## Abstract

Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors 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 the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.

## 1. Introduction

Crisis bifurcations are well known since the fundamental publications (Grebogi *et al*. 1982, 1983) and investigated in several works theoretically and experimentally (see for instance Grebogi *et al*. (1986) and Ditto *et al*. (1989)). However, in contrast to individual crisis bifurcations, complete bifurcation scenarios formed by several types of crises bifurcations in the parameter region of chaotic dynamics without periodic inclusions (denoted as robust chaos according to Banerjee *et al*. 1998) are still insufficiently investigated. In the previous works (Avrutin *et al*. 2007*a*; Avrutin & Schanz in press), the so-called bandcount (the number of bands or strongly connected components of chaotic attractors)-adding scenario is reported, which forms a complex self-similar structure in the two-dimensional parameter space. This scenario is formed by multi-band chaotic attractors (MBCAs) and occurs in the region of chaotic dynamics close to its boundary with the region of periodic dynamics. Hereby the regions of specific bandcounts are bounded by the bifurcation curves of interior crises. Since these crises are caused by unstable periodic orbits, the described structure is connected with the bifurcation structure of the adjacent region of periodic dynamics. In the case considered in the cited works, the structure of this area is determined by the period-adding scenario of stable periodic orbits. In this scenario between regions with periods *p*_{1} and *p*_{2} in a two-dimensional parameter space, there exists a region with period *p*_{1}+*p*_{2}. Since the existence regions of these periodic orbits are typically continuous, they cause MBCAs to emerge beyond the boundary where they become unstable whereby the bandcounts are given by *p*_{1}+1, *p*_{2}+1 and *p*_{1}+*p*_{2}+1, respectively. Hence the bandcounts in the chaotic domain reflect the periods in the periodic domain. However, it is also shown in the cited work that the correspondence between this overall bandcount-adding structure and the period-adding structure is not one-to-one. In fact, the bandcount-adding structure is much more complex than the period-adding structure since it is caused not only by the periodic orbits involved in the period-adding scenario, which may be both stable and unstable, but also by further orbits not involved in the period-adding scenario and which are unstable for all parameter values. These orbits lead to bandcount doubling and further bandcount-adding scenarios nested ad infinitum in each region of the overall bandcount-adding scenario and its sub-regions resulting in a self-similar structure. Consequently, the question arises how the chaotic domain is structured in the case when the area of periodic dynamics is organized by another phenomenon than the period-adding scenario.

This work is split into three parts. In *part I* (this paper) we introduce the investigated system and the used notation and summarize the results concerning the periodic dynamics of this system necessary for understanding its chaotic behaviour. Then, the structure of the chaotic domain in the two-dimensional parameter space is investigated in detail. The basic structure of this domain is explained by a novel bifurcation scenario denoted as the *bandcount increment scenario*. It will be shown that this scenario explains most but not all results obtained numerically. In order to bridge this gap, we investigate in *part II* the interior structures of the regions organized by the overall bandcount increment scenario. Our investigations reveal that these structures can be explained taking into account the already known bifurcation scenarios, namely the bandcount-adding scenario and the bandcount-doubling scenario. Finally, in *part III* we will report, how the detected structure of the two-dimensional parameter space changes under the variation of a third parameter. In this way, a complete description of the chaotic domain in the three-dimensional parameter space is obtained. With respect to practical applications, this part is most important since it explains the structure of the parameter space for many cases. Furthermore, we will present examples for applications from the field of power electronics.

## 2. Investigated system

Let us consider the following one-dimensional piecewise-linear discontinuous map(2.1)already investigated in many publications (for references see Di Bernardo *et al*. 2007), which represents some kind of canonical form of one-dimensional maps with a discontinuity.

Note that the theory of chaotic behaviour in piecewise-smooth discontinuous systems like (2.1) goes back to Keener (1980). However, the number of connected components of the MBCAs has not been investigated in detail until now.

The number of parameters in system (2.1) can be reduced by using a suitable scaling. As shown for instance in the study by Avrutin *et al*. (2007*b*) for the parameter *l*, only three characteristic cases have to be considered, namely . The case of the continuous map (*l*=0) is already well investigated for both periodic and chaotic domains (Maistrenko *et al*. 1993; Nusse & Yorke 1995; Di Bernardo *et al*. 1999; Zhusubaliyev & Mosekilde 2003; Avrutin *et al*. 2007*a*). From the two cases of positive discontinuity (*l*=1) and negative discontinuity (*l*=−1), the first one is more complex due to a greater number of codimension-3 bifurcations. Therefore, we consider in the current work the case of negative discontinuity, leaving the more complex case of positive discontinuity for future investigations. As shown in the study by Avrutin *et al*. (2007*b*), in this case the region of periodic dynamics of system (2.1) is organized mainly by two codimension-3 big bang bifurcations occurring at the points and . Owing to the symmetry property of system (2.1) given by(2.2)it is sufficient to consider only one of them, for instance the one at the point (0, 0, 1). These bifurcations initially reported by Avrutin & Schanz (2006) are characterized by two manifolds in the three-dimensional parameter space: one-dimensional and two-dimensional. The bifurcation structures above the two-dimensional manifold are formed by the period-adding phenomenon and below this manifold by the phenomenon of period increment with the coexistence of attractors.

As shown in the cited works, the structure of the periodic domain of system (2.1) can be explained almost completely by the codimension-3 big bang bifurcations mentioned above. However, the question arises what happens outside this region. Typically, the periodic orbits that are stable within this region become unstable outside and the dynamic behaviour becomes chaotic. Since the structure of the periodic domain is already investigated, the remaining task is to discover the organizing principles of the chaotic domain. In particular, we want to examine how the structures of the periodic and chaotic domains in the parameter space are related. In fact, the periodic orbits becoming unstable at the boundary of the periodic domain play an important role in the chaotic domain by organizing chaotic attractors. Typically they serve as skeletons of these attractors, cause several crises and so on. For this reason, it is not surprising that the chaotic domain is organized differently below (that means for *a*<0) and above (*a*>0) the two-dimensional manifold (*a*=0) of the codimension-3 bifurcation mentioned above. In Avrutin & Schanz (in press), we considered the simpler case, namely the structure above the two-dimensional manifold. In this case, the structure of the chaotic domain is given by a complex nested (presumably self-similar) pattern caused by interior crises. In the following we investigate the more complex case, namely the structures below the two-dimensional manifold.

## 3. Crises

According to Ott (2002), there are several types of crises bifurcations that can be distinguished by the underlying mechanisms leading to these crises and the change of the chaotic attractor at the bifurcation point. The following three types of crises are important for our work.

*Boundary crises* occur when the minimal distance between a chaotic attractor and the boundary of its basin of attraction decreases to zero as a parameter is varied. Since this boundary typically coincides with the stable manifold of an unstable invariant set like a fixed point or a periodic orbit, boundary crises are in fact collisions of the chaotic attractor with this stable manifold and hence with the unstable invariant set itself.

*Interior crises* are caused by the collision of a chaotic attractor with an unstable invariant set located within its basin of attraction. Consequently, the chaotic attractor will not be destroyed at this crisis bifurcation; instead, the bifurcation leads to a sudden change of the size, shape or topology of the chaotic attractor.

*Merging crises* occur when the bands of a chaotic attractor merge. Sometimes these bifurcations are also denoted as crisis-like transitions since the geometric form of the attractor is not significantly changed at the bifurcation point but only its topological structure. Merging crises are well known as a period-doubling cascade is typically followed by a band-merging cascade like it is the case, for instance, in the logistic map. However, a period-doubling cascade is not a necessary condition for a band-merging cascade. The tent map for instance shows a band-merging cascade solely. Remarkably, the boundaries of a chaotic attractor collide at a merging crisis not only pairwise with each other but also additionally with an unstable periodic orbit.

All three types of crises mentioned above represent codimension-1 bifurcations, which means they can be observed under variation of one parameter. In a two-dimensional parameter space, it is also possible that two or more codimension-1 crises curves intersect, leading to a codimension-2 crisis. Especially, in Stewart *et al*. (1995), the bifurcation pattern caused by an intersection of a boundary crisis and an interior crisis curve is described in detail. In the same work, it is mentioned that interior crises curves may intersect as well and that this type of codimension-2 bifurcation is already observed (Ueda *et al*. 1990). However, no further details about codimension-2 interior crisis are reported.

Codimension-2 interior crises can be easily confused with codimension-1 merging crises. Let us consider a typical region in a two-dimensional parameter space *a*×*b* bounded by two curves *η*_{1}, *η*_{2} of interior crises. For simplicity, let us assume that the attractors in this region have two bands and that the crises are caused by an unstable fixed point located in the gap between these bands. In figure 1*a*, this region is marked with ^{2} and originates from a point at the boundary between the domain ^{1} of a stable fixed point and the domain ^{1} of one-band attractors. Then, varying the parameters along the line marked with B, we observe a bifurcation diagram like that shown schematically in figure 1*b*. The interior crisis occurs at the point where the line B intersects the curve *η*_{1}. As one can see, the unstable fixed point collides at this point with the lower boundary of the upper band of the attractor. This causes a sudden change of the size of the attractor and leads to a vertical edge in the corresponding bifurcation diagram. A similar situation takes place by the variation of the parameters along the line marked in figure 1*a* with C. The only difference is that in this case the fixed point collides with the upper boundary of the lower band of the attractor as shown in figure 1*c*. Remarkably, in both cases the length of the vertical edge decreases when we move the intersection point of B with *η*_{1} (respectively C with *η*_{2}) towards the intersection point of D with *η*_{1} and *η*_{2}.

Hence, varying the parameters along the line marked in figure 1*a* with D, we observe that the fixed point collides simultaneously with the lower boundary of the upper band and with the upper boundary of the lower band of the attractor. As one can see in figure 1*d*, this codimension-2 interior crisis cannot be distinguished from a usual merging crisis. The situation we observe here is in some sense similar to the one of the well-known codimension-2 cusp bifurcation. This bifurcation is defined as the intersection point of two codimension-2 saddle–node bifurcations, but varying the parameters across the cusp point one observes not a saddle–node but a pitchfork bifurcation.

In the following we will discuss the regions in two-dimensional parameter space bounded by both types of crises curves. Hereby one should keep in mind that if the bounding curves represent interior crises, the situation is similar to the one shown in figure 1*b* or *c*. In the other case, which means the bounding curves represent merging crises, any intersection of these curves looks like figure 1*d*.

## 4. Periodic orbits, kneading orbits and symbolic sequences

Let us summarize the notation used throughout this series of papers. Following one of the standard symbolic dynamics (Bai-Lin 1989), we denote a point *x*<0 by the symbol and a point *x*>0 by . A periodic orbit will be denoted by , where *σ* is the symbolic sequence corresponding to one period of the orbit and the superscript ‘s’ or ‘u’ is used when it is necessary to indicate whether the orbit is stable or unstable, respectively. The same superscripts are used in the notation that refers to the regions in the parameter space where stable or unstable orbits , respectively, exist. These regions are bounded by the curves of border collision bifurcation where the corresponding orbit collides with the border *x*=0 and will be destroyed. The border collision bifurcation curves are denoted as , whereby the index refers to the fact that the *i*th point of the orbit collides with the border *x*=0. The superscripts ‘*ℓ*’ and ‘r’ represent the direction of the collision, that is whether the *i*th point of the orbit O_{σ} collides with the border *x*=0 from the left (*ℓ*) or the right (r).

As already mentioned, there exist regions of MBCAs in the chaotic region that are bounded by the curves of several crises induced by an unstable periodic orbit . For the analytical calculation of these regions, the intersection of with specific points of a kneading orbit has to be traced. In general, a kneading orbit is defined as itinerary of some critical point of the system function. For system (2.1), the critical point is the point of discontinuity *x*=0. Therefore, there are two kneading orbits representing the iterations of the point *x*=0 using the function *f*_{ℓ} or *f*_{r}, respectively, in the first step. In each subsequent iteration step, the application of the functions *f*_{ℓ} and *f*_{r} is unambiguous. The application sequence of these functions defines the notation for specific points of the kneading orbit. As an example, the point is denoted as .

The interior crisis curves given by the collision of a specific point of an unstable periodic orbit with a specific point of a kneading orbit are denoted as , where *ℓ* and r depend on the first symbol of the used kneading orbit. For the curves of merging crises, we use the notation with the same meaning of the sub- and superscripts. The existence region of an MBCA is denoted as , whereby is the number of bands and *σ* is the sequence of the unstable periodic orbit that undergoes the collision with the kneading orbit. Note that the upper index is not redundant because it does not always hold that . This is because for a certain MBCA more than one unstable periodic orbit may be involved determining the overall number of gaps and consequently the number of bands.

## 5. Periodic dynamics and boundaries of the chaotic domain

Although the main topic of this paper is given by the bifurcations involving MBCAs, we need some results related to periodic orbits in order to explain these bifurcations. It follows directly from the results presented in Avrutin *et al*. (2007*b*) that in the part of the parameter space we consider here, the only possible stable periodic orbits are . Hereby for each *n* the existence region of the orbit in the three-dimensional parameter space is bounded by two border collision bifurcation surfaces(5.1)(5.2)which can be straightforward calculated using the conditions that the first (respectively last) point of the orbit collides with the boundary *x*=0 from the left (respectively right) side. Additionally, it can be easily shown that the stability region is bounded by the surfaces(5.3)where the orbits become unstable. Note that for each two successive indices the regions and overlap leading to the coexistence of the stable (*n*+1)- and (*n*+2)-periodic attractors as shown in figure 2.

As an example, in figure 3*a* the region is shown. This region is bounded from above by the curve , below by and to the right by . In the upper part, this region overlaps with the region and in the lower part with . Note that in figures 2–9 we use the following scaling of the system parameters:(5.4)Since the tangent function increases monotonously, the scaling preserves the topological structure of the parameter space. This scaling is used to achieve a better graphical representation of the overall parameter space especially for large parameter values.

From equation (5.3) it follows directly that there are three characteristic cases for the location of these regions with respect to each other. For −1<*a*<0 the boundary is located on the right side of the boundary as shown in figure 3*b*. For *a*<−1 it is located on the left side of as we can see in figure 3*d*. In the special case *a*=−1, the boundaries and coincide so that the transition to chaos occurs over a bifurcation where two coexisting attractors simultaneously lose their stability (figure 3*c*).

Now we can identify the boundary between the region of chaotic dynamics _{ch} and the region of periodic dynamics. As one can see from figure 3*b*, in the case −1<*a*<0 this boundary is formed by the curves and . In the case *a*=−1 all boundaries coincide at the line *b*=1 (hence, in figure 3*c*). Consequently, the boundary of the region _{ch} is in this case given by this straight line. Finally, for *a*<−1 the boundary of _{ch} is formed by the curves and (figure 3*d*).

Additionally, we can identify the boundary between the region _{ch} and the region of divergent behaviour _{div}. This boundary is formed by the surface where system (2.1) undergoes a boundary crisis. Taking into account that the boundary of the basin of attraction is given by the unstable fixed point and the relevant upper boundary of the chaotic attractor can be calculated as *f*_{ℓ}(0), we obtain the surface of the boundary crisis(5.5)marked in figure 3*a*. So far we have described the boundaries of the region _{ch}. As a next step we have to discover the interior structure of this region.

## 6. Bandcount increment scenario

In order to explain the structure of the chaotic region _{ch}, let us first consider the most simple case *a*=−1. As already mentioned, in this case the boundary of the region _{ch} is given by the straight line *b*=1. Hereby we have to keep in mind that this case is not generic, since for any value *a*≠−1 the boundary of the region _{ch} has a more complex shape. Nevertheless we will see that the results we obtain for *a*=−1 are useful to understand the more complex cases −1<*a*<0 and *a*<−1.

The structure of the region _{ch} shown in figure 4 is calculated numerically for *a*=−1 using a bandcounting method reported in Avrutin *et al*. (2007*a*) and implemented in the software package AnT *4.669* (www.ant4669.de). As one can see, this structure is dominated by overlapping triangle-like shapes with complex interior structures. Let us first consider these triangle-like shapes. Therefore, we choose a curve in the parameter space, for instance the curve *μ*=3−2*b* that intersects the upper and lower boundaries of all these triangle-like shapes, as shown by the dashed curve in figure 4. The bifurcation diagram and the corresponding bandcounts along this curve are shown in figure 5. As one can see in the bandcount diagram, one-band attractors are interrupted by a sequence of multi-band windows whereby the bandcounts in these windows form an increasing sequence starting with bandcount 4 which is incremented by 1 in each step(6.1)

Owing to the analogy of this sequence with the sequence of periods occurring in the period increment scenario, we denote the bifurcation scenario shown in figure 5 as the *bandcount increment* scenario. Now our task is to explain how this scenario emerges.

As one can see in figure 5*a*, the transitions between one- and multi-band attractors in the presented case take place via merging crises. In order to explain these crises, we have to determine which unstable orbits are involved in the crises and are therefore responsible for the formation of the bandcount increment scenario. Since the involved attractors have _{m}−1=3, 4, 5, ⋯ gaps and in merging crises each gap contains one point of the responsible unstable periodic orbit, we expect that these orbits have the periods 3, 4, 5, ⋯. Fortunately, we know already the family of orbits with these periods, namely the family . It can be easily verified that these orbits are in fact responsible for the formation of the bifurcation scenario presented in figure 5*a*. As shown in figure 6*b*, the period-3 orbit is involved in the crises where the four-band attractor emerges. The same holds for the period-4 orbit and the five-band attractor (figure 6*a*), as well as in general for any orbit and the corresponding (*n*+2)-band attractor. Recall that these orbits determine the structure of the periodic part of the parameter space (see §5). Now we state that they play an important role in the chaotic domain too.

In order to illustrate the influence of a specific unstable orbit on the structure of the chaotic domain _{ch}, let us first consider the most simple example, namely the orbit . As already mentioned, this orbit is involved in the merging crises where four-band attractors emerge. The curves of these crises can be calculated from the intersections of specific points of the involved unstable periodic orbit with the corresponding points of the kneading orbit.

In principle, for the calculation of the merging crises caused by the orbit , each of its points can be used. For instance, the last point of the orbit collides at one of the merging crises simultaneously with the points and and at the other merging crisis simultaneously with the points and , as shown in figure 6*b*. Choosing a different point of the unstable orbit will result in the collision with different points of the kneading orbit but of course the same merging crisis curve. Straight forward calculations show for *a*=−1 that(6.2)This leads together with the above-mentioned conditions and to the curves of the merging crises caused by the orbit .

Similar calculations can be performed for an arbitrary orbit . Calculating the merging crises curves involving from the conditions and , we obtain(6.3)(6.4)Note that equation (6.3) is valid for *n*≥2 while equation (6.4) is valid for *n*≥1.

Let us denote the region bounded by the merging crises curves and the stability boundary of the orbit as with *n*≥2. The region represents an exception and is bounded by the merging crisis curve , the stability boundary and the line *μ*=0. Of course each of the regions is a subset of the corresponding region , where the orbit exists and may have an influence on the dynamics. More precisely, represents a part of where MBCAs are influenced by the unstable orbits . In fact, there are other regions within also influenced by , which will be discussed in the second part of this work. Because the regions are influenced by the unstable orbits , one could assume that the MBCAs in these regions must have *n*+2 bands. In fact, the situation is more complex. In figure 7 the regions calculated analytically by equations (6.3) and (6.4) are shown for *n*=2, …, 7. As one can see, the regions , and overlap pairwise exactly in the same way as the periodic regions , and do. In the overlapping region , the attractors have 2*n*+1 gaps and hence 2*n*+2 bands. Hereby *n* gaps are occupied by the orbit and *n*+1 gaps by the orbit . According to our notation (see §4), this region is in the following denoted as . Similarly, in the other overlapping region , the attractors have bands. In the remaining part of , which means in the region(6.5)the attractors have *n*+2 bands. To clarify the described structure, let us consider as an example the region marked bold in figure 7. As one can see, this region overlaps with and . As a consequence, we observe the regions influenced only by as well as the regions (influenced by and ) and (influenced by and ). Hence, the structures of periodic and chaotic domains are strongly connected with each other since both of them are influenced by the sequence of pairwise overlapping regions and with *n*=1, …, ∞. However, there is a significant difference between both structures. In the region of the periodic dynamics, the overlapping of and leads to the coexistence of the corresponding stable orbits and . In contrast to this, the overlapping of and in the region _{ch} leads to the fact that the MBCA existing in this region is structured by both coexisting unstable orbits and simultaneously. This is illustrated in figure 8 for the unstable orbits and . In this figure, the coexisting unstable orbits and and the MBCAs are shown. In the middle part of the figure, both unstable orbits are located outside the chaotic attractor (region ) so that the attractor has six bands. In contrast to this, in both adjacent regions and , one of the unstable orbits is located inside the attractor and the other one outside so that the attractor is structured by only one of the orbits, which consequently leads to the bandcounts 3 and 4.

## 7. Transitions to chaos

The results presented above explain not only the structure of the chaotic domain but also the possible transitions to chaos. Some examples for these bifurcations in system (2.1) are shown in figure 9. As a typical situation, figure 9*a* demonstrates the transition from the region to chaos. At the bifurcation point, both orbits and lose the stability and the dynamics becomes chaotic. After the bifurcation, the attractors have six bands (region ) until the merging crisis where the three-band attractor emerges (region ). This attractor persists until the next merging crisis occurring at and leading to a one-band attractor. It can be shown that the gaps where the points of the period-3 orbit are located become smaller for decreasing *μ* so that the bifurcation diagram becomes eventually the one shown in figure 9*d*.

A similar situation is shown in figure 9*b*. Like the previous case, the attractor has six bands after the bifurcation. However, in contrast to the previous case the next crisis occurs here at and not at . Therefore, the attractor existing after this crisis has four bands. Additionally, one can clearly see that the existence region of this attractor is interrupted by some further regions that we did not consider in this part of the work. The same situation occurs for each pair of the coexisting periodic orbits and . For instance figure 9*c* shows the bifurcation where the orbits and become unstable. Correspondingly, after the bifurcations the attractors have eight or five bands, respectively.

The only situation, which is more complex than discussed above, occurs in the case presented in figure 9*d*. Here the boundary between the periodic and chaotic domains is crossed at the point where the curve intersects the curve . Consequently, this point is located within the region and at the boundary of both adjacent regions and . As one can see in figure 7, under variation of parameters along a horizontal line across this point, the following phenomena occur simultaneously: (i) the orbit becomes unstable, (ii) the orbit is destroyed, (iii) the unstable orbit emerges, and (iv) the dynamics becomes chaotic, the attractors belong to the region . Of course, the same situation can be observed not only for these orbits but also for each three subsequent orbits , and .

The difference between the transitions to chaos shown in figure 9*a–d* is difficult to explain when dealing only with the parameter plane *b*×*μ*. In this plane all four cases represent codimension-2 bifurcations. However, considering the overall three-dimensional parameter space *a*×*b*×*μ*, we state that in this space the first three cases are still codimension-2 bifurcations whereas the last one represents a codimension-3 bifurcation. For details we refer to the forthcoming parts of this paper.

## 8. Summary and outlook

In this work we have investigated a piecewise-linear map in the region of chaotic dynamics. The observed dynamic behaviour represents robust chaos in the sense of Banerjee *et al*. (1998), which means it is not interrupted by any periodic inclusions (windows). Nevertheless, system (2.1) undergoes several crises in this region, which change the topological structure of the attractors and form a complex bifurcation scenario. It is shown how this scenario reflects the structure of the adjacent periodic domain. In the considered case, the investigated system shows in this domain an infinite sequence of periodic attractors organized by the period increment scenario with coexisting attractors. As a consequence, in the chaotic domain we detected a novel bifurcation scenario denoted as the bandcount increment scenario. The bifurcations forming this scenario are the merging crises caused by unstable periodic orbits whose existence regions originate from the periodic domain.

We demonstrated that for each region involved in the period increment scenario with coexisting attractors that organize the periodic domain, there exists a corresponding region in the chaotic domain. Schematically this correspondence can be represented as shown in table 1. However, the coexistence of attractors in the periodic domain does not lead to the coexistence of multi-band attractors in the chaotic domain but to the emergence of a single attractor with a higher number of bands. Since the period increment scenario with coexisting attractors represents a generic scenario, which can be observed in many dynamical systems, we conclude that the bandcount increment scenario is a generic phenomenon as well.

## Footnotes

- Received September 25, 2007.
- Accepted March 4, 2008.

- © 2008 The Royal Society