The bandcount increment scenario. III. Deformed structures

Transcription

1 Proc. R. Soc. A (009) 65, 1 57 doi: /rspa Published online 7 August 008 The bandcount increment scenario. III. Deformed structures BY VIKTOR AVRUTIN, BERND ECKSTEIN AND MICHAEL SCHANZ* Institute of Parallel and Distributed Systems (IPVS ), University of Stuttgart, Universitätstrasse 8, Stuttgart, Germany 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 ind 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. Remarably, 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. Keywords: bandcount increment; bandcount adding; bandcount doubling; interior crises; merging crises; piecewise-linear discontinuous maps 1. Introduction Investigating the chaotic domain in a multidimensional parameter space represents a challenging tas 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 now how the variation in parameters influences the dynamics. For example, one is interested to now 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 stic slip effects, switching electronic circuits, power converters and others, the situation that so-called robust chaos occurs is well nown (see references in Banerjee & Verghese (001), Zhusubaliyev & Moseilde (00) and Bernardo et al. (007)). 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 * Author for correspondence (michael.schanz@informati.uni-stuttgart.de). Received June 008 Accepted July This journal is q 008 The Royal Society

2 V. Avrutin et al. previous publications and especially in Parts I and II of this wor (Avrutin et al. 008a,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 wor, we investigated the piecewise-linear discontinuous map given by ( x nc1 Z f [ ðx n Þ Z ax n Cm; for x n!0; ð1:1þ f r ðx n Þ Z bx n Cm Cl; for x n O0: This map represents a special case of a well-nown two-dimensional piecewiselinear normal form investigated by many authors (see Bernardo et al. 007). 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 lzk1 by a suitable scaling) and the case a!0, where the periodic domain is organized by the period increment scenario (Avrutin et al. 007b) with coexisting attractors, sometimes also referred to as multi-stability. This scenario is formed by a sequence of periodic orbits O s with Z1,,., whose stability regions P s 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 azk1. It was shown that the main structure-forming component is given by a sequence of regions of multi-band chaotic attractors organized by the bandcount increment scenario, as shown in figure 1. 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 wor, namely SðnÞ : ðkn;nþ1 K p ; p ; with SðnÞ Z arctanðnþ; for n fa; b; mg: ð1:þ The boundaries between the regions involved in the bandcount increment scenario represent curves of merging crises, caused by the orbits O u. 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 P u, the bandcount increment scenario includes regions of two types, namely triangle-lie regions Q C and trapezoidal regions Q C. As indicated by the hc1 sub- and superscripts, each region of the first type is influenced by two unstable periodic orbits O u and O u, and contains (C)-band attractors. By contrast, each region of the second C1 type is influenced by only one orbit O u,sothat the corresponding attractors have C bands. 1 For details related to the notation used, see Part I of this wor. To avoid confusion, we emphasize that the term merging crisis refers in this wor 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 00). Proc. R. Soc. A (009)

3 Bandcount increment: deformed structures (a) 1 (b) div S(b) S(b) 1.0 Figure 1. Bandcount increment structures and the embedded bandcount-adding substructures calculated (a) numerically and (b) analytically for azk1. The region of divergent behaviour is shown in grey. The triangle-lie region mared bold in (b) represents the region U considered in and. 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 azk1. 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 Q C for Z1,,. ( ), as well as on the bandcount-adding hc1 substructures located within the regions Q C ( 5), will be described. We hc1 will see that all these structures are organized by the same unstable orbits as in the case azk1, 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 wor 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 azk1.. Deformation of the parameter space for asl1 Examples for numerically calculated bifurcation structures in the case ask1 are shown in figure. 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 vp ch Proc. R. Soc. A (009)

4 V. Avrutin et al. (a) S(b) (b) S(b) Figure. Numerically detected bifurcation structures in the characteristic cases (a) K1!a!0 and (b) a!k1. Light grey shading indicates periodic behaviour and dar grey divergent behaviour. The inset in (a) shows the mared rectangle enlarged. An analytical explanation of the presented structures is shown in figures and. Parameter values: (a) azk0.566, (b) azk under variation of a, since all structures we described so far (both the bandcount increment scenario and its interior substructures) originate from the boundary vp ch between the periodic and chaotic domains. Figures a and a show the analytically determined boundaries of the regions of periodic dynamics for the case K1!a!0 and a!k1, respectively. Each of the regions P s is bounded from above and below by the border-collision bifurcation curves x 0;[ and x ;r, respectively, defined by the condition that the first (respectively last) point of the orbit O collides with the boundary xz0. From the right-hand side, the region P s is bounded by the bifurcation curve q where the orbit O becomes unstable. The overlapping of the Proc. R. Soc. A (009)

5 Bandcount increment: deformed structures 5 (a) (b) S(b) Figure. Analytically calculated bifurcation structures in the (a) periodic and (b) chaotic domains in the case K1!a!0, corresponding to figure a. Shading is similar to that described in the caption of figure. The bold grey curves in (a,b) mar the region P s, and the bold blac curves in (b) mar the region U. Parameter value: azk subsequent regions P s, P s and P s leads to the fact that for each we K1 C1 observe not only the region where the orbit O represents the unique attractor but also the regions P s hp s and P s hp s where attractors with K1 C1 periods and C1 (respectively C1 and C) coexist. Let us now turn to the boundary vp ch between the periodic and chaotic domains. In the case azk1, this boundary is given by the straight line bz1 where the stability boundaries q of all orbits O are located. For ask1, this is not the case. Recall (see Part I, 5) that for K1!a!0 the line q is located on the right-hand side of q C1, as shown in figure a, and for a!k1 on Proc. R. Soc. A (009)

6 6 V. Avrutin et al. (a) (b) S(b) Figure. Analytically calculated bifurcation structures in the (a) periodic and (b) chaotic domains in the case a!k1, corresponding to figure b. Shading is similar to that described in the caption of figure. The bold grey curves in (a,b) mar the region P s, and the bold blac curves in (b) mar the region U. Parameter value: azk the left-hand side, as shown in figure a. Therefore, varying a from the value K1to some values K1G a, we observe that the boundary of the chaotic domain P ch becomes a non-smooth curve composed of several pieces. In the case K1!a!0, this curve consists of the parts of the curves q where the orbit O becomes unstable, and the border-collision bifurcation curves x 0;[ (Part I, eqn (5.1)) where the first point x 0 collides with the boundary xz0. Hence, the boundary vp ch is defined by the curves q upward from the intersection points q hx 0;[ and by K1 the curves x 0;[ rightward from the intersection points q C1 hx 0;[ (figure a). Similarly, in the case a!k1, the boundary consists of the same curves q and the Proc. R. Soc. A (009)

7 Bandcount increment: deformed structures 7 border-collision bifurcation curves x ;r (Part I, eqn (5.)) where the last point x collides with the boundary xz0. Hence, the boundary vp ch is defined by the curves q downward from the intersection point x C1;r hq C1 and by the curves x ;r rightward from the intersection point x ;r hq K1 (figure a). Consequently, the question arises of how the structure of the region P ch beyond the boundary vp ch is organized in the case ask1. As shown in figure, this structure is more complex than that in the case azk1. However, we will demonstrate that almost all results obtained for azk1 are still valid for ask1. As we will see, the structure of the region P ch for ask1 can be described as a deformation of the structure of this region for azk1. Recall that the bandcount increment scenario taes place within the chaotic domain P ch, but is influenced by a structure that is organizing the adjacent periodic domain, namely by a sequence of pairwise overlapping regions P s with Z1,,. forming the period increment scenario with coexisting attractors. In the chaotic domain, each of the involved unstable periodic orbits O u causes merging crises, which separate three multiband regions Q C, Q C and Q C. Note that the region Q C K1 h hc1 K1 h also represents the region Q 0 C 0 h0 C1 for the predecessor orbit O u with 0 0 Z also represents the region Q 00 C 00 K1 h00 with 00 ZC1. This is a consequence of the overlapping structure. The important fact is that one particular orbit O u is involved in crisis bifurcations of three regions. In and, we describe how the regions Q C and Q C are K1 h hc1 deformed for ask1. The description of the regions Q C is not within the scope K1. Analogously, the region Q C hc1 for the successor orbit O u 00 of this wor for two reasons. First, these regions are not deformed for ask1, 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.. Structure of the parameter space for L1!a!0 As in the case azk1, the codimension- bifurcation points x 0;[ hq lie on the boundary vp ch. In fact, the curves x 0;[ and q compose the boundary as described in. By contrast, the codimension- bifurcation points x ;r hq lie no longer on the boundary vp ch but for each in the periodic domain within the region P s. Furthermore, the fact that for each the stability boundary q K1 C1 is located on the left-hand side of the stability boundary q leads to additional codimension- bifurcation points x 0;[ hq C1. To demonstrate the influence of the deformation of the boundary vp ch on the bifurcation structure within P ch, let us consider as an example the curves of crisis bifurcations involving the unstable orbit O u. Recall (see Part I, fig. 7) that for azk1 the region U where the attractors are directly influenced by this orbit has a triangle-lie shape and consists of three parts, namely Q 8 h (influenced by O u and O u ), Q 10 (influenced by O u h and O u ) and Q 5 (influenced by O u only), as shown in figure 1b. The boundaries of the Proc. R. Soc. A (009)

8 8 V. Avrutin et al. region U are in this case given by two curves of crisis bifurcations g [ =r and the boundary vp ch, where the stability boundaries q, q and q are located. As shown in Part I of this wor, the curves g [ =r can be calculated for arbitrary using the equations and x Z x o r[ r ð:1þ x Z x o [ rk; ð:þ respectively. Here, x denotes the th point of the orbit O, whereas x o [ r and x o K r[ r represent the corresponding points of the neading orbit (see Part I, ). Solving equations (.) and (.1), we obtain g [ Z g r Z ða; b; mþ m ZK abkab Kb C1 ðab Ka Cb K1Þb ; ð:þ ða; b; mþ m Z a b C1 Ka b C Cab KabKab C1 Ca Cb K1 a b C1 Ka b C Cab KabKab C Cb CaKb : ð:þ Note that eqns (6.) and (6.) in Part I represent the special case of equations (.) and (.), respectively, for azk1. As shown in figure b, for K1!a!0, the bold-framed region U consists of the same three parts as for azk1, namely Q 5, Q 8 and Q 10. h h However, the shape of U is now more complex. Namely, it is bounded not only by the merging crisis curves g [ =r and the stability boundary q, but also by the curves bounding the adjacent region P s, namely by those parts of the curves x 0;[ and q which define the boundary vp ch, as described in. Let us now consider the lower corner of the region U, where the curve g [ originates from, or, in other words, where it hits the boundary vp ch. In the case azk1, this point is given by the intersection point of the curves x ;r and q, i.e. by the lower corner of the region P s. By contrast, in the case K1!a!0, the lower corner of the region P s, i.e. the intersection point of the curves x ;r and q, is located in the periodic domain within the region P s, as shown in figure a. Therefore, the crisis bifurcation curve g [ representing the lower boundary of the region U or, more precisely, of the regions Q 8 and h Q 5, cannot emerge from this intersection point. Instead, the codimension- bifurcation point where the curve g [ originates from is in this case the intersection point x 0;[ hq. However, neither the orbit O nor the orbit O is involved in the crisis bifurcation g [. Remarably, at the codimension- bifurcation point where the curve g [ originates from, not only do the curves x 0;[ and q intersect but also the border-collision bifurcation curve x;r (dashed curve, figure b) of the orbit O is responsible for this crisis. Let us next consider the upper corner of the region U, where the curve g r originates from, or, in other words, where it hits the boundary vp ch. Solving equation (.1) for Z, we obtain a curve r, which originates from the intersection point of the curves x 0;[ and q, but of course this curve corresponds to the merging crisis curve g r only in the chaotic domain, i.e. on the right-hand Proc. R. Soc. A (009)

9 Bandcount increment: deformed structures 9 side of the boundary vp ch, which is in that part of the parameter space given by the stability boundary q. This can be easily explained by taing into account that the condition (.1), used for the calculation of the curve r, represents the fact that the point of discontinuity belongs to the stable manifold of the unstable periodic orbit O u. If this situation occurs in the chaotic domain, it corresponds in fact to the merging crisis caused by the unstable orbit O u. However, in the periodic domain (that is, if the unstable orbit O u coexists with a stable periodic orbit), this situation does not have any further consequences for the asymptotic dynamics. Consequently, the curve r becomes a merging crisis curve g r at that point where it intersects the boundary vp ch. The curve g r originating at this point represents rightwards the upper boundary of the region U or, more precisely, of the regions Q 10 and Q 5. The lower boundary of the h region Q 10 is given by the merging crisis curve g [, which originates h from the point x 0;[ hq. The bifurcation structure at this point is similar to the structure at the point x 0;[ hq described above. After considering the outer boundaries of the region U, let us now consider the boundaries between the regions Q 5, Q 8 and Q 10. These boundaries h h are located within U and define, together with the outer boundaries of the region U, the region Q 5. Recall that, in the case azk1, this region has a trapezoidlie shape and is bounded from the right and from below by the curves g r and g [, respectively, and from the left and from above by the curves g r and g[, respectively. For K1!a!0, the situation with the curve g r is the same as described above for the curve g r. For the same reason, it originates from a point at the boundary vp ch (which is in this specific part of the parameter space given by the stability boundary q ) where the solution of equation (.1) (for Z) reaches the chaotic domain. Therefore, the boundary of the region Q 5 is given by the curves g [, g r, g r and g [ as in the case azk1 and additionally by q and x 0;[, as they define the boundary vp ch in this part of the parameter space. In one sense, one can say that some part of Q 5 is partially covered by the region P s. Hereby, there are two possible situations dependent on the relative location of the upper right corner of the region P s and the curve g r. If the point x 0;[ hq is located below the curve g r, the region Q 5 is not convex but still connected. In the opposite case, the region Q 5 would split into two parts, as shown in figure b, for the region Q. As one can see, the region Q is partially covered by P s and, therefore, consists of two parts, one of them located above x 0;[ and the other located on the right of q.. Structure of the parameter space for a!l1 The situation in the case a!k1 is similar to but in some ways also the reverse of the situation in the case K1!a!0. As in the case azk1, for each, the codimension- bifurcation point x ;r hq lies on the boundary vp ch as described in. By contrast, and in reverse of the situation in the case K1!a!0, the codimension- Proc. R. Soc. A (009)

10 50 V. Avrutin et al. bifurcation point x 0;[ hq lies no longer on the boundary vp ch but in the periodic domain within the region P s. Furthermore, the fact that for each the C1 stability boundary q K1 is located on the left-hand side of the stability boundary q leads to the additional codimension- bifurcation points x ;r hq K1. To demonstrate the influence of the deformation of the boundary vp ch on the bifurcation structure within P ch also in the case a!k1, let us consider again the curves of crisis bifurcations involving the unstable orbit O u. As shown in figure b, for a!k1 the bold-framed region U consists again of the three parts Q 5, Q 8 and Q 10. In this case, it is bounded by the merging crisis h h curves g [ =r, the stability boundary q and, additionally, by the curves bounding the adjacent region P s, namely by those parts of the curves x ;r and q that define the boundary vp ch, as described in. Let us first consider the upper corner of the region U where the curve g r originates from, or, in other words, where it hits the boundary vp ch. In this case, the codimension- bifurcation point is given by the intersection point hq 5. Analogous to the previous case, neither the orbit O nor the orbit O 5 is involved in the crisis bifurcation g r and the border-collision bifurcation curve x 0;[ (dashed curve, figure b) of the orbit O responsible for this crisis intersects the codimension- bifurcation point x 5;r hq 5. Let us next consider the lower corner of the region U, where the curve g [ originates from, or, in other words, where it hits the boundary vp ch. Solving equation (.) for Z, we obtain a curve [, which originates from the intersection point of the curves q and x ;r x 5;r. Similar to the case K1!a!0, this curve becomes a merging crisis curve g [ at the point where it intersects the boundary vp ch.the curve g [ originating at this point represents rightwards the lower boundary of the region U or, more precisely, of the regions Q 5 and Q 8. The upper h boundary of the region Q 8 is given by the merging crisis curve g r h,which originates from the point x ;r hq. The bifurcation structure at this point is similar to the structure at the point x 5;r 5 hq described above. Let us finally also consider in this case the boundaries between the regions Q 5, Q 8 h and Q 10 h.fora!k1, the situation with the curve g [ is the same as described above for the curve g [. For the same reason, it originates from a point at the boundary vp ch (which is in this part of the parameter space given by the stability boundary q ) where the solution of equation (.) (for Z) reaches the chaotic domain. Therefore, the boundary of the region Q 5 is given by the curves g [, g r, g r and g [ as in the case azk1 and, additionally, by q and x ;r, as they define the boundary vp ch in this part of the parameter space. In this case, one can say that some part of Q 5 is partially covered by the region P s. Hereby, there are two possible situations dependent on the relative location of the lower right corner of the region P s and the curve g [. If the point x ;r hq is located above the curve g [, the region Q 5 is not convex but still connected. In the opposite case, the region Q 5 would split into two parts. Proc. R. Soc. A (009)

11 Bandcount increment: deformed structures 51 As in the case azk1, the structure described in and is repeated for all regions U, which causes the complex topological structure of the chaotic domain close to the boundary vp ch. As can be demonstrated by analytical calculation of the involved crisis curves, the bandcount increment structure, developed in the case azk1 in its most simple and most definitive form, becomes in both cases K1!a!0 and a!k1 more complex owing to the nontrivial shape of the boundary vp ch between the periodic and chaotic domains, as well as to the partial covering of the regions Q C1. Nevertheless, the basic component of this structure given by the sequence of overlapping regions U, which reflects the sequence of overlapping regions P s, 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 azk1 within each of the regions Q C. As shown hc1 in Part II of this wor, within each region Q C there exist two families of hc1 subregions Q K ;m s ;m with s ;m Zð Þ mc ðc1 Þ and s ;m Zð Þ ðc1 Þ mc, respectively. The bandcounts in these regions are given by K ;m Z js ;m j Cj j CjC1 j C1 Z js ;m j C C; ð5:1þ and can be explained by taing into account that from K ;m K1gapsoftheK ;m -band attractors C1 gaps are occupied by the unstable orbit O, C gaps by the unstable orbit O C1 and the remaining js ;m jzðc1þðmcþcc and js ;m jzccðcþðmcþ, respectively, by the orbit O s;m. These regions serve as a first layer of the infinite bandcount-adding scheme described in detail by Avrutin & Schanz (008) and organized in the same way as the well-nown Farey tree- or Stern Brocot tree-lie period-adding scheme. Note that, according to the period-adding scheme, between the existence regions of the unstable periodic orbits O s;m and O s;mc1, the existence region of the orbit O s;m s ;mc1 is located. As a direct consequence of this, between the regions Q K ;m s ;m and Q K ;mc1 s ;mc1, there exists a region with bandcount js ;m s ;mc1 jcc and so on ad infinitum. As an example, let us consider the subregions located within the region Q 10, as shown in figure 1. The two families of subregions mentioned h above are given by Q 8mC8 and Q 10mC8 in this case. Consider the ð Þ mc ð Þ ð Þ ð Þ mc first family, then between the regions Q 6 and Q there exists ð Þ ð Þ ð Þ 6 ð Þ the region Q 70 and so on. Remarably, in the case azk1, the ð Þ ð Þ ð Þ 6 ð Þ origins of the bandcount-adding structure within the region Q C are hc1 distributed along a specific part of the boundary vp ch. More precisely, for each unstable periodic orbit O s involved in a crisis bifurcation forming this inner bandcount-adding structure, the point z s Zxs 0;[ hxs jsjk1;r Zg s [ hg r s where the corresponding region Q jsjcc s originates from lies on this boundary between the intersection points g r hvp ch and g [ C1 hvp ch. Varying a from the value K1 to any other value K1G a, we observe that the bandcount-adding structure Proc. R. Soc. A (009)

12 5 V. Avrutin et al. (a) 0.65 = (b) 0.65 = (c) 0.65 = (d) 0.65 = (e) 0.65 = S(b) 0.9 Figure 5. (a e) Analytically calculated bifurcation structure of the chaotic domain P ch at the parameter values azk(1c a ). Shading is similar to that described in the caption of figure. For a detailed description, see text. undergoes a dramatic change. By calculating the boundaries of the regions involved in the bandcount-adding scenario within a region Q C, one can hc1 prove that the origins of all these regions collapse to a single point, as shown in figure 5. In the case K1!a!0, this point is given by the lower corner of the Proc. R. Soc. A (009)

13 Bandcount increment: deformed structures 5 region Q C, i.e. the intersection point g [ hvp hc1 C1 ch, whereas in the case a!k1 this point is given by the upper corner of the region Q C, i.e. the hc1 intersection point g r hvp ch. Obviously, each of these points represents a codimension- 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. (007a). However, in the cited wor, no analytical explanation of the observed phenomenon was presented. Note that the size of the bandcount-adding structures emerging at the point g [ hvp C1 ch in the case K1!a!0 and the point g r hvp ch in the case a!k1 decreases with increasing distance to the plane azk1. 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 bandcountadding scenario within the region Q 6. As shown in Part II of this wor, this h region is given by Q 16. The boundaries h [ =r of this region for the ðþ ð Þ ðþ ð Þ special case azk1 are given by eqns (.) and (.5) in Part II. However, eqn (.) in Part II allows us to calculate these surfaces for arbitrary values of a and b. As a result, we obtain 9 h [ ðþ ð Þ Z fða; b; mþjm Z ðab C1Þða5 b 8 Ca b 6 Ca b Ca b Kab CaK1Þ= ða 6 b 10 Ca 6 b 9 Ca 5 b 9 Ca 5 b 8 Ca 5 b 7 Ca b 7 Ca b 6 Ca b 5 Ca b 5 Ca b Ca b Ka b Ca b Kab Ca bkab CaKbÞg; >= h r ðþ ð Þ Z fða; b; mþjm Z ðab C1Þða6 b 8 Ca 5 b 6 Ca b Ka b Cab Ka C1Þ= ða 7 b 9 Ca 6 b 9 Ca 6 b 8 Ca 6 b 7 Ca 5 b 7 Ca 5 b 6 Ca b 6 Ca b 5 Ca b 5 Ca b Ka b Ca b Ka b Cab Ka b Cab Ka CbÞg: >; ð5:þ Hence, we can calculate the area of the region Q 16 in the plane b!m as ðþ ð Þ a function of a Q 16 ðþ ð Þ ðaþ Z ð b b 1 h r ðþ ð Þ ða; bþkh[ ðþ ð Þ ða; bþ db; ð5:þ where h [ =r ða; bþ denotes the values of m at the surfaces h [ =r as a ðþ ð Þ ðþ ð Þ function of a and b. Here, the lower limit b 1 corresponds to the stability boundary q for a!k1 andq for K1!a!0. The upper limit b corresponds to the rightmost point of the region Q 16 given by the intersection point of the curves ðþ ð Þ Proc. R. Soc. A (009)

14 5 V. Avrutin et al S(a) Figure 6. Area of the region Q 16 ðþ ð in the plane b!m dependent on a. Þ h [ =r ða; bþ. Figure 6 shows the area Q 16 calculated according ðþ ð Þ ðþ ð Þ to equation (5.). As one can see, this area is maximal at some value a close to azk1 and then decreases monotonously in both directions. Hereby, it becomes clear why the region Q 16 is difficult to observe numerically for values of a far from ðþ ð Þ a. For example, at S(a)ZK1.1 (that is, azk1.96) the area of the region Q 16 ðþ ð Þ is approximately.05!10 K8. However, the area of the region Q 16 tends to ðþ ð Þ zero asymptotically for a/0 anda/kn. Consequently, the region Q 16 ðþ ð Þ exists for any value of a(kn,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(kn,0), although their areas may decrease rapidly. 6. Summary In this Part III of our wor about the bandcount increment scenario, we described the generic case ask1. The bifurcation structures in the domain of robust chaos can be explained as the deformations of the bifurcation structures for the non-generic case azk1 described in Parts I and II of this wor. In particular, the regions U influenced by specific unstable periodic orbits O u, which possess a triangle-lie shape in the non-generic case azk1, possess a non-convex pentagonal shape in the generic case ask1. For values of a far away from K1, 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 Q C located within the regions U hc1. It was demonstrated in Part II of this wor that these substructures are self-similar and organized by the bandcount-adding scenario as described in Avrutin & Schanz (008). Hereby in the non-generic case azk1, the origins of the regions involved in this scenario are distributed along the boundary vp ch. Now we have shown that, in the generic case ask1, 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 azk1, so that they become difficult to detect numerically. However, we showed analytically that these regions persist for any a. Proc. R. Soc. A (009)

15 Bandcount increment: deformed structures 55 Remarably, the bandcount increment structures occur not only in the onedimensional 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. 008) demonstrates the bandcount increment scenario as presented in this wor. Furthermore, the continuous normal form (Banerjee & Grebogi 1999; Bernardo et al. 1999; Zhusubaliyev et al. 008) shows similar, although not completely identical, bifurcation structures. The investigation of these bifurcation structures represents a challenging tas, 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 neading orbit responsible for the boundary of an attractor represents a cumbersome tas. For details, we refer to Mira et al.(1996). 7. Conclusion The wor 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 001; Zhusubaliyev & Moseilde 00; Bernardo et al. 007), there was a lac of results about the bifurcation structures in the chaotic domain, except it was nown that they are formed by interior and merging crisis bifurcations (Grebogi et al. 198; Maistreno 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. 007a). 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 (008). In this wor, 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 wor. As in the case of the bandcount-adding Proc. R. Soc. A (009)

16 56 V. Avrutin et al. 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 wor 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 wor, we focused on the interior substructures of the overall bandcount increment scenario and demonstrated that they are formed by the already nown bandcount-adding phenomenon. The complete self-similar structure as described in Avrutin & Schanz (008) occurs here within each of the regions involved in the overall bandcount increment structure. The results reported in Parts I and II of this wor 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 wor 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 now the structure of the multi-band windows in order to avoid them. References Avrutin, V. & Schanz, M. 008 On the fully developed bandcount adding scenario. Nonlinearity 1, (doi: / /1/5/010) Avrutin, V., Ecstein, B. & Schanz, M. 007a On detection of multi-band chaotic attractors. Proc. R. Soc. A 6, (doi: /rspa ) Avrutin, V., Schanz, M. & Banerjee, S. 007b Codimension- bifurcations: explanation of the complex 1-, - and D bifurcation structures in nonsmooth maps. Phys. Rev. E 75, 1 7. (doi: /PhysRevE ) Avrutin, V., Ecstein, B. & Schanz, M. 008a The bandcount increment scenario. I. Basic structures. Proc. R. Soc. A 6, (doi: /rspa ) Avrutin, V., Ecstein, B. & Schanz, M. 008b The bandcount increment scenario. II. Interior structures. Proc. R. Soc. A 6, 7 6. (doi: /rspa ) Banerjee, S. & Grebogi, C Border collision bifurcation in two-dimensional piecewise smooth maps. Phys. Rev. E 59, (doi:10.110/physreve.59.05) Banerjee, S. & Verghese, G. (eds) 001. Nonlinear phenomena in power electronics attractors, bifurcations, chaos, and nonlinear control. New Yor, NY: IEEE Press. Banerjee, S., Yore, J. & Grebogi, C Robust chaos. Phys. Rev. Lett. 80, (doi: /PhysRevLett.80.09) Bernardo, M. D., Feigen, M., Hogan, S. & Homer, M Local analysis of C-bifurcations in n- dimensional piecewise smooth dynamical systems. Chaos Solitons Fractals 10, (doi: /s (98)0017-8) Bernardo, M. D., Budd, C., Champneys, A. R. & Kowalczy, P. 007 Piecewise-smooth dynamical systems: theory and applications. Berlin, Germany: Springer. Proc. R. Soc. A (009)

17 Bandcount increment: deformed structures 57 Dutta, P. S., Routroy, B., Banerjee, S. & Alam, S. S. 008 On the existence of low-period orbits in n-dimensional piecewise linear discontinuous maps. Nonlin. Dyn. 5, (doi: / s y) Grebogi, C., Ott, E. & Yore, J. A. 198 Chaotic attractors in crisis. Phys. Rev. Lett. 8, (doi:10.110/physrevlett ) Maistreno, Y. L., Maistreno, V. L. & Viul, S. I Bifurcations of attracting cycles of piecewise linear interval maps. J. Tech. Phys. 7, Milnor, J On the concept of attractor. Commun. Math. Phys. 99, (doi: / BF01180) Mira, C., Gardini, L., Barugola, A. & Cathala, J.-C Chaotic dynamics in two-dimensional noninvertible maps. Singapore: World Scientific. Ott, E. 00 Chaos in dynamical systems. Cambridge, UK: Cambridge University Press. Zhusubaliyev, Z. T. & Moseilde, E. 00 Bifurcations and chaos in piecewise-smooth dynamical systems. Singapore: World Scientific. Zhusubaliyev, Z. T., Moseilde, E., De, S. & Banerjee, S. 008 Transitions from phase-loced dynamics to chaos in a piecewise-linear map. Phys. Rev. E 77, (doi:10.110/physreve ) Proc. R. Soc. A (009)