosekilde (2003) and Bernardo *et al*. (2007)). Typically, this notion refers to the fact that the chaotic domain does not contain any periodic inclusions (Banerjee *et al*. 1998). However, it was shown in previous publications and especially in Parts I and II of this work (Avrutin *et al*. 2008*a*,*b*) that this domain can, nevertheless, possess a complex structure formed by crisis bifurcations. At these 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).

In Parts I and II of this work, we investigated the piecewise-linear discontinuous map given by(1.1)This map represents a special case of a well-known two-dimensional piecewise-linear normal form investigated by many authors (see Bernardo *et al*. 2007). Concerning the periodic solutions, we considered the characteristic case of a negative jump (*l*<0, whereby the investigated system can always be reduced to the case *l*=−1 by a suitable scaling) and the case *a*<0, where the periodic domain is organized by the period increment scenario (Avrutin *et al*. 2007*b*) with coexisting attractors, sometimes also referred to as ‘multi-stability’. This scenario is formed by a sequence of periodic orbits with *k*=1, 2, …, whose stability regions overlap pairwise.1 The question we investigate in this series of papers is how these orbits influence the dynamics after the transition to chaos.

The first step towards understanding the structure of the chaotic domain was done by investigation of the special case *a*=−1. 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. Note hereby that, for a better graphical representation, we use in all figures throughout this paper the same topology preserving scaling of parameters as in Parts I and II of this work, namely(1.2)The boundaries between the regions involved in the bandcount increment scenario represent curves of merging crises,2 caused by the orbits . In this sense, both the periodic and chaotic domains 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 , the bandcount increment scenario includes regions of two types, namely triangle-like regions and trapezoidal regions . As indicated by the sub- and superscripts, each region of the first type is influenced by two unstable periodic orbits and , and contains (2*k*+4)-band attractors. By contrast, each region of the second type is influenced by only one orbit , so that the corresponding attractors have *k*+2 bands.

So far, the overall structure of the chaotic domain given by the bandcount increment scenario as well as its substructures given by the bandcount-adding and -doubling scenarios are explained in the case *a*=−1. In the following, we will demonstrate how these results can be generalized. Hereby, the influence of the variation in *a* on the overall bandcount increment structure formed by the regions for *k*=1, 2, … (§§2–4), as well as on the bandcount-adding substructures located within the regions (§5), will be described. We will see that all these structures are organized by the same unstable orbits as in the case *a*=−1, although the shapes of the bifurcation structures change significantly as soon as *a* is varied. Owing to this, one can say that the structures reported in the previous Parts I and II of this work are deformed under variation in the parameter *a*. As we will demonstrate in the following, this deformation preserves the relative location of regions leading to specific bandcounts with respect to each other. Moreover, it is topology preserving for any value of *a* except for *a*=−1.

## 2. Deformation of the parameter space for *a*≠−1

Examples for numerically calculated bifurcation structures in the case *a*≠−1 are shown in figure 2. As one can see, there are significant differences compared with the bifurcation structures shown in figure 1. To explain these differences, let us first consider the deformation of the periodic domain and the boundary under variation of *a*, since all structures we described so far (both the bandcount increment scenario and its interior substructures) originate from the boundary between the periodic and chaotic domains.

Figures 3*a* and 4*a* show the analytically determined boundaries of the regions of periodic dynamics for the case −1<*a*<0 and *a*<−1, respectively. Each of the regions is bounded from above and below by the border-collision bifurcation curves and , respectively, defined by the condition that the first (respectively last) point of the orbit collides with the boundary *x*=0. From the right-hand side, the region is bounded by the bifurcation curve where the orbit becomes unstable. The overlapping of the subsequent regions , and leads to the fact that for each *k* we observe not only the region where the orbit represents the unique attractor but also the regions and where attractors with periods *k* and *k*+1 (respectively *k*+1 and *k*+2) coexist.

Let us now turn to the boundary between the periodic and chaotic domains. In the case *a*=−1, this boundary is given by the straight line *b*=1 where the stability boundaries of all orbits are located. For *a*≠−1, this is not the case. Recall (see Part I, §5) that for −1<*a*<0 the line is located on the right-hand side of , as shown in figure 3*a*, and for *a*<−1 on the left-hand side, as shown in figure 4*a*. Therefore, varying *a* from the value −1 to some values −1±*ϵ*_{a}, we observe that the boundary of the chaotic domain becomes a non-smooth curve composed of several pieces. In the case −1<*a*<0, this curve consists of the parts of the curves where the orbit becomes unstable, and the border-collision bifurcation curves (Part I, eqn (5.1)) where the first point collides with the boundary *x*=0. Hence, the boundary is defined by the curves *upward* from the intersection points and by the curves *rightward* from the intersection points (figure 3*a*). Similarly, in the case *a*<−1, the boundary consists of the same curves and the border-collision bifurcation curves (Part I, eqn (5.2)) where the last point collides with the boundary *x*=0. Hence, the boundary is defined by the curves *downward* from the intersection point and by the curves rightward from the intersection point (figure 4*a*).

Consequently, the question arises of how the structure of the region beyond the boundary is organized in the case *a*≠−1. As shown in figure 2, this structure is more complex than that in the case *a*=−1. However, we will demonstrate that almost all results obtained for *a*=−1 are still valid for *a*≠−1. As we will see, the structure of the region for *a*≠−1 can be described as a deformation of the structure of this region for *a*=−1. Recall that the bandcount increment scenario takes place within the chaotic domain , but is influenced by a structure that is organizing the adjacent periodic domain, namely by a sequence of pairwise overlapping regions with *k*=1, 2, … forming the period increment scenario with coexisting attractors. In the chaotic domain, each of the involved unstable periodic orbits causes merging crises, which separate three multi-band regions , and . Note that the region also represents the region for the predecessor orbit with *k*′=*k*−1. Analogously, the region also represents the region for the successor orbit with *k*″=*k*+1. This is a consequence of the overlapping structure. The important fact is that one particular orbit is involved in crisis bifurcations of three regions.

In §§3 and 4, we describe how the regions and are deformed for *a*≠−1. The description of the regions is not within the scope of this work for two reasons. First, these regions are not deformed for *a*≠−1, although they undergo a similar change to the other two regions, i.e. they are partially covered by the periodic regime. Second, their inner structure is detected mainly numerically, and a complete analytic description is not available as not all details are clear yet.

## 3. Structure of the parameter space for −1<*a*<0

As in the case *a*=−1, the codimension-2 bifurcation points lie on the boundary . In fact, the curves and compose the boundary as described in §2. By contrast, the codimension-2 bifurcation points lie no longer on the boundary but for each *k* in the periodic domain within the region . Furthermore, the fact that for each *k* the stability boundary is located on the left-hand side of the stability boundary leads to additional codimension-2 bifurcation points .

To demonstrate the influence of the deformation of the boundary on the bifurcation structure within , let us consider as an example the curves of crisis bifurcations involving the unstable orbit . Recall (see Part I, fig. 7) that for *a*=−1 the region where the attractors are directly influenced by this orbit has a triangle-like shape and consists of three parts, namely (influenced by and ), (influenced by and ) and (influenced by only), as shown in figure 1*b*. The boundaries of the region are in this case given by two curves of crisis bifurcations and the boundary , where the stability boundaries , and are located.

As shown in Part I of this work, the curves can be calculated for arbitrary *k* using the equations(3.1)and(3.2)respectively. Here, denotes the *k*th point of the orbit , whereas and represent the corresponding points of the kneading orbit (see Part I, §4). Solving equations (3.2) and (3.1), we obtain(3.3)(3.4)Note that eqns (6.3) and (6.4) in Part I represent the special case of equations (3.3) and (3.4), respectively, for *a*=−1.

As shown in figure 3*b*, for −1<*a*<0, the bold-framed region consists of the same three parts as for *a*=−1, namely , and . However, the shape of is now more complex. Namely, it is bounded not only by the merging crisis curves and the stability boundary , but also by the curves bounding the adjacent region , namely by those parts of the curves and which define the boundary , as described in §2.

Let us now consider the lower corner of the region , where the curve originates from, or, in other words, where it hits the boundary . In the case *a*=−1, this point is given by the intersection point of the curves and , i.e. by the lower corner of the region . By contrast, in the case −1<*a*<0, the lower corner of the region , i.e. the intersection point of the curves and , is located in the periodic domain within the region , as shown in figure 3*a*. Therefore, the crisis bifurcation curve representing the lower boundary of the region or, more precisely, of the regions and , cannot emerge from this intersection point. Instead, the codimension-2 bifurcation point where the curve originates from is in this case the intersection point . However, neither the orbit nor the orbit is involved in the crisis bifurcation . Remarkably, at the codimension-2 bifurcation point where the curve originates from, not only do the curves and intersect but also the border-collision bifurcation curve (dashed curve, figure 3*b*) of the orbit is responsible for this crisis.

Let us next consider the upper corner of the region , where the curve originates from, or, in other words, where it hits the boundary . Solving equation (3.1) for *k*=3, we obtain a curve , which originates from the intersection point of the curves and , but of course this curve corresponds to the merging crisis curve only in the chaotic domain, i.e. on the right-hand side of the boundary , which is in that part of the parameter space given by the stability boundary . This can be easily explained by taking into account that the condition (3.1), used for the calculation of the curve , represents the fact that the point of discontinuity belongs to the stable manifold of the unstable periodic orbit . If this situation occurs in the chaotic domain, it corresponds in fact to the merging crisis caused by the unstable orbit . However, in the periodic domain (that is, if the unstable orbit coexists with a stable periodic orbit), this situation does not have any further consequences for the asymptotic dynamics. Consequently, the curve becomes a merging crisis curve at that point where it intersects the boundary . The curve originating at this point represents rightwards the upper boundary of the region or, more precisely, of the regions and . The lower boundary of the region is given by the merging crisis curve , which originates from the point . The bifurcation structure at this point is similar to the structure at the point described above.

After considering the outer boundaries of the region , let us now consider the boundaries between the regions , and . These boundaries are located within and define, together with the outer boundaries of the region , the region . Recall that, in the case *a*=−1, this region has a trapezoid-like shape and is bounded from the right and from below by the curves and , respectively, and from the left and from above by the curves and , respectively. For −1<*a*<0, the situation with the curve is the same as described above for the curve . For the same reason, it originates from a point at the boundary (which is in this specific part of the parameter space given by the stability boundary ) where the solution of equation (3.1) (for *k*=2) reaches the chaotic domain. Therefore, the boundary of the region is given by the curves , , and as in the case *a*=−1 and additionally by and , as they define the boundary in this part of the parameter space.

In one sense, one can say that some part of is partially covered by the region . Hereby, there are two possible situations dependent on the relative location of the upper right corner of the region and the curve . If the point is located below the curve , the region is not convex but still connected. In the opposite case, the region would split into two parts, as shown in figure 3*b*, for the region . As one can see, the region is partially covered by and, therefore, consists of two parts, one of them located above and the other located on the right of .

## 4. Structure of the parameter space for *a*<−1

The situation in the case *a*<−1 is similar to but in some ways also the *reverse* of the situation in the case −1<*a*<0. As in the case *a*=−1, for each *k*, the codimension-2 bifurcation point lies on the boundary as described in §2. By contrast, and in *reverse* of the situation in the case −1<*a*<0, the codimension-2 bifurcation point lies no longer on the boundary but in the periodic domain within the region . Furthermore, the fact that for each *k* the stability boundary is located on the left-hand side of the stability boundary leads to the additional codimension-2 bifurcation points .

To demonstrate the influence of the deformation of the boundary on the bifurcation structure within also in the case *a*<−1, let us consider again the curves of crisis bifurcations involving the unstable orbit . As shown in figure 4*b*, for *a*<−1 the bold-framed region consists again of the three parts , and . In this case, it is bounded by the merging crisis curves , the stability boundary and, additionally, by the curves bounding the adjacent region , namely by those parts of the curves and that define the boundary , as described in §2.

Let us first consider the upper corner of the region where the curve originates from, or, in other words, where it hits the boundary . In this case, the codimension-2 bifurcation point is given by the intersection point . Analogous to the previous case, neither the orbit nor the orbit is involved in the crisis bifurcation and the border-collision bifurcation curve (dashed curve, figure 4*b*) of the orbit responsible for this crisis intersects the codimension-2 bifurcation point .

Let us next consider the lower corner of the region , where the curve originates from, or, in other words, where it hits the boundary . Solving equation (3.2) for *k*=3, we obtain a curve , which originates from the intersection point of the curves and . Similar to the case −1<*a*<0, this curve becomes a merging crisis curve at the point where it intersects the boundary . The curve originating at this point represents rightwards the lower boundary of the region or, more precisely, of the regions and . The upper boundary of the region is given by the merging crisis curve , which originates from the point . The bifurcation structure at this point is similar to the structure at the point described above.

Let us finally also consider in this case the boundaries between the regions , and . For *a*<−1, the situation with the curve is the same as described above for the curve . For the same reason, it originates from a point at the boundary (which is in this part of the parameter space given by the stability boundary ) where the solution of equation (3.2) (for *k*=4) reaches the chaotic domain. Therefore, the boundary of the region is given by the curves , , and as in the case *a*=−1 and, additionally, by and , as they define the boundary in this part of the parameter space. In this case, one can say that some part of is partially covered by the region . Hereby, there are two possible situations dependent on the relative location of the lower right corner of the region and the curve . If the point is located above the curve , the region is not convex but still connected. In the opposite case, the region would split into two parts.

As in the case *a*=−1, the structure described in §§3 and 4 is repeated for all regions , which causes the complex topological structure of the chaotic domain close to the boundary . As can be demonstrated by analytical calculation of the involved crisis curves, the bandcount increment structure, developed in the case *a*=−1 in its most simple and most definitive form, becomes in both cases −1<*a*<0 and *a*<−1 more complex owing to the non-trivial shape of the boundary between the periodic and chaotic domains, as well as to the partial covering of the regions . Nevertheless, the basic component of this structure given by the sequence of overlapping regions , which reflects the sequence of overlapping regions , persists in both cases.

## 5. Bandcount adding

The next question we have to deal with concerns the bandcount-adding structures, which we detected in the case *a*=−1 within each of the regions . As shown in Part II of this work, within each region there exist two families of subregions with and , respectively. The bandcounts in these regions are given by(5.1)and can be explained by taking into account that from gaps of the -band attractors *k*+1 gaps are occupied by the unstable orbit , *k*+2 gaps by the unstable orbit and the remaining and , respectively, by the orbit .

These regions serve as a first layer of the infinite bandcount-adding scheme described in detail by Avrutin & Schanz (2008) and organized in the same way as the well-known Farey tree- or Stern–Brocot tree-like period-adding scheme. Note that, according to the period-adding scheme, between the existence regions of the unstable periodic orbits and , the existence region of the orbit is located. As a direct consequence of this, between the regions and , there exists a region with bandcount and so on ad infinitum. As an example, let us consider the subregions located within the region , as shown in figure 1. The two families of subregions mentioned above are given by and in this case. Consider the first family, then between the regions and there exists the region and so on. Remarkably, in the case *a*=−1, the origins of the bandcount-adding structure within the region are distributed along a specific part of the boundary . More precisely, for each unstable periodic orbit O_{σ} involved in a crisis bifurcation forming this inner bandcount-adding structure, the point where the corresponding region originates from lies on this boundary between the intersection points and . Varying *a* from the value −1 to any other value −1±*ϵ*_{a}, we observe that the bandcount-adding structure undergoes a dramatic change. By calculating the boundaries of the regions involved in the bandcount-adding scenario within a region , one can prove that the origins of all these regions collapse to a single point, as shown in figure 5. In the case −1<*a*<0, this point is given by the lower corner of the region , i.e. the intersection point , whereas in the case *a*<−1 this point is given by the upper corner of the region , i.e. the intersection point . Obviously, each of these points represents a codimension-2 bifurcation with quite unusual properties. Namely, at this point, one stable periodic orbit is destroyed and the other one becomes unstable. Additionally, at the same point, an infinite number of everywhere unstable periodic orbits emerge, leading to the fact that from this point an infinite number of interior crisis curves originate, bounding the regions with different bandcounts. Numerically observed, this bifurcation was reported for the first time in Avrutin *et al*. (2007*a*). However, in the cited work, no analytical explanation of the observed phenomenon was presented.

Note that the size of the bandcount-adding structures emerging at the point in the case −1<*a*<0 and the point in the case *a*<−1 decreases with increasing distance to the plane *a*=−1. Therefore, these regions become very small and hard to detect numerically. Consequently, the question arises whether they persist for arbitrary values of *a*.

As an example let us consider the largest region involved in the bandcount-adding scenario within the region . As shown in Part II of this work, this region is given by . The boundaries of this region for the special case *a*=−1 are given by eqns (2.4) and (2.5) in Part II. However, eqn (2.3) in Part II allows us to calculate these surfaces for arbitrary values of *a* and *b*. As a result, we obtain(5.2)

Hence, we can calculate the area of the region in the plane *b*×*μ* as a function of *a*(5.3)where denotes the values of *μ* at the surfaces as a function of *a* and *b*. Here, the lower limit *b*_{1} corresponds to the stability boundary for *a*<−1 and for −1<*a*<0. The upper limit *b*_{2} corresponds to the rightmost point of the region given by the intersection point of the curves . Figure 6 shows the area calculated according to equation (5.3). As one can see, this area is maximal at some value *a*^{*} close to *a*=−1 and then decreases monotonously in both directions. Hereby, it becomes clear why the region is difficult to observe numerically for values of *a* far from *a*^{*}. For example, at *S*(*a*)=−1.1 (that is, *a*≈−1.96) the area of the region is approximately 2.0345×10^{−8}. However, the area of the region tends to zero asymptotically for *a*→0 and *a*→−∞. Consequently, the region exists for any value of *a*∈(−∞,0). Furthermore, there is numerical evidence that, for all other regions involved in the bandcount-adding structures, the dependency of the area on *a* is similar. In other words, all these regions exist at any value of *a*∈(−∞,0), although their areas may decrease rapidly.

## 6. Summary

In this Part III of our work about the bandcount increment scenario, we described the generic case *a*≠−1. The bifurcation structures in the domain of robust chaos can be explained as the deformations of the bifurcation structures for the non-generic case *a*=−1 described in Parts I and II of this work.

In particular, the regions influenced by specific unstable periodic orbits , which possess a triangle-like shape in the non-generic case *a*=−1, possess a non-convex pentagonal shape in the generic case *a*≠−1. For values of *a* far away from −1, the non-convexity leads these regions to split into two parts.

Another significant change in the bifurcation structure is related to the substructures of the regions located within the regions . It was demonstrated in Part II of this work that these substructures are self-similar and organized by the bandcount-adding scenario as described in Avrutin & Schanz (2008). Hereby in the non-generic case *a*=−1, the origins of the regions involved in this scenario are distributed along the boundary . Now we have shown that, in the generic case *a*≠−1, these origins collapse to singular points at this boundary. The size of the specific regions forming the bandcount-adding scenario decreases with increasing distance from the plane *a*=−1, so that they become difficult to detect numerically. However, we showed analytically that these regions persist for any *a*.

Remarkably, the bandcount increment structures occur not only in the one-dimensional map we investigated but also in the multidimensional case. There is at least numerical evidence that the two-dimensional discontinuous normal form (Dutta *et al*. 2008) demonstrates the bandcount increment scenario as presented in this work. Furthermore, the continuous normal form (Banerjee & Grebogi 1999; Bernardo *et al*. 1999; Zhusubaliyev *et al*. 2008) shows similar, although not completely identical, bifurcation structures. The investigation of these bifurcation structures represents a challenging task, as there is a significant difference between one- and multidimensional maps regarding the complexity of the analytical calculation of the crisis curves. This is not only because the analytical calculation of unstable periodic orbits is more difficult in this case, but also, and mainly, because the determination of a kneading orbit responsible for the boundary of an attractor represents a cumbersome task. For details, we refer to Mira *et al*. (1996).

## 7. Conclusion

The work presented here was motivated by a seemingly simple question, namely how is the domain of so-called robust chaos organized? The phenomenon of robust chaos, often observed when dealing with piecewise-smooth models, is characterized by the absence of periodic inclusions, but shows typically complex structures formed by multi-band chaotic attractors. Whereas several bifurcation structures in the periodic domain have been investigated (see Banerjee & Verghese 2001; Zhusubaliyev & Mosekilde 2003; Bernardo *et al*. 2007), there was a lack of results about the bifurcation structures in the chaotic domain, except it was known that they are formed by interior and merging crisis bifurcations (Grebogi *et al*. 1982; Maistrenko *et al*. 1996). This situation changed when some simple but efficient methods for the numerical investigation of multi-band attractors were developed (Avrutin *et al*. 2007*a*). Using these methods, we were able to discover regularities in the occurrence of multi-band attractors numerically. These discoveries served us as the basis for our analytical investigations.

Since the bifurcations forming the structures in the chaotic domain are crisis bifurcations caused by unstable periodic orbits, the structures of adjacent periodic and chaotic domains are related. After a periodic orbit becomes unstable at the boundary of the periodic domain, it can cause a crisis in the chaotic domain. This was the reason why we considered two typical bifurcation scenarios that are often observed in piecewise-smooth systems, namely the period adding and period increment scenarios with coexisting attractors.

The influence of the period-adding scenario on the structure of the adjacent chaotic domain is reported in Avrutin & Schanz (2008). In this work, we presented the bandcount-adding bifurcation structure, which is formed by interior crises and leads to a self-similar structure of the chaotic domain. The connection between periodic and chaotic domains allowed us to describe the unstable periodic orbits responsible for the formation of this scenario in terms of the corresponding symbolic sequences and to calculate the scaling constants in the underlying two-dimensional parameter space.

The situation in the case of the period increment scenario with coexisting attractors turned out to be significantly more complex, so that its explanation required the three parts of this work. As in the case of the bandcount-adding scenario, the structure of the periodic domain influences the chaotic domain where we discovered an infinite sequence of multi-band attractors with increasing bandcounts. This sequence forms a novel bifurcation scenario, which we have named the bandcount increment scenario and described in detail.

We demonstrated in Part I of this work that, in contrast to the overall bandcount-adding scenario, the overall structure of the bandcount increment scenario is formed by merging crises. Additionally, owing to the fact that the regions of existence of orbits with subsequent periods overlap, we detect a sequence of regions in the chaotic domain where the bandcounts result from this overlapping. In Part II of this work, we focused on the interior substructures of the overall bandcount increment scenario and demonstrated that they are formed by the already known bandcount-adding phenomenon. The complete self-similar structure as described in Avrutin & Schanz (2008) occurs here within each of the regions involved in the overall bandcount increment structure. The results reported in Parts I and II of this work were obtained for a non-generic case where the bifurcation structure exists in its most simple and definitive form. Finally, in this Part III, we turned to the generic case and showed that the reported bifurcation structure persists, but will be deformed and partially covered by the periodic domain.

The results presented in this work seem to be purely theoretical and not directly applicable. However, when dealing with technical devices operating within the chaotic regime, it is often important to guarantee broadband chaos. A typical example for such an application is secure communication using chaotic attractors as signal carriers. In this case, one has to know the structure of the multi-band windows in order to avoid them.

## Footnotes

↵For details related to the notation used, see Part I of this work.

↵To avoid confusion, we emphasize that the term ‘merging crisis’ refers in this work to bifurcations where some of the bands of a multi-band chaotic attractor collide pairwise, and not to the bifurcation where the two coexisting chaotic attractors collide (Ott 2002).

- Received June 4, 2008.
- Accepted July 24, 2008.

- © 2008 The Royal Society