## Abstract

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.

## 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 one-band 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 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*. (2007*a*). 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(1.1)This map represents a special case of a well-known two-dimensional piecewise-linear 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 *l*=−1 by 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 denoted as ‘multi-stability’). This scenario is formed by a sequence of periodic orbits with *k*=1, 2, …, whose stability regions (in the parameter space) 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 *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. As already mentioned, the numerical results are obtained by a box-counting-based 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(1.2)The boundaries between the regions involved in this scenario represent curves of merging crises2 caused by the orbits . 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 , the bandcount increment scenario includes regions of two types, namely triangle-like regions and trapezoidal regions . As indicated by the subscripts 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.

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 and the other within the regions . Again, 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

Let us start with the structure of the regions and consider as a first step the case *k*=1. The numerically calculated structure of the region is shown in figure 2*a*. Comparing this structure with the *bandcount adding* structures reported in Avrutin & Schanz (in press), we recognize an unambiguous similarity. As shown in figure 2*a*, the region located in the middle of has bandcount 16, and above this region we observe a sequence of regions with bandcounts _{m}=20, 24, 28, 32, …=16+4*m*, *m*∈, whereas below we detect a sequence of regions with _{m}=22, 28, 34, 40, …=16+6*m*, *m*∈. 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 2*c*, where the parameter *μ* is varied along the dashed line (*b*)=0.7895 in figure 2*a*. 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 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 . As a consequence, each multi-band attractor within this region has two gaps occupied by the unstable orbit and three gaps occupied by the unstable orbit . In particular, for the region of 16-band attractors located in the middle of , 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 . The border-collision curves bounding the region of existence of this orbit (figure 2*b*) can easily be calculated,(2.1)and(2.2)Note that the region originates from the point *b*=1, *μ*=1/4. 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 . The curves of interior crises caused by the orbit result from the intersections of specific points of this orbit with suitable points of the kneading orbit. Solving the equations(2.3)we obtain the two curves(2.4)and(2.5)bounding the region . As one can see from figure 2*b*, this region originates from the same point *b*=1, *μ*=1/4 as the region . 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 with bandcounts(2.6)shown in figure 2*b*, 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 and the second family by interior crisis bifurcations of the sequence of completely unstable orbits . The bandcounts of the corresponding attractors result from 4*m*+6 and 6*m*+4 gaps, respectively, occupied by the points of these orbits and five further gaps where the orbits and are located. The boundaries of the corresponding regions and can be calculated analytically for an arbitrary *m*. By solving the equations(2.7)and(2.8)we obtain the curves bounding the family of regions . The boundaries of the regions can be calculated analogously.

Comparing the numerical and analytical results shown in figure 2, we state that the substructures within the region include many more regions than the two families and . In fact, these families 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 . Then, between each two subsequent regions and of the first generation, we observe the region of the second generation. As one can see for *m*=0 and 1, this results in the bandcounts 30 and 38 marked in figure 2*c*. Between the regions and , there exists the region , which belongs to the third generation (e.g. the bandcount 52 shown in figure 2*c*). This process continues ad infinitum and explains the bandcounts shown in figure 2*c* on the r.h.s. of the region . The bandcounts shown in figure 2*c* on the l.h.s. of the region can be explained in exactly the same way using the sequences . In both cases, each orbit , involved in this bandcount adding scenario, is a completely unstable orbit responsible for the emergence of the region with . Furthermore, the relative location of the regions _{σ} and is the same as shown in figure 2*b* for the orbit . Namely, both regions originate from a point at the line *b*=1 and the curves are tangent to the boundaries of _{σ} 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.

## 3. Bandcount doubling

In figure 2*a*, it is clearly visible that the regions forming the bandcount adding structure within the region 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 this cascade is formed 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 2+3+10+20 gaps occupied by the orbits , , and , respectively. The next bandcount 76 is explained by 2+3+10+20+40 gaps, and in general we obtain(3.1)

The symbolic sequences *σ*_{n} for *n*=0–3 corresponding to the unstable periodic orbits leading to this cascade are presented in table 1. These sequences and all further sequences *σ*_{n} for *n*>3 can be generated using the technique presented in Avrutin & Schanz (in press). Based on these sequences, the regions forming the bandcount doubling cascade in the middle of can be calculated. In figure 4, the boundaries of these regions (the curves of interior crises caused by the orbits ) are shown for the sequences *σ*_{i}=*σ*_{0}–*σ*_{3} given in table 1. In figure 4*a*, the regions and are presented as they appear in the parameter space *b*×*μ*. However, a better graphical representation of the results is possible by introducing a sequence of local coordinate systems with respect to the middle lines of the regions . Hence, in figure 4*b*–*d* the pairs of regions , are shown using these local coordinate system. It is clearly visible that in each pair the next region is slightly displaced with respect to the middle curve of the previous region . Furthermore, it can be clearly seen that the areas become very narrow for increasing *n*. This behaviour is 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 *δ* of the period doubling cascade (Feigenbaum 1979). Hereby, one of the scaling constants has a finite value, whereas the other one has the ‘value’ ∞. Therefore, the width of the regions 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 of the region for the calculation of figure 3.

The same bandcount doubling scenario takes place not only within the region , but also within each of the regions involved in the bandcount adding scenario described above. For instance, within the region , located between the regions and this cascade leads to the bandcounts given by(3.2)Again, the unstable periodic orbits responsible for this cascade have doubled periods, namely 40, 80, 160 and so on.

## 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 . 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 along the arc marked in figure 4*a*. 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 to the outside, which means 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 and with (4.1)Note that these sequences are derived from the symbolic sequence of the ‘host region’ and the symbolic sequence of the ‘first region’ from the bandcount doubling cascade, which corresponds to the case *j*=0 in both families. The periods of the orbits are the same in both the cases, namely 20+10*j*=30, 40, 50, … Consequently, the bandcount 46 shown on the l.h.s. of figure 5 is structured by 2 gaps occupied by , 3 gaps occupied by , 10 gaps occupied by and finally by 30 gaps occupied by . By contrast, bandcount 46 shown on the r.h.s. of figure 5 is structured by 15 gaps occupied by the same orbits , , and by a further 30 gaps where the other 30-periodic orbit, namely , is located.

The regions and , influenced by the families of orbits and , 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 regions and , we find the region , 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 and , as well as the bandcounts 86 (located between the bandcounts 46 and 56), corresponding to the regions and . Note that there are four further regions with the bandcount 86, which are explained differently. Two of them, namely and belong to the first generation and the other two ( and ) belong to the third generation.

As one can see, the bandcount scenario within the region involves an infinite number of regions . Recall that each of these regions is embedded into the region of existence of the corresponding completely unstable periodic orbit . Note that all these regions originate from that point at the *b*=1 line, where the region originates from. Consequently, we state that the codimension-2 bifurcation occurring at this point is quite complex. At this point, two former stable periodic orbits and 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.

Remarkably, the results we have obtained so far are valid not only for the region , but also for any region involved in the bandcount adding scenario within the region , 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 _{ch}, which means the line *b*=1, 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 regions

So far the structure of the region is explained. It can be verified both numerically and analytically that all regions for *k*>1 have the same structure. In particular, in the middle of each of these regions there exists a region of (6*k*+10)-band attractors. Of course, this number of bands is explained by (*k*+1)+(*k*+2)+(4*k*+6) gaps, occupied by the orbits , and . Above this region, we observe the family of regions and below the family of regions , which together form the first generation of the bandcount adding scenario within this specific region . Remarkably, the boundaries of these regions can still be calculated analytically, even for large values of *k* and *m*. Examples of these structures for *k*=2 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’ are organized in the same way by the orbits of the 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 reflect the fact that these completely unstable orbits are forming a self-similar structure like the family.

## 6. Overlapping structures within

As already mentioned, the fine substructures within the regions and are different. After the first type of these substructures is explained, let us consider the second one by focusing first on the case *k*=2. Figure 8*a* shows the structure of the region 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 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 confirm that the regions of 6- and 10-band attractors also lie within the region of the unstable period-2 orbit , and the regions of 8- and 10-band attractors within the region of the unstable period-4 orbit . 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 *k*>1 the orbit is involved in six different merging crises, as shown in figure 9*a* for the special case *k*=3. Two of the merging crises were already calculated and presented in part I of this work. Their corresponding curves in the parameter space form the triangle-like regions in the centre part of figure 9*a*, shown also in figures 1, 2 and 7. The remaining four crises bifurcation curves and confine the ‘rabbit ear’-like regions above and below this triangle-like centre region. The calculation of these curves is similar to the calculation of , namely the curves can be obtained using the conditions(6.1)and the curves using the conditions(6.2)Hereby, the upper curve is at the *b*=1 tangent to the upper existence boundary of the orbit (the curve , where this unstable orbit is destroyed by a border-collision bifurcation). Similarly, the lower merging crisis curve is at the *b*=1 tangent to the other border-collision curve (figure 9*a*). In the following, let us denote the upper rabbit ear-like region (confined by the curves ) by and the lower rabbit ear-like region (confined by the curves ) by .

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 *k*>1 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, the region overlaps with the within a part of the triangle-like centre region influenced by . As an example, in figure 9*b* the overlapping of the region (dashed curves) and the region (thick grey curves) is shown. As one can see, this overlapping is located within the region . Similarly, within the region , the regions (black curves) and (thin grey curves) overlap. This explains the bandcounts 6, 8 and 10 mentioned at the beginning of this section. As one can see, the results obtained numerically are explained by the analytically calculated curves and , as shown in figure 8*b*. Remarkably, the regions and , as well as their overlap , possess some further interior substructures. The investigation 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 9*b*. The presented transitions between several multi-band chaotic attractors (except the fine substructures mentioned above) can be explained easily using the knowledge about the bifurcation structures in the two-dimensional parameter space. The transition between the six- and four-band attractors at is caused by the unstable orbit . Between and , this orbit is located within the four-band attractor until it causes the next crisis at , where a four-band to a six-band transition occurs. The next crisis at is induced by the unstable orbit , which emerges at the border-collision bifurcation closely before (figure 9*a*). This crisis increases the number of bands from 6 to 10. The next crisis occurs at and decreases the number of bands from 10 to 8. It is the last crisis, which is caused by the unstable orbit , that will be destroyed closely after that by the border-collision bifurcation . The last two crises at and are induced by the orbit . As one can see, the beginning and end of the regions , and in the bifurcation diagram presented in figure 10 can be explained by the six crisis curves mentioned above.

## 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 , 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 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 *n*−1, *n* and *n*+1, which cause the bandcounts 2*n*, 2*n*+2 and 3*n*+1 to emerge.

So far, we have described almost all components that form the bifurcation structure in the plane *b*×*μ*. 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 *a*=−1. As a next step, in part III of our work we will 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 *a*=−1. 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.

## Footnotes

↵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).

↵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 and tent maps. Here, the unstable orbits with periods

*p*_{i}=2^{i}lead to the bandcounts . In this case, bandcounts and periods of the responsible 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.- Received November 2, 2007.
- Accepted March 18, 2008.

- © 2008 The Royal Society