Research Article Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios

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1 Discrete Dynamics in Nature and Society Volume 211, Article ID , 3 pages doi:1.1155/211/ Research Article Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios Viktor Avrutin, Michael Schanz, and Björn Schenke IPVS, University of Stuttgart, Universitätstrasse 38, 7569 Stuttgart, Germany Correspondence should e addressed to Björn Schenke, joern.schenke@ipvs.uni-stuttgart.de Received 18 May 21; Revised 8 January 211; Accepted 9 January 211 Academic Editor: J. Kurths Copyright q 211 Viktor Avrutin et al. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. We investigate the structure of the chaotic domain of a specific one-dimensional piecewise linear map with one discontinuity. In this system, the region of roust chaos is emedded etween two periodic domains. One of them is organized y the period-adding scenario whereas the other one y the period-increment scenario with coexisting attractors. In the chaotic domain, the influence of oth adjacent periodic domains leads to the coexistence of the recently discovered andcount adding and andcount-increment scenarios. In this work, we focus on the explanation of the overall structure of the chaotic domain and a description of the andcount adding and andcount increment scenarios. 1. Introduction Piecewise smooth maps are in the meanwhile known to e appropriate models of many dynamical processes in science and technology as mentioned already in textooks on this topic 1 4. Standard examples mentioned in these ooks are from the fields of electronics, mechatronics, and mechanics like power converters, impacting systems, systems with suspensions, gears, transmissions, or all earings. Additionally to these usual applications further examples range from systems in micro- and nanotechnology, in particular piezoelectric energy harvesting devices and devices using piezoelectric or other micro-machined actuators 5, 6 to generate movement or propulsion in a very precise manner, up to several models 7 9 of the famous Fermi accelerator 1 as a possile explanation of high-energetic cosmic radiation. Furthermore they occur naturally as oneor two-dimensional Poincaré maps of corresponding time continuous flows The inherent state space reduction allows in many cases a much more efficient investigation of the dynamic ehavior than the original flow. In this work we investigate a specific

2 2 Discrete Dynamics in Nature and Society one-dimensional piecewise linear map derived from the canonical form 14 of onedimensional piecewise linear maps with a single discontinuity. Since the pioneer works 15, 16, it is known that the system f l x n ax n, if x n <, x n 1 f r x n x n l, if x n > 1.1 represents some kind of discontinuous canonical form for order-collision ifurcations. It can e shown 17 that for the dynamic ehavior of this system the exact value of the parameter l is not significant ut only the sign of l, so there are three characteristic cases l<, l>and l. The first two cases represent discontinuous maps having a positive or negative jump, respectively. The third case represents the skew tent map, a continuous map with different slopes of the system function on the left- and right-hand side of the point of discontinuity. This map was investigated since the pioneer works y Maistrenko et al. 18, 19 and y many authors see, e.g., references in 3, 4. The case of the negative jump has already een studied in several works, oth in the regular periodic or aperiodic ut not chaotic and in the chaotic domain. In these works, it was discovered that an extended part of the regular domain of the negative jump case is structured y a ifurcation scenario denoted as period increment scenario with coexisting attractors while the adjacent chaotic domain is structured y the andcount-increment ifurcation scenario The first scenario is formed y order collision ifurcations where an attracting periodic orit collides with the point of discontinuity and disappears, so that after the ifurcation the system evolves to a different attractor. By contrast, the second scenario is formed y crisis ifurcations. Here, efore and after the ifurcation the system shows chaotic ehavior. However, at the ifurcation point the structure of the chaotic attractor, especially the numer of ands connected components, also known as cyclic chaotic intervals changes. The remaining part of the periodic domain of the negative jump case was found to e structured y the period adding ifurcation scenario and the adjacent chaotic domain y the andcount-adding ifurcation scenario 17, 23. To avoid misunderstandings, note that the notation period adding is used in the literature for different ifurcation scenarios. In this work we use a notation ased on the fact that there exists a ifurcation structure where the periods of susequent periodic orits form an arithmetic progression: p n 1 p n Δp p nδp. It is known that three different cases are possile. In the first case the existence regions of the orits with periods p n and p n 1 overlap see Figure 1 a and we denote this ifurcation scenario as period increment scenario with coexistence of attractors. In the second case the existence regions of the orits with periods p n and p n 1 adjoin each other without overlapping and without a gap etween them. Accordingly, we denote this situation as pure period increment scenario. In the last case the situation is more complex. There is a gap etween the existence regions of two susequent periodic orits and the ehavior in these gaps can e either periodic or chaotic. In the first case one well-known situation is given y the self-similar Farey-tree like ifurcation structure where etween the existence regions of two periodic orits with periods p n and p n 1 there exists a region with period p n p n 1 andsoonadinfinitum see Figure 1. In the following we denote this scenario as period adding scenario.

3 Discrete Dynamics in Nature and Society 3 Period 2 Period 2 and 4 Period x p a Period 2 Period 6 Period x p Figure 1: The period increment scenario with coexisting attractors a and the period adding scenario in map 1.3. For each scenario three sketches of the map, a ifurcation diagram and a period diagram are shown. Parameter values are: a.42, 3.5 a and a.42, 3.5. Their locations are shown in Figure 2 with gray lines marked with D and E. In this work we study the case of the positive jump, which can e transformed into a standard form, which simplifies analysis. For any l> the transformation ( ) x, a,, ( lx,a,,l ) - 1.2

4 4 Discrete Dynamics in Nature and Society leads us y dropping the prime to the map f l x n ax n, if x n <, x n 1 f r x n x n 1, if x n >, 1.3 which we will investigate in the following and which can e seen as a canonical form of discontinuous linear maps with a positive jump. The domain of regular dynamics of map 1.3 was investigated in 24. Inthiswork it was shown that this system has a domain of roust chaos in the sense of Banerjee et al. 25 emedded etween the two regular domains. However, the structure of the chaotic domain of this system was not investigated. The first results related to ifurcations occurring in the chaotic domain are going ack to the works y Mira and Gumowski Already in the 6s they discovered a large class of ifurcations, called contact ifurcations, which result from contacts of two invariant sets. These ifurcations were later rediscovered y Greogi et al. in the eginning of the 8s 31, 32 and denoted as crises ifurcations. Oviously, if one of the involved invariant sets is given y a chaotic attractor, then a contact ifurcation represents a crisis in the sense mentioned aove, so the crises represent a suclass of contact ifurcations. Since then crises ifurcations were oserved and investigated in several application fields However, the ifurcation structure formed only y crises ifurcations and organizing the roust chaotic domain are arely investigated so far. Examples of such ifurcation structures were recently reported in There it is shown, that the structure of the chaotic domain is strongly influenced y the structure of the adjacent periodic domain. If this adjacent periodic domain is structured y the period adding scenario in the sense mentioned aove, then the roust chaotic domain is structured y the andcount-adding scenario. By contrast, in the case of an adjacent periodic increment scenario with coexistence of attractors the chaotic domain is structured y the andcountincrement scenario. By contrast to the situations investigated in the cited works where the domain of roust chaos is located etween regular and divergent domains, the results shown in 24 imply that in map 1.3 the chaotic domain is sandwiched etween two regular domains, one of them organized y the period adding scenario and the other one y the periodic increment scenario with coexistence of attractors. Therefore, one can assume that in this case the chaotic domain is organized y oth, andcount-adding and andcount-increment scenarios. This fact will e demonstrated in the paper. Furthermore, the question arises whether there is any interaction etween the scenarios. Is it possile, that the unstale periodic orits which induce one of the andcount scenarios lead to changes in the other andcount scenario? Can the scenarios overlap? Is the complete chaotic domain etween period adding and period increment covered y their andcount counterparts, or is there a region which is structured y neither of them? Also, apart from any possile interaction, it is interesting to see if there are significant differences in the andcount scenarios as they occur in system 1.3, and the previously discovered ones. In this work, we examine in which regions in parameter space the andcount-adding and increment scenarios exist and we examine each scenario to determine if any interaction takes place. This work is structured as follows. After this introduction, we explain in Section 2 symolic sequences and kneading orits, which are two important technical concepts

5 Discrete Dynamics in Nature and Society 5 required for the understanding of this work. In Section 3 we descrie the investigated region in parameter space and the ig ang ifurcations structuring this region. In Section 4 we summarize the structure of the periodic domain y explaining in short the period adding and the period increment scenarios. In Section 5 we give an overview over the structure of the chaotic domain. It consists of three different regions, Q add, Q inc,andq 1. The oundaries separating these three regions are explained in Section 5.1 and their properties are descried in Section 5.2, whereas the following sections descrie the ifurcation structure inside the regions Q add and Q inc.insection 6, the andcount-adding region Q add is descried, wherey Section 6.1 is devoted to the overall andcount-adding structure and Section 6.2 to its nestedsustructures. The oundary etween period adding and andcount-adding shows some interesting ehavior, which is descried in Section 7. Finally, in Section 8, we give an overview over the region Q inc and the andcount increment structure located therein. 2. Symolic Sequences and Kneading Orits Important for all descriptions that involve periodic orits is the concept of symolic sequences 39. A symolic sequence is a sequence of the letters L and R, each representing a single point of an orit. L stands for a point x < andrfor a point x >. Note that when dealing with periodic orits, only the symols necessary for descriing one period will e written and hence the symolic sequences are shift-invariant. We denote a periodic orit which corresponds to the symolic sequence σ as O σ. The chaotic domain of map 1.1 in the case l 1 is organized in a similar way as in the aove-mentioned case l 1. This means it is structured y interior and andmerging crises. Note that we refer to the and-merging crisis as crisis, even though there is no discontinuous change in the overall size of the attractor. Instead, there is a discontinuous change in the numer of ands of the attractor and consequently also in its topological structure. Both interior and merging crisis share the property that they can e determined y calculating the intersection of a point of the colliding unstale periodic orit with the oundary of the chaotic attractor. For that, one has to determine the oundary of a chaotic attractor. This is done using kneading theory y following up kneading orits, that means the forward iterates or itineraries of the critical points. We remark that this idea is going ack to the works y Gumowski and Mira 28, 29. Already in 1977 they recognized that the oundaries of chaotic attractors in 2D maps are given y itineraries of some critical lines lignes critiques. For details we refer to 4. Of course, when dealing with 1D maps, these critical lines represent critical points and their itineraries define the kneading orits. Following this idea, a kneading orit is an orit started at a critical value of the system function, which is given for system 1.3 y x. It has the property of jumping etween the maxima and minima of the ands of a chaotic attractor, until it reaches a specific length, after which it leaves the oundaries of the attractor. Because system 1.3 is discontinuous at the critical point x, there exist two different kneading orits, one eginning with f l and the other eginning with f r. For the kneading orits we use the following compact notation. A point of a kneading orit given y f a f f c f d f z, witha,, c, d, z {r/l}, is written as x ko z dca.this means that the sequence of iteration steps leading to this point starts with f z and ends with f a x ko. This is similar to the symolic sequences used for periodic orits. Note that z dc the sequences used for kneading orits are not shift-invariant.

6 6 Discrete Dynamics in Nature and Society 3. Investigated Parameter Space Region and Big Bang Bifurcations The periodic dynamics of system 1.3 is investigated in 24. It is shown in this work that the periodic domain of system 1.3 is organized y four codimension-3 ig ang ifurcations. A ig ang ifurcation point in a n-dimensional parameter space is an intersection point of infinitely many n 1 -dimensional ifurcation surfaces. The periods and symolic sequences of the periodic orits existing etween these ifurcation surfaces are organized y a rule specific to the type of the ig ang ifurcation. A ig ang ifurcation point has an influence region which is composed of the union of the existence regions of all its periodic orits. So y determining the location and type of a ig ang ifurcation point, the structure of its complete influence region can e determined as well. The codimension-3 ig ang ifurcation points found in system 1.3 are of the specific type initially descried in 41. A ifurcation point B of this type possesses an extended influence region Ω B in the 3D parameter space, which is organized y two manifolds, a 2D manifold M 2D B and a 1D manifold M 1D B. Within the influence region Ω B the ifurcation structure aove the 2D manifold M 2D B is given y the period adding phenomenon. We denote the corresponding part of Ω B y Ω add B. In the 2D manifold the structure is given y the pure period increment phenomenon and elow M 2D B y the period increment phenomenon with coexisting attractors. The part of Ω B organized y the period increment phenomenon with coexisting attractors is denoted in the following y Ω inc B. Herey the terms aove and elow refer to the direction of the 1D manifold M 1D B. An inherent property of system 1.3, which is reflected in the structure of its parameter space, is given y the symmetry f ( a,,, x ) f (, a, ( 1 ), x ). 3.1 This symmetry implies that the ifurcations occur in this system pairwise and generate identical structures in the parts of the parameter space, which can e mapped onto each other y 3.1. Therefore, it is sufficient to consider only two of the codimension-3 ig ang ifurcations organizing the 3D parameter space of system 1.3. In 24 it is shown that the region in the parameter space given y P {( a,, ) a < 1, < 1, } 3.2 is organized y two codimension-3 ig ang ifurcations occurring at the points B 1, 1,, B 2, 1,. 3.3 The two other codimension-3 ig ang ifurcations occur at the points B 1 1,, 1, B 2 1,, 3.4 and organize the part of the parameter space symmetric to P with respect to the symmetry 3.1. If a > 1, there exist no stale periodic orits for all values of and in P. Aswe concern ourselves in this work with the part of the parameter space which is located etween

7 Discrete Dynamics in Nature and Society E Ω add (B 2 ) τ env 3 5 Q ch a a< Ω inc (B 1 ) 3.6 B M 2D (B 2 ) τ a B 1 M 2D (B 1 ) 3.6 D Ω inc (B 2) 3 5 Q ch 2 A B C c a> 1 τ env Ω add (B 1 ) Figure 2: Numerically calculated ifurcation structures in the, -plane for a.42 a, a, a.42 c, and 2. 1., 3.6. The gray areas in a and c mark the chaotic domain Q ch. The dashed lines in a and c show the locations of the ifurcation scenarios, shown in Figures 1, 3, and 5.

8 8 Discrete Dynamics in Nature and Society 2.35 τ env x 1.95 a τ env x.85 3 Figure 3: Numerically calculated ifurcation diagrams for a.42, 1.8, 1.5 a and a.42, 1.195, 3.. Both show similar structures in the left part, while they are significantly different in the right part. The location of these scans in the, -plane is shown in Figure 2 c with dashed lines. The line A corresponds to a and the line C to. the two periodic scenarios in system 1.3, we do not investigate this part of the parameter space and restrict our investigation to region 3.2. Figure 2 shows the ifurcation structure of the region P in three characteristic cases a<, a anda>. As one can see, in the case a the complete, -plane is covered y the periodic domain, which is in this case formed y M 2D B 1 and M 2D B 2. By contrast, in oth cases a<anda> there is a chaotic domain Q ch characterized y a positive Lyapunov exponent located etween the influence regions of B 1 and B 2. As one can see in Figure 3, the chaotic domain Q ch has a complex interior structure formed y oth oneand and multiand chaotic attractors. However, in 24 this structure is not investigated. Therefore the question arises, which ifurcation phenomena form the interior structure of the chaotic domain. It is worth noticing that the aove-mentioned question in the case of the negative jump of the system function l 1 is already investigated. The ehavior close to the oundary etween the period adding structure and the chaotic domain is reported in 23. The map considered in this work is equivalent to a special case of map 1.1 with identical slopes a and. However, the otained results are also valid for any values of a and. Itis shown in the cited work that the chaotic domain is organized in this case y the andcountadding scenario, which is formed y interior crises 31. A part of these crises is caused

9 Discrete Dynamics in Nature and Society 9 y the unstale periodic orits originating from the periodic domain where they are stale and organized y the period adding scenario. Additionally, the interior sustructures of the andcount-adding scenario are formed y interior crises caused y periodic orits, which are everywhere unstale. Since map 1.3 shows also a transition from period adding to chaos, it can e expected, that the chaotic domain is in this case also organized y the andcountadding scenario. The ehavior close to the transition from the periodic domain organized y the period increment scenario with coexisting attractors to chaos for map 1.1 in the case l 1 is investigated in a series of works It is shown that the structure of the chaotic domain is organized in this case y two types of crises, namely and-merging crises and interior crises. The unstale periodic orits originating from the periodic domain lead to andmerging crises organized y the andcount-increment scenario, whereas the everywhere unstale orits lead to interior crises forming the interior sustructures of this scenario. The andcount-increment scenario is significantly more complex than the andcount-adding scenario and is closely related to the composite and nonsmooth shape of the oundary etween periodic and chaotic domain. In contrast to the case of the andcount-adding scenario where this oundary represents a smooth surface, in the case of the andcountincrement scenario the oundary is given y segments of ifurcation surfaces of two types order-collision ifurcations and degenerated flip ifurcations. Therefore, the question arises, up to which extent the results otained for the case l 1 can e adopted also for the case l 1 considered in the present work. 4. Periodic Domain As the ifurcation structure of the chaotic domain is influenced y the periodic orits, which exist in the adjacent periodic domain, let us recall some results related to the ifurcation structure of this periodic domain. For a more detailed description we refer to 24. As mentioned aove, the periodic domain of system 1.3 in the region P of the parameter space is organized y two codimension-3 ig ang ifurcation points, B 1 and B 2 see Figure 2. In the case a the influence region of each of these points consists of a twodimensional manifold, which together cover the complete, -plane and are oth organized y the pure period increment scenario. The oundary etween the manifolds M 2D B 1 and M 2D B 2 is given y the curve τ { } (a, ) 1, a, < 1,, 4.1 which is shown in Figure 2. Below τ the influence region of the ig ang point B 1 is located. It consists of an infinite numer of staility regions of periodic orits with odd periods, eginning with P s LR 2 and continuing with P s LR 2n for n > 1. Aove τ, the influence region of the ig ang ifurcation point B 2 is located. It is structured similarly to the influence region of B 1, consisting of an infinite numer of staility regions of periodic orits with even periods, eginning with P s LR and continuing with P s LR 2n 1 for n>1. In the case a /, Ω B 1 as well as Ω B 2 are no longer structured y the pure period increment scenario. Note that, however, oth regions still contain the staility regions of the same orits as efore. In the case a>, Ω B 2 is structured y the period increment scenario

10 1 Discrete Dynamics in Nature and Society with coexisting attractors. Since system 1.3 is piecewise linear and has only one point of discontinuity, at most two attractors may coexist. In this case, this means that the regions P s overlap pairwise. The region Ω B LR 2n 1 1 is structured y the period adding scenario, that means etween two susequent regions P s and P s there is some free space where the LR 2n LR 2n 2 staility regions of the orits are located, which can e generated y the infinite symolic sequence adding scheme. Note that the infinite symolic sequence adding scheme can e considered as a symolic representation of the well-known Farey-tree 39, 42, 43, which is a sutree of the Stern-Brocot-tree 44, 45. In the case a<the situation is reversed, with Ω B 2 structured y the period adding scenario and Ω B 1 structured y the period increment scenario with coexisting attractors. In oth cases, the chaotic domain appears etween the two influence regions, centered at the former location of the curve τ. For a 1, oth influence regions Ω B 1 and Ω B 2 shrink and their oundaries move away from each other in the direction of the respective ig ang ifurcation points. Since the ifurcation scenarios in oth cases a>anda<are identical, we consider in the following only the case a>. For the analytical calculations in the following we will need some periodic orits of system 1.3. It can e easily seen that system 1.3 has at most two fixed points, however only the fixed point x R exists in the parameter space region P. Since we consider < 1, the fixed point x R is unstale. The existence regions of the n 1 -periodic orits O LR n are ounded y the surfaces of order collision ifurcations where the involved orit collides with the oundary x fromtheleftwithitsfirstpointx LRn and from the right with its next to last point x LRn n 1.In the following these surfaces will e denoted y ξ,l and ξ n 1,r, respectively. As system 1.3 LR n LR n is piecewise linear, the orits O LR n and hence the order collision ifurcation surfaces can e calculated analytically for all n>. The colliding points x LRn and x LRn n 1 are: x LRn n 1 ( n 1 1 ) n 1 x LRn a n, 1 1 ( n 1 1 ) ( n 2 1 ) an 2 x see 24 The conditions x LRn andx LRn n 1 lead us to the following expressions for the order collision ifurcation surfaces ξ,l and ξ n 1,r. These expressions are valid regardless to LR n LR n which influence region Ω B 1,2 the corresponding order collision ifurcation surfaces elong and in which ifurcation scenarios these surfaces are involved in ξ n 1,r LR n { (a, ξ,l ) n } 1 LR, a < 1, < 1,, n n { } (a, ) n 2 1 a 1 a n, a < 1, < 1,. n 2 a 1 a n 4.5

11 Discrete Dynamics in Nature and Society 11 The orits O LR n are not everywhere stale in their existence regions n. The oundary etween the region in which an orit O LR n is stale and the region in which it is unstale is given y one of the ifurcation suspaces θ ± LR {( a,, ) a n ±1 }. 4.6 n Note that an orit O LR n involved in the period increment scenario with coexisting attractors that means with odd n for a>and even n for a< ecomes unstale via degenerated flip ifurcation 46 at the oundary θ. By contrast, an orit O LR n LR n involved in the period adding scenario with odd n for a<and even n for a> ecomes unstale at the oundary θ.notethatθ represents a codimension-2 ifurcation given y the intersection curve LR n LR n of the ifurcation surfaces ξ,l and ξ n 1,r located at the surface τ env for details see 47. LR n LR n When dealing with the period increment scenario with coexisting attractors, it is necessary to know, which oundaries of its regions correspond to which order collision ifurcations. In the case a>the upper oundary of the region 2n 1 is given y the surface ξ n 1,r and the lower oundary y the surface ξ,l. LR n LR n As one can see in Figure 2 c the surfaces detected so far form the oundary etween Ω inc B 2 and the chaotic domain. In the case a>this oundary consists of pieces of the order collision ifurcation surfaces ξ,l and pieces of the staility oundaries θ.the LR n LR 2n 1 oundary etween the region Ω add B 1 and the chaotic domain is given y the smooth surface τ env { } (a, ) 1 a, a < 1, < 1,, 4.7 a which follows from the condition f l f r f r f l.notethatτ env τ for a. 5. General Properties of the Chaotic Domain Continuing the work reported in 24, we investigate the structure of, parameter suspace of system 1.3 for specific values of a. Typical examples for the ifurcation scenarios calculated numerically y varying are shown in Figure 3. The presented examples correspond to the lines marked with A and C in Figure 2 c. Note that the ifurcation scenarios, which can e otained for other values of a and are in the most cases similar to the presented examples. As one can see, oth diagrams have parts that are similar to each other, and parts which differ. In the left part of oth diagrams we oserve the period adding structure already mentioned in Section 4. For increasing values of there is a region which apparently contains a two-and chaotic attractor, that is interspersed with chaotic attractors with higher andcounts. Increasing further, we oserve a crisis ifurcation where the ands of the two-and attractor merge and an one-and attractor emerges. Afterwards the two diagrams egin to differ. In Figure 3 a, the one-and attractor persists until the dynamic ecomes periodic again at the oundary etween chaotic domain and the region organized y the period increment scenario with coexisting attractors as mentioned in Section 4.InFigure 3, however, one more crisis ifurcation leads to the appearance of another two-and chaotic attractor. Like the two-and attractor next to the period adding scenario, this attractor is interrupted y chaotic attractors with higher andcounts. The crises, which induce these

12 12 Discrete Dynamics in Nature and Society B A a a> a< Figure 4: Numerically calculated andcount diagrams in the 2D parameter space for a.42 a, a.42 and 2. 1., 3.6, showing the asic ifurcation structure of the chaotic domain. The periodic domain is colored white, the chaotic domain in various shades of gray. The gray tones represent the andcount of the regions, calculated using the numeric approach reported in 48. The lighter the gray of a chaotic region the higher its andcount is. The white lines in a show the locations of two ifurcation scenarios, shown in Figures 9 and 14. high-andcount attractors look different than the crises, which induce the high-andcount attractors in the left part of the figure. Also, the whole structure is more complex. Based on these oservations, we divide the chaotic domain into three suregions. We call the region adjacent to the period adding scenario Q add, the single-and region Q 1 and the region next to the period increment scenario Q inc. It seems to e a hard if not an unsolvale task to explain the presented ifurcation scenarios ased on the one-dimensional ifurcation diagrams only. Since a one-dimensional

13 Discrete Dynamics in Nature and Society 13 ifurcation diagram is limited in the amount of information it can reveal aout twodimensional structures in the parameter space, we determined andcounts numerically using the algorithm reported in 48 in relevant parts of a, -plane see Figure 4. Note that this figure shows the same region in the parameter space as Figure 2. In this figure, the three regions we have identified aove are clearly visile, as are the structures within and the oundaries etween them. Furthermore it ecomes clearly visile that the chaotic domain shown as a homogeneous gray region in Figure 2 has a well-organized interior structure Region Boundaries The outer oundaries of Q ch are already given through the oundaries of the periodic domain. One of these oundaries is defined through the surface τ env. The other oundary is defined through the union of parts of the lower oundaries of the regions P s and parts LR 2n 1 of the staility oundaries θ see Section 4. LR 2n 1 Our task is now to find the oundaries separating the region Q 1 from Q add and Q inc. As one can see in Figure 3, the ifurcation which defines these oundaries represents a andmerging crisis. Given the numer of gaps in the attractor efore the ifurcation one the unstale periodic orit involved in this crisis is the unstale fixed point x R, see 4.2, as shown in Figure 5. To calculate the crisis surfaces, the points of the appropriate kneading orits are needed. For the and-merging crisis at the smaller value of, these are x ko and l x ko 1. This and-merging crisis can thus e calculated y any of the conditions lr x R x ko and x R x ko. This leads to the oundary surface l lr γ add { } (a, ) 1, < a < 1, < 1, 5.1 which separates the region Q 1 from the region Q add. It is worth noticing that for the calculation of the other and-merging crisis surface caused y the unstale fixed point x R we have to use points of the other kneading orit. As one can see in Figure 5, the kneading orit points relevant for this crisis ifurcation are a 1 1 and xko a Hence, the andmerging crisis at the higher value of can e calculated y any of the conditions x R x ko r 2 r 2 lr l x ko r 2 l and x R x ko. This leads to the oundary surface r 2 lr γ inc { } (a, ) a a 2 1, < a < 1, < 1, a a which separates the region Q 1 from the region Q inc. The location of the surfaces γ add and γ inc in the plane, is shown in Figure Boundary Surface Properties Let us consider some properties of the crisis surfaces γ add and γ inc. It is noticeale that the surface γ add can e seen as some kind of continuation of the curve τ for the case a /. Both

14 14 Discrete Dynamics in Nature and Society 4 τ env Qadd Q γ add Q 1 ξ,l γ inc inc LR 3 x ko rrl x x ko lr x ko l x R x ko rrlr Figure 5: Numerically calculated ifurcation diagram for a.42, 1.37, 3.4, showing a typical part of the chaotic domain. The location of this scenario in the, -plane is marked with B in Figure 2 c PLR 5 Q inc γ inc Q 1 Qadd γ add τ env 2 1 Figure 6: Numerically calculated structure of the, -plane for a.42, 2. 1., 3.6. The two curves γ add and γ inc are analytically calculated and show the and-merging crisis curves structuring the chaotic domain. In the periodic domain, the curve γ inc is shown dotted. are defined through 1/, anddiffer only in the value of a at which they exist. Since the value of in 5.1 does not depend on a, the location of the surface γ add is the same in all, -planes. Considering τ env, this means that the area of the chaotic region etween γ add and τ env tendstozerofor a and increases for a 1. Note that γ add and τ env never intersect, since they can only coincide at a. The ehavior of γ inc is more complicated. Like γ add, it tends to τ env for a andis located far away from it for a 1. However, unlike γ add its interaction with the oundary etween the periodic and the chaotic domain is more sophisticated. As descried in Section 4, the periodic regions shrink for a 1. It is not readily apparent whether their oundary moves faster away from γ add than γ inc, so it is possile that γ inc will intersect some periodic regions P s LR n for some values of a. That it indeed does so can

15 Discrete Dynamics in Nature and Society 15 e seen, for example, in Figure 6, which shows an intersection of γ inc and the lower oundary of a periodic region,toeprecise for a.42. So the question arises whether it is possile that γ inc will intersect every periodic region 2n 1 for appropriate values of a. To confirm this, let us calculate the intersection of γ inc with the lowest point of a staility region P s. If this point can e found, γ LR n inc will intersect that region for smaller asolute values of a, and will not intersect it for larger values. To locate this point, we need the order collision ifurcation surfaces ξ,l and the corresponding staility oundaries θ. LR n LR n In the case of a >, the lowest value of of a staility region P s is at the LR 2n 1 intersection of its lower oundary with the staility oundary. We get that value of as a function of a and n y solving θ for and inserting it into ξ,l. Since we want γ LR n LR n inc to intersect P s at the same point, we insert the same value into γ LR 2n 1 inc as well. This leads to: ξ,l a LR 1 a ( ), n 1 1/n a a n 1 /n 1 γ inc a 1 1/n a a 1 2/n 1 a a 1 2/n a 1/n. 5.3 Note that 5.3 is valid only for a>incomination with odd values of n as well as for a< in comination with even values of n. By solving ξ,l LR n a 1 γ inc 1/n a 1 1/n for a,weget: a n γ inc 1 n 1 2 1/2 n, 5.4 which is the value of a at which γ inc intersects the lowest point of the staility region P s. LR n The largest value a n γ inc can take is a 1 γ inc 2 1/ Because a n γ inc decreases monotonically for increasing n, approaching for n, we can conclude that γ inc intersects every region P s for small enough asolute values of a. Consequently, a transition from the period- n 1 LR n orit O LR n to one-and chaos as shown in the right part of Figure 3 a for n 1 is possile for any n. Since γ inc is the oundary etween Q 1 and Q inc, an intersection of γ inc with a periodic region P s alters the general structure of Q ch. Recall, that the intersection point of γ LR n inc with the oundary of the periodic domain Ω inc B 2 depends on the parameter. For larger values of than at the point of intersection, there is the region Q inc etween Q 1 and the period increment regions. For smaller values of than at the point of intersection, the periodic domain overlaps with Q 1 so that there is no Q inc for the values of etween an intersection with a periodic region P s and its staility oundary θ. This is illustrated in Figure 7. LR n LR n This explains the difference etween the two ifurcation diagrams shown in Figure 3. Figure 3 a corresponds to a value of, which lies etween the intersection of γ inc with the oundary of a periodic region and the staility oundary elonging to that periodic region. At values of, which are not etween such a point of intersection and the corresponding staility oundary, all ifurcation diagrams of the chaotic domain calculated for constant values of a and will look similar to Figure 3. They will contain at least a two-and attractor, which is adjacent to the region of period increment with coexisting attractors scenario.

16 16 Discrete Dynamics in Nature and Society 1.8 γ inc Q inc A 3 B Q inc Q Figure 7: Analytically determined structure of the, -plane for a.23, , The and-merging crisis curve γ inc intersects the periodic region P s LR 3. Between the two intersection points marked with A and B, Q 1 is adjacent to the periodic region. For values of smaller than A or larger than B, Q 1 is adjacent to Q inc instead. 6. Bandcount Adding Region At this point the question may arise, why we distinguish etween the two regions Q add and Q inc. The main reason for that is that the ifurcation structure of these regions emerges at the oundary to the corresponding adjacent periodic regions and depends strongly on the fact how these periodic regions are structured. In this section we turn to the ifurcation structure of the region Q add,thatis,the ifurcation structure occurring at the oundary to the adjacent period adding scenario. We will demonstrate elow, that the ifurcation structure of this region is organized y the socalled andcount-adding scenario, initially reported in 23. In general, the region Q add is ounded y the surfaces γ add and τ env this can e seen in Figure 6 and in Figure 8 elow. The distance etween these surfaces shrinks with decreasing asolute value of a, so that the complete region of the andcount-adding scenario shrinks more and more. It turns out, however, that this is the only effect of the parameter a. Note that the ifurcation structure of the chaotic domain near the oundary to a periodic region organized y the period adding scenario in a different piecewise linear map is investigated in detail in 23. In the cited work the andcount-adding scenario was descried and it is remarkale, although somehow expected, that the ifurcation structure in Q add is organized similarly Overall Bandcount Adding Structure The ifurcation structure of the region Q add is shown in Figure 8. As it was demonstrated in 23, the overall andcount-adding scenario is composed of two recursive, self-similar structures, the actual andcount-adding scenario and the andcount douling scenario nested within. Note that we will denote the regions that form these scenarios as andcountadding regions and andcount douling regions, respectively. The structure of the overall andcount-adding scenario is closely connected to the structure of the adjacent period adding scenario. This is due to the fact that the stale orits forming the period adding scenario in the periodic domain are not destroyed at the oundary to the chaotic domain. Instead, at this

17 Discrete Dynamics in Nature and Society Q 11 LR 8 Q 1 Q 9 LR 6 Q 7 LR 4 Q 14 LR 4 LR 6 γ add Q add τ env Figure 8: Numerically calculated andcount diagram in the 2D parameter space for a.2, , Both the period adding gray lines and the andcount-adding gray areas scenarios can e seen, separated y the oundary curve τ env. The white curves inside the white ox show the analytically calculated interior crisis curves η r and η l, which are the oundaries of the andcountadding region Q 7. The other white curves show the interior crises corresponding to the respective laels. LR 4 LR 4 LR 4 The and-merging crisis curve γ add separates Q add from Q 1. Note that the region in the upper part is an irregular region inside Q 1. The marked rectangle is shown enlarged in Figure 1. oundary the orits ecome unstale and are responsile for the crisis ifurcations forming the overall andcount-adding scenario. As an example in Figure 8 the region 4 is shown, where the 5-periodic orit O LR 4 is stale. The oundaries of this region are given y the order-collision ifurcation surfaces ξ,l and ξ 3,r of this orit. As one can see, oth curves intersect the oundary τ env at the same LR 4 LR 4 point. Remarkaly, at this point the region Q 7 originates, which is ounded y the crisis LR 4 ifurcation surfaces caused y the same orit O LR 4 ut which is now unstale. As one can clearly see in Figure 8, the descried phenomenon occurs for each periodic orit, which is stale within the periodic domain. Since these orits are organized y the period adding scenario, the regions of multiand chaotic attractors are organized in a similar way. Recall, that the infinite symolic sequence adding scheme which organizes this scenario, contains not only the first generation sequences, which are given in the considered case y LR n with n even, ut also of sequences of further generations, which result from the concatenation of their corresponding parent sequences. For example, the second generation sequences LR n LR n 2 result from the concatenation of their parent sequences LR n and LR n 2. Consequently, the overall structure can e defined recursively and consists therefore of an infinite numer of generations. This fact is reflected in the structure of the chaotic domain. This means that there is a first generation of regions that stretches over the complete andcount-adding region Q add namely the regions of the orits with the sequences LR n, n even and there are regions of higher generations, which are located etween the regions of the lower generations. For example, the second generation regions are located etween

18 18 Discrete Dynamics in Nature and Society the first generation regions, the third generation regions are located etween the second generation regions, and so on, ad infinitum. Figure 8 shows this. The procedure for the analytical calculation of the regions forming the overall andcount-adding scenario is descried in detail in 23. In this work it is shown that the oundaries of these regions are given y interior crisis ifurcations, which occur at parameter values where the corresponding unstale periodic orits collide with the chaotic attractor. An example for this is shown in Figure 9, y the collision of the period-5 orit O LR 4 with the 7- and chaotic attractor. Note that the figure shows also that the orit ecomes virtual which means that one of its points leaves its domain of definition according to to the symolic sequence of the orit, and therefore the orit disappears at the parameter values where one of its points is located on the oundary of the chaotic attractor. The explanation for that lies in the definition of order collision ifurcations, namely that one point of the orit hits the oundary x. This means that another point is given y f d, withd {l, r}. However, this is also a point of a kneading orit of the system given y 1.3, which has the property to jump etween the oundaries of the chaotic attractor. Therefore, for system 1.3 at a order collision ifurcation, which happens within the chaotic domain, one point of the colliding orit is always located at a oundary of the chaotic attractor. Consequently, the oundaries of the region Q 7 can e determined using the LR 4 condition that a point of the orit O LR 4 hits the oundary of the chaotic attractor, which is given y an appropriate point of the kneading orit. Since the investigated system is piecewise linear, similar calculations can e performed for all asic orits O LR n analytically. In this way we otain the oundaries of all regions Q n 3, which form the first generation of LR n the andcount-adding scenario: η r LR n { ( (a, ) n 2 1 ) } a 2 1, < a < 1, < 1,, 6.1 n 3 1 a 3 η l LR n { (a,, ) < a < 1, < 1, ( n 2 n) a 2 ( n ) } a 1 2 n 2 n a 2 ( n ). a As Figure 8 shows, the numer of ands in each region Q n 3 is equal to p 2 where p is the LR n period of the periodic orit O LR n causing the crisis ifurcations. This represents a difference to the scenario descried in 23, since in this work this numer is equal to p 1. The difference is easily explained taking into account that the p-periodic orit determines the numer of gaps etween the ands, and hence the numer of ands is p 1. In the case of system 1.3 considered here, there is one additional gap see Figure 9 occupied y the unstale fixed point x R, everywhere in Q add. In summary, a first generation region Q n 3 of the overall andcount-adding scenario LR n is induced y the orit O LR n and hence ounded y the interior crisis surfaces η l, η r.it LR n LR n is located next to the staility region P s and has a andcount of n 3. LR n The regions of the higher generations have similar properties. They are also induced y unstale orits that are stale in the adjacent periodic domain and they are also located

19 Discrete Dynamics in Nature and Society τ env η l LR 4 Q 7 LR 4 ηr LR 4 x Figure 9: Numerically calculated ifurcation diagram for a.42, 1.32,.3.5. The curves show the unstale periodic orit O LR 4 lack lines, which collides with the chaotic attractor in two interior crisis ifurcations, and the unstale fixed point x R gray line, which occupies the remaining gap. Note that the orit is destroyed through order collisions at x. Its nonexistent or virtual parts are shown dashed. The location of this scenario in the, -plane is marked with A in Figure 4 a. next to the staility region of these orits. The symolic sequences of these orits can e determined using the infinite sequence adding scheme, as descried in 24. As an example, for each n, etween the first generation regions Q n 3 and Q n 5 there is a region Q 2n 6 of LR n LR n 2 LR n LR n 2 the second generation, and so on, ad infinitum. As an example for the regions of the second generation the region Q 14 is laeled in Figure 8. LR 4 LR Nested Sustructures Within each of the regions forming the overall andcount-adding structure, there is a nested sequence of regions with higher andcounts organized y the andcount douling scenario. The oundaries of the regions in this scenario are induced y interior crisis ifurcations as already pointed out in 23. As an example, Figure 1 shows the ifurcation structures located within the region Q 7. The first two regions of the andcount douling scenario are the LR 4 region Q 17 located in the middle part of Q 7 and the region Q 37 located in LR 2 LR 6 LR 4 LR 6 LR 4 LR 2 LR 4 the middle part of Q 17. LR 2 LR 6 The regions forming the andcount douling scenario originate at the same point the intersection of the oundary surfaces with τ env as their surrounding regions and are located, roughly speaking, in the middle of them. Each region in the andcount douling scenario has the property that it is induced y an orit with a period that is twice as high as the period of the orit that induces the region surrounding it. Also, each of these regions contains the next region, following the same rules concerning andcount and location, and so on, ad infinitum. Note that most of the regions in such a douling-cascade are very hard to detect numerically ecause their size decreases rapidly.

20 2 Discrete Dynamics in Nature and Society.536 Q 7 LR 4 Q 17 LR 2 LR 6 Q 37 LR 6 LR 4 LR 2 LR 4 Q 22 LR 2 LR 6 LR 4 Q 27 LR 2 LR 6 (LR 4 ) 2 Q 32 LR 2 LR 6 (LR 4 ) 3 Q add Q 22 LR 6 LR 2 LR 4 Q 27 LR 6 LR 2 (LR 4 ) 2 Q 32 LR 6 LR 2 (LR 4 ) τ env Figure 1: Numerically calculated andcount diagram in the 2D parameter space for a.2, , The andcount-adding region Q 7 is shown enlarged. The andcount LR 4 douling region Q 17 is visile in the middle. The first three regions of oth nested andcount-adding LR 2 LR 6 structures to the left and to the right of Q 17 are laeled. The second andcount douling region LR 2 LR 6 Q 37 is also laeled. The rectangle in Figure 8 shows where this figure is located in the overall LR 6 LR 4 LR 2 LR andcount-adding 4 scenario. The analytical Figure 11 corresponds to this image. This shrinking of the regions is analogous to the ehavior of the andcount douling investigated in 23. In this work, the scaling constant for the length of the regions is analytically determined to e 2, whereas the scaling constant for the width of the regions does not exist, ecause its value tends to explaining the rapid shrinking of the size of the regions. Since the expressions of the crisis ifurcations involved in the andcount douling scenario in system 1.3 are significantly more complicated than it is the case for the system investigated in 23, it is very difficult to determine the scaling constants analytically. However, since the andcount douling regions in system 1.3 decrease in size in a similar way as the regions in the cited work with the length of the regions decreasing y an approximately constant factor in each step in the sequence and the width of the regions decreasing stronger with each step, we can assume that the scaling constants are similar too. This means, that there exists a finite scaling constant corresponding to the length of the regions and there exists no finite scaling constant corresponding to the width of the regions. The andcount douling regions Q 17 LR 2 LR 6 and Q 37 shown in Figure 1 illustrate this. LR 6 LR 4 LR 2 LR 4 The symolic sequences of the orits that induce the andcount douling regions can e determined using the creation rules presented in 23. Note that the orits corresponding to these sequences do not occur in the adjacent period adding scenario ut emerge at the oundary τ env etween the periodic and the chaotic domain. Their regions of existence are ounded y two order-collision ifurcation surfaces and can e determined in the same way as it can e done for the orits of the period adding scenario. To give an example for the andcount douling scenario, we descrie the first steps of the scenario inside the region Q 7 shown in Figure 1. The first region inside Q 7 has LR 4 LR 4

21 Discrete Dynamics in Nature and Society 21 a andcount of 17 and is induced y the orit O LR 2 LR6. The numer of ands in this region results from the fact, that 5 gaps of the attractor are occupied y the orit O LR 4, 1 further gaps y the O LR 2 LR 6, and one gap y the fixed point xr. Using the creation rules for the symolic sequences mentioned aove, it can e easily shown that the next two steps of the scenario are caused y the orits O LR 6 LR 4 LR 2 LR 4 and O LR 6 LR 4 LR 2 LR 6 LR 2 LR 4 LR 6 LR 2 with periods 2 and 4, respectively. The andcounts in the corresponding regions are 37 and 77, respectively, and can e explained y the numer of gaps of the multiand attractors which are occupied y the unstale periodic orits involved in the cascade up to a certain period including the unstale fixed point. Hence the andcounts are given y and , respectively. This concludes the description of the andcount douling scenario. However, there are still nested sustructures left to descrie. As can e seen in Figure 1, there are several regions located within the region Q 7 on oth sides of the andcount douling region Q 17. LR 4 LR 2 LR 6 These regions form a nested andcount-adding scenario. The symolic sequences of the orits that induce these additional nested regions can e determined using the infinite sequence adding scheme. The starting sequences of this adding scheme correspond to the sequences of the andcount douling region and its surrounding region. Note that the sequences on one side of the region Q 17 are slightly different from the sequences on the other side. In LR 2 LR 6 oth cases, the sequence of the surrounding region Q 7 is concatenated several times, ut LR 4 the sequence of the andcount douling region Q 17 in one case is shifted with respect to LR 2 LR 6 the other case. This can e seen in Figure 1: while the symolic sequences to the left of the andcount douling region Q 17 form the family LR 6 LR 2 LR 4 n, the sequences to the LR 2 LR 6 right form the family LR 2 LR 6 LR 4 n. These families form the first generation of the nested andcount-adding scenario. Further generations result from them in the same way as descried for the overall andcountadding scenario. Note that these nested andcount-adding regions have all the properties of the andcount-adding regions descried earlier. In particular, this means that each of these regions contains its own andcount douling cascade, and each of these andcount douling regions induce further nested andcount-adding scenarios, which in turn contain andcount douling regions, and so on, ad infinitum. Additionally, each andcount douling region nested inside another region contains a nested andcount-adding scenario, in a similar way as descried aove. This process continues ad infinitum, leading to a self-similar structure of the region Q add. 7. Dynamics on the Boundary etween the Period Adding Region and the Chaotic Domain Using the analytical formulations of the interior crisis surfaces confining the overall andcount-adding regions Q σ 2 σ, we can oserve that for each σ their right intersection point denoted in the following y ζ σ elongs to the oundary surface τ env. Recall that σ refers here to any asic sequence LR n with n odd for a>, respectively even for a<, as well as to any sequence which can e derived from a pair LR n, LR n 2 using the sequence adding scheme. It can also e verified analytically that the surfaces of order collision ifurcation of the orit O σ which causes these crises, intersect the oundary surface τ env at the same point. This is illustrated y Figure 8 which shows the oundaries of the andcount-adding region Q 7 LR 4 and the period adding region 4. Since the orit O σ is stale within the periodic domain and unstale in the chaotic one, the staility oundary θ σ intersects τ env at the same point as

22 22 Discrete Dynamics in Nature and Society well. This means for example that ζ LR n η r LR n η l LR n τ env ξ,l LR n ξ n 1,r LR n τ env θ LR n τ env. 7.1 Already this leads us to conclude that the points ζ σ represent ifurcations with a codimension larger than one. Next let us consider a point ζ σ as a function of the third parameter a. Recall that the variation of a in the intervals 1, and, 1 does not change the topological structure of the regions Ω add B 1 and Q add. Hence, in the 3D parameter space a,, each point at the curves ζ σ a with a 1, and a, 1 represents a codimension-2 ifurcation Furthermore, it turns out that for any a 1, and a, 1 the points ζ σ a represent codimension- 2 ig ang ifurcations. Recall that the region Q σ 2 σ contains an infinite numer of nested suregions with higher andcounts andcount douling and nested andcount-adding regions. All interior crisis surfaces ounding these regions originate from the point ζ σ a. In fact, not only the interior crisis surfaces ut also the order collision ifurcation surfaces of the orits responsile for these crises originate from the point ζ σ a. As a consequence, from each point ζ σ a at the oundary τ env originates an infinite numer of existence regions of everywhere unstale periodic orits. As an example, Figure 11 shows a few of the curves forming the ifurcation structure emerging at the point ζ LR 4 compare the the corresponding numeric results shown in Figure 1. The figure demonstrates the analytically calculated interior crises curves ounding the region Q 7 and its nested sustructures, namely the first two andcount LR 4 douling regions Q 17 LR 2 LR 6, Q 37, as well as the first nested andcount-adding LR 6 LR 4 LR 2 LR 4 regions Q 5n 17 and LR 2 LR 6 LR 4 n Q5n 17 for n 1, 2, 3. Additionally to these interior crises LR 6 LR 2 LR 4 n curves the existence oundaries or the corresponding unstale periodic orits are shown as dashed curves. For sake of clarity only the order collision ifurcations involving the orits responsile for crises ounding the douling regions are laeled. The remaining 12 dashed curves define the order collision ifurcations involving the orits responsile for crises ounding the nested andcount-adding regions. Note that only the curves ξ,l and ξ 3,r exist LR 4 LR 4 on the right side of the staility oundary θ where they confine the period adding region LR 4 n. All other order collision ifurcation curves originate from the point ζ LR 4 exist only on the left side of θ. LR 4 The question arises, how the system 1.3 ehaves at the codimension-2 ig ang ifurcations point ζ σ. Calculating analytically the points of the orit O σ one otains that at the point ζ σ the expressions for these points ecome / and hence indeterminate. As a consequence of this, there exists an asoring set with positive Leesgue measure, where each point elongs to an orit O σ. Since the point ζ σ elongs to the staility oundary θσ, each of these infinite numer of coexisting orits O σ is neutral neither stale nor unstale. Among these orits there are two special cases, namely the orits undergoing the order collision ifurcations, which confine the region P σ. These two orits represent oth kneading orits which are in this case cyclic and represent the oundaries of the the asoring set, so that the points of the other orits which coexist with them are located etween them. This is illustrated y Figure 12 where the system function and its fifth iterated are shown at the point ζ LR 4 which is given for a.2 y 4 1/a , 1 a / a see 4.6 and 4.7. In oth Figures 12 a and 12 the asoring set is marked on the horizontal axis. As one can see, this set consists of two intervals which are ounded y the points of oth kneading orits starting with f l and f r, as shown in

23 Discrete Dynamics in Nature and Society ξ 3,r LR 4 ξ 18,r LR 6 LR 4 LR 2 LR 4 ξ,l LR 6 LR 4 LR 2 LR 4 8,r ξ LR 2 LR 6 ξ,l LR 2 LR 6 θ + LR 4 ξ,l LR 4 ζ LR τ env 4 Figure 11: Analytically calculated ifurcation structure close to the ig ang ifurcation point ζ LR 4. Solid curves except the oundary curve τ env and the staility oundary θ refer to the interior crisis LR ifurcation curves, dashed curves to the order collision ifurcation curved. 4 For more details see text. The numerical Figure 1 corresponds to this image. Figure 12 a. The ehavior on the asoring set ecomes immediately clear y considering the fifth iterated function. As one can see in Figure 12, on this set the fifth iterated function coincide with the angle s isector. Hence, each point on the asoring set is a neutral fixed point for the fifth iterated and elongs to a neutral period-5 orit of the original map. All these period-5 orits correspond to the same symolic sequence LR 4 as indicated in Figure 12 a. Note finally that there is only a countale numer of regions P σ involved in the period adding scenario, and hence only a countale numer of curves ζ σ on the oundary surface τ env. In the remaining noncountale set the ehavior is aperiodic, corresponding to the limiting case infinite period of the period adding structure. 8. Bandcount Increment Region In this section we now turn to the ifurcation structure of the region Q inc,thatis,the ifurcation structure occurring at the oundary to the adjacent period increment scenario. As in the case of the andcount-adding scenario, the parameter region Q inc containing the so-called andcount-increment scenario is ounded y the surfaces γ inc and the envelope confining the period increment region, which is now a composite surface and more complicated than the envelope τ env confining the period adding region. The distance etween these surfaces shrinks with decreasing asolute value of a, so that also the complete region of the andcount-increment scenario shrinks more and more. As in the case of the andcountadding scenario, this is the only effect of the parameter a. The andcount-increment scenario is much more complex than the andcount-adding scenario and is investigated in detail in 2 22 for a different piecewise linear map, namely

24 24 Discrete Dynamics in Nature and Society 1 1 f(x) f [5] (x) 1 1 x ko rr x ko x ko l lr x 1 x ko r 1 1 x ko rr x ko x ko l lr x 1 x ko r a Figure 12: Behavior at the ig ang ifurcation point ζ LR 4: system function a and its fifth iterated. On the horizontal axis the asoring set containing an infinite numer of coexisting neutral orits O LR 4 is marked. In a three orits are shown: two kneading orits in other words, the orits O LR 4 undergoing order collision ifurcation and one orit located etween them, which does not undergo a order collision ifurcation. for the map 1.1 in the case l 1. The andcount-increment scenario in the cited works and the scenario investigated in this work share some properties, namely the triangular form of the regions, the overlap of adjacent regions, and some of the structures occurring inside the triangular regions. The andcount-increment scenario consists of triangular regions existing mainly etween two consecutive periodic regions P s and P s.infigure 13 a some triangular LR n 2 LR n regions are laeled with their andcounts 8, 1, and 12, using lack numers. There are also some additional, smaller regions, which are disconnected from the igger triangles and which have also triangular shape. They are located directly elow periodic regions and aove the larger triangles. In Figure 13 a some of these regions are laeled with their andcounts 6, 8 and 1, using white numers. To explore the scenario, we start y counting the andcounts in a sequence of large triangular regions the results calculated numerically for varied along the line marked with B in Figure 4 a are shown in Figure 14, together with an appropriate ifurcation diagram. As can e seen, the andcount of each region equals that of the preceding region increased y 2. The reason for that lies in the orits that induce the crisis surfaces ounding the regions. As Figure 14 shows, these orits are O u and collide with the attractor in and-merging crises LR n not in interior crises, as it was the case for the andcount-adding scenario. Note that a region, which is induced y such an orit O u is not located adjacent to LR n the periodic region n. Instead, it is located next to the periodic region n 2 one step to the left. This is illustrated in Figure 13, withq 8 eing located next to P LR 5 LR 3, Q 1 next to LR 7 5,andsoon. The procedure to determine these crises analytically is similar to the determination of the interior crises of the andcount-adding scenario. We need to find appropriate kneading orits that form the oundary of the chaotic attractor at the collision point. Using these kneading orits, we can determine the collisions of a point of the unstale orit with the chaotic attractor. Proceeding in this way for all orits O LR n we get the oundaries of all large

25 Discrete Dynamics in Nature and Society 25 triangular regions of the andcount-increment scenario: γ l LR n { } (a, ) a n a 2 2 1, a < 1, < 1,, a n a 2 1 γ r LR n { (a,, ) a < 1, < 1, 8.1 ( n 4 n 2) a 2 ( n ) } a 2 1 ( n 4 n 2) a 2 ( n ). a 3 These two sets of and-merging crisis surfaces give us the upper and lower oundaries of the large triangular regions. The remaining right vertical oundaries are given y the staility oundaries of the adjacent periodic regions, so that the regions Q n 3 are ordered y the LR n oundaries θ. These orders are laeled in Figure 13 for the triangular region Q 8. LR n 2 LR 5 This descries the larger triangular regions, ut the smaller regions shown in Figure 13 remain unexplained. Considering the lower oundaries of these regions, we see that they are induced y the same and-merging crises as the lower oundaries of the larger triangles. Therefore, we conclude that the aove-mentioned smaller triangular regions are not really a separate phenomenon. In fact, they are part of the igger triangles, which get overlapped y periodic regions. So the oundaries of the smaller regions consist entirely of already known ifurcation surfaces. A smaller region induced y the orit O u has a lower oundary γ r,a LR n LR n right oundary γ l which are oth and-merging crisis surfaces and an upper oundary LR n 2 ξ,l which is a order collision surface. This is also shown in Figure 13, with the smaller LR n 2 region Q 8 in the middle part of the image. Since the andcount-increment scenario is LR 5 formed completely y the triangular regions, we will refer to them as andcount-increment regions. A andcount-increment region, which is not overlapped completely y its adjacent periodic region consists simply of the igger triangular part of the region connected to the smaller part. The region Q 12 shown in Figure 13 is an example for that. LR 9 If a andcount-increment region is overlapped y its adjacent periodic region or not depends on the value of a. For increasing a, less regions will e overlapped and vice versa. It is possile for every andcount-increment region to find feasile values of a, so that it gets overlapped, and feasile values of a, so that it does not get overlapped. This concludes the description of the andcount-increment scenario. However, the andcount-increment region Q inc is far more complex than that. Explaining all existing sustructures is eyond the scope of this work. For this reason, further exploration of the andcount-increment region is left for susequent works. 9. Summary In this work the question, whether the andcount-increment and the andcount-adding scenario in system 1.3 interact with each other, is answered. It is demonstrated that this is not the case y determining that oth scenarios exist in their own regions of the parameter space, Q add and Q inc, separated y the one-and attractor region Q 1. This oservation is then followed y a detailed descriptions of the complete andcount-adding scenario and the

26 26 Discrete Dynamics in Nature and Society a 3.5 θ LR Q 6 LR 3 γ l LR 5 Q 8 LR 5 γ r LR 5 ξ,l LR 3 γl LR 7 Q 8 LR 5 Q 1 LR 7 Q 1 LR 7 Q 12 LR 9 γ r LR 7 Q 1 Q inc γ inc Figure 13: Parameter space diagram for a.42, , a shows the numerically determined andcounts, while shows the corresponding analytically determined diagram. In a, regions are laeled with their andcounts. The orders of two regions Q 8 are completely laeled, LR 5 consisting of the and-merging crises γ l and γ r, the staility order θ and the order collision ξ,l. LR 5 LR 5 LR 3 LR Dotted lines show and-merging crisis curves inside periodic regions. 3 main structure of the andcount-increment scenario. These descriptions confirm that no orit involved in the andcount-increment scenario influences the andcount-adding scenario, and vice versa. The descriptions of oth scenarios also showed, that the andcount-adding scenario in system 1.3 ehaves exactly in the same way as it does in the previously investigated case see 23. However, for the andcount-increment scenario the situation is more complicated. Although the main structure of the andcount-increment scenario in system 1.3 is very

27 Discrete Dynamics in Nature and Society 27 3 x R x LR 9 LR 11 LR 13 LR 15 BC Figure 14: Bifurcation and andcount diagrams for a.42, , 1.7. Each curve shows a single point of unstale periodic orits of the form O LR 2n 1, each colliding with the attractor in two andmerging crises. Only one point per orit is shown, for the sake of clarity, ut the other points of the orits also fit the gaps. Note that all orits shown in this image, are destroyed in order collisions at x. Their virtual parts are shown dashed. The location of this scenario in the, -plane is marked with B in Figure 4 a. similar to the previously investigated case see 2 22 there are remarkale differences especially with respect to the sustructures. The investigation of these sustructures is intricate and eyond the scope of this paper and is therefore left for future work. List of Symols L/R: Symols corresponding to points of a periodic orit smaller/larger than, respectively O σ : Periodic orit corresponding to a symolic sequence σ x σ k : Thekth point of a periodic orit O σ

28 28 Discrete Dynamics in Nature and Society Q ch : Q add : Q inc : Q 1 : Qσ: k x ko c a : Ω B : The chaotic domain The part of the chaotic domain organized y the andcount-adding scenario The part of the chaotic domain organized y the andcount-increment scenario The part of the chaotic domain containing only a one-and attractor A region in the chaotic domain with andcount k, which is ounded y crisis ifurcations caused y the collision of the orit O σ with the chaotic attractor A point of a kneading orit corresponding to the sequence of iteration f a f f c, witha,, c {r/l} The 3D influence region of the codimension-3 ig ang ifurcation point B see Section 3 M 2D B : The 2D manifold of the codimension-3 ig ang ifurcation point B see Section 3 M 1D B : The 1D manifold of the codimension-3 ig ang ifurcation point B see Section 3 Ω add B : The part of the influence region Ω B which is organized y the period adding scenario see Section 3 Ω inc B : The part of the influence region Ω B which is organized y the period increment scenario see Section 3 τ : The oundary etween M 2D B 1 and M 2D B 2 see 4.1 P: The parameter space region investigated in this work see 3.2 P σ : A parameter space region in which the periodic orit O σ exists Pσ: s A suset of P σ in which the periodic orit O σ is stale ξσ k,d : A order collision caused y the periodic orit O σ, in which the kth point of the orit collides with the order from the left/right, with d {l, r}, respectively, see 4.4 θσ: A oundary at which the orit O σ ecomes unstale via the eigenvalue 1. The oundary is defined y 4.6 and is relevant for the period increment scenario with coexisting attractors θσ: A oundary at which the orit O σ ecomes unstale via the eigenvalue 1. The oundary is defined y 4.6 and is relevant for the period adding scenario τ env : The order etween the region Ω add B 1 and the chaotic domain see 4.7 γ add : The and-merging crisis which separates the region Q 1 from the region Q add see 5.1 γ inc : The and-merging crisis which separates the region Q 1 from the region Q inc see 5.2 γσ d : A and-merging crisis caused y the orit O σ,withd {l, r} see 6.1 ησ: d An interior crisis caused y the orit O σ,withd {l, r} see 6.1 a n γ inc : The value of a at which γ inc intersects the lowest point of the staility region P s LR n see 5.4 ζ σ : A point in parameter space at which the oundaries of the andcount-adding region corresponding to O σ intersect the oundary surface τ env, as well as many other ifurcation curves see 7.1. Acknowledgment This work was supported y the DFG project Organizing centers in nonsmooth dynamical systems: ifurcations of higher codimension in theory and applications.

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