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Original Research ARTICLE

Front. Appl. Math. Stat., 29 May 2017 | https://doi.org/10.3389/fams.2017.00007

Bifurcation Structures in a Bimodal Piecewise Linear Map

  • 1Department of Differential Equations and Oscillation Theory, Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
  • 2Instituts für Systemtheorie und Regelungstechnik (IST), University of Stuttgart, Stuttgart, Germany

In this paper we present an overview of the results concerning dynamics of a piecewise linear bimodal map. The organizing principles of the bifurcation structures in both regular and chaotic domains of the parameter space of the map are discussed. In addition to the previously reported structures, a family of regions closely related to the so-called U-sequence is described. The boundaries of distinct regions belonging to these structures are obtained analytically using the skew tent map and the map replacement technique.

1. Introduction

Various bifurcation scenarios have always been in focus of many researchers from different theoretical and applied fields. It is know that bifurcation sequences often differ for piecewise smooth (PWS) systems with respect to smooth ones (see, e.g., [1]). In particular, this happens due to border collision bifurcations (BCB). This specific notion was initially introduced Nusse and Yorke in [2], but similar phenomena were also investigated earlier in Leonov [3] and Feigin [4] (see also [5, 6] and references therein). Recall that due to a BCB an attracting fixed point may be translated to an attracting cycle of any period or even to cyclic chaotic intervals. The simplest example of such behavior is the skew tent map, whose bifurcation scenarios are completely described due to intensive studies [710]. Therefore, the skew tent map is often used as a BCB normal form when studying a generic 1D continuous PWS map [5, 1113]. Another peculiar situation for PWS maps can happen when an eigenvalue (or a conjugate pair of complex eigenvalues) of a cycle crosses the unit circle. Because of certain degeneracy of the map at the bifurcation value, the standard theorems for flip, fold, or Neimark-Sacker bifurcations cannot be applied, and moreover, the result of such a crossing can be atypical. For example, an eigenvalue of a fixed point becomes −1, but after the bifurcation one observes not a 2-cycle, but a chaotic attractor [14]. Further, the bifurcation structure of the skew tent map also illustrates such a phenomenon as robustness of chaotic attractors [15]. That is, in the parameter space there exist connected domains that are characterized by chaotic attractors, that never happens in smooth maps.

In the present paper we continue studying the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map f. In our previous works basic families of regions related to regular and chaotic dynamics were investigated. In Panchuk et al. [16] we disclosed three basic bifurcation structures related to attracting cycles: the skew tent map (STM) structure, the period adding (PA) structure, and the fin structure. The STM structure concerns cycles whose points are located on the two adjacent branches only (the left and the middle or the middle and the right). The PA structure is composed by regions associated with cycles placed on the two outermost branches (the left and the right). The fin structure is related to periodicity regions having one common boundary with a PA-region.

In Panchuk et al. [17] we described certain parameter regions corresponding to chaotic attractors (chaoticity regions) and related bifurcation structures. These chaoticity regions neighbor PA and fin regions, and their boundaries are given by homoclinic bifurcations of repelling cycles. In general, if a cycle has negative multiplier, then a merging bifurcation occurs, and the number of pieces of a chaotic attractor is halved. If a cycle has positive multiplier, an expansion bifurcation occurs, and one observes an abrupt increase in size of the attractor with uneven density of points after the bifurcation. For more details see, e.g., [18].

In the current paper we summarize the results of previous works and describe new family of periodicity regions observed in the parameter space of the bimodal map. They emerge from the border that separates the parameter region D1/D2 where asymptotic dynamics is associated with two adjacent branches (STM structure) from the region D0 where all three branches can be involved. For parameters belonging to D1/D2 symbolic sequences of the cycles form the so-called U-sequence (see, e.g., [19, 20]), among which only basic cycles can be stable while all others are unstable. For parameters in D0 every cycle with symbolic sequence belonging to U-sequence has a new complementary cycle (whose symbolic sequence differs by one symbol) having a single point on the third branch. If the slope of the third branch is small enough, this new cycle is stable. We refer to this situation as the “stabilization” of a U-sequence cycle.

The paper is organized as follows. In Section 2 we introduce the main definitions and notations used throughout the paper. In Section 3 we recall how the STM structure is formed. Section 4 is devoted to PA and fin regions related to attracting cycles. In Section 5 we explain the phenomena that occur when the invariant absorbing interval involving only one border point collides with the second border point and expands over the third branch. In Section 6 the chaoticity regions surrounding the PA and the fin structures are studied. Section 7 concludes.

All results described are obtained in form of explicit analytic expressions, which can be used for various applied problems. For such a practical usage see, for instance, [21, 22].

2. Preliminaries

2.1. Basic Notations

Let us consider a family of 1D continuous piecewise linear maps f : ℝ → ℝ such that

f:xf(x)={fL(x)=aLx+μL,xdL,fM(x)=aMx+μM,dL<xdR,fR(x)=aRx+μR,x>dR,    (1)
with fL(dL)=fM(dL),fM(dR)=fR(dR).    (2)

Here the slopes aL, aM, aR, the offsets μL, μM, μR, and the border points dL,dR are real parameters, and we suppose that dL<dR. By using the conditions (2) any two of eight parameters can be eliminated, but for the sake of generality, we keep all of them.

In Panchuk et al. [16, 17] the regular and chaotic dynamics of map (1) is studied in the cases (i) |aL|<1,|aR|<1 and (ii) 0<aL<1, aR>1. In the present work we consider the case aL>0, aR0.

Let us first introduce the notations used throughout the paper.

• We denote by p a point in the parameter space of the map f.

• The values of f at the border points, :=f(dL) and r:=f(dR), are called (following the tradition of French school on Iteration Theory [23]) critical points. Successive images of ℓ and r are denoted as i:=fi() and ri:=fi(r), i ⩾ 1.

• The term partition refers to the intervals IL=(-,dL], IM=(dL,dR], and IR=(dR,). With each partition we associate the corresponding symbol, L, M, and R, respectively. That is, every point xIs is coupled with the symbol s, where s{L,M,R}.

• A cycle {xi}i=0n-1 of f is associated with the symbolic sequence σ = s0sisn−1, |σ| = n, si{L,M,R}. The cycle is denoted by Oσ.

• Consider a cycle Oσ. Clearly, any cyclic shift of σ: σi = sisn−1s0si−1 corresponds to the same cycle Oσ. Resting on the sequence σi one can construct a function composition fn=fsifsn-1fs0fsi-1:=fσi. Then every point xi satisfies fσi(xi) = xi: = xσi. For instance, the points of the 3-cycle {x0, x1, x2} with x0IL, x1IM, x2IR can be written as x0=xLMR, x1=xMRL, x2=xRLM. Clearly, in the main notation for the cycle Oσ any σi can be used, and we choose either the shortest one or the most suitable for a particular purpose.

• νσ denotes the multiplier of Oσ, given by νσ:=i=0n-1asi.

• The periodicity region associated with the stable cycle Oσ is denoted by Pσ.

• Besides fixed points and cycles the map (1) can also have attracting n-cyclic chaotic intervals denoted as Qn={Ji}i=0n-1. Hereby every Ji is a closed interval such that (i) fn is chaotic on Ji, and (ii) f(Ji) = Ji+1, i=0,n-2¯, f(Jn−1) = J0. For convenience, J0 refers to the leftmost interval.

• The symbol Cn denotes a region in the parameter space related to Qn. We also refer to Cn as a chaoticity region.

The regions Pσ and Cn are confined by boundaries related to certain bifurcations. The boundaries of periodicity regions Pσ are usually associated with one of the following bifurcations: a border collision bifurcation (BCB) occurring when a point of a cycle collides with one of the border points; a degenerate flip bifurcation (DFB) which occurs when the multiplier of a cycle equals minus one; a degenerate +1 bifurcation (DB1) associated with the multiplier being equal plus one. For details on these bifurcations we refer to Sushko and Gardini [14], Nusse and Yorke [2], Banerjee et al. [5], and Zhusubaliyev and Mosekilde [6].

The boundaries of chaoticity regions are often related, besides BCBs and DFBs, to homoclinic bifurcations of a repelling cycle located at the immediate basin boundary of the attractor. For instance, a merging bifurcation or an expansion bifurcation leads to changing the number of intervals of a chaotic attractor. Whereas a final bifurcation (known also as a boundary crisis) implies a transformation from a chaotic attractor to a chaotic repeller (see, e.g., [18]). A homoclinic bifurcation is always associated with an unstable cycle Oσ and is defined by the condition cj = xσi, where xσi is the appropriate point of Oσ and cj=fj(c), c ∈ {ℓ, r}, is the critical point of the proper rank j ⩾ 0.

For the bifurcation boundaries of periodicity and chaoticity regions we use the following notations:

ξσid denotes a BCB of the cycle Oσ, which occurs when xσi = d with d{dL,dR}.

• ησ and θσ are used for a DFB and a DB1 boundaries, respectively, associated with νσ = −1 and νσ = +1.

γσicj, ζσicj and χσicj refer to homoclinic bifurcations: a merging bifurcation, an expansion bifurcation, and a final bifurcation, respectively.

2.2. Main Characteristic Regions in the Parameter Space

At first we describe the regions characterized by simple dynamics, that is, either an attracting fixed point or divergence to infinity. Then a brief description of regions related to more complex behavior follows (see Figure 1).

FIGURE 1
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Figure 1. The parameter plane (μL,μR) of the map f for 0<aL<1 and aR>1. The regions PL, PM are related to stable fixed points. The regions S1, S2, S3 are related to diverging orbits. The regions D1, D2, and D0 are associated with asymptotic orbits located in ILIM, IMIR, and ILIMIR, respectively.

The function f can have at most three fixed points1, one in each partition Is, s{L,M,R}. A fixed point Os, s{L,R}, given by xs = μs/(1 − as) appears/disappears through collision with the border point ds. The related bifurcation boundary is

ξs={p:μs=(1-as)ds}.    (3)

Notice that the upper index ds is omitted in the notation ξs because it is clear that at ξL the collision occurs with dL while at ξR it occurs with dR. Below, for shortness, we drop upper indices in notations for BCB boundaries whenever it is clear which border point is involved in the bifurcation (namely, when the first symbol of σ is L or R the related border point is dL or dR, respectively).

From 0 ⩽ as < 1 the stability regions of OL and OR are obtained:

PL={p:μL<(1aL)dL, 0aL<1},PR={p:μR>(1aR)dR, 0aR<1}.    (4)

The stability region Ps is separated from the divergence region Ss by the boundary as = 1, where

SL={p:μL<(1-aL)dL,μR<(1-aR)dR,aL>1},SR={p:μR>(1-aR)dR,μL>(1-aL)dL,aR>1}    (5)

(see Figure 1 where PL and SR are shown).

The region

PM={p:μL>(1aL)dL,  μL+(1+aL)            dL(1+aR)dR<μR<(1aR)dR},    (6)

associated with the attracting fixed point OM given by xM=μM/(1-aM), is confined by three boundaries. Namely, the BCB boundary ξL, the BCB boundary ξR, and the DFB boundary

ηM={p:μR=μL+(1+aL)dL-(1+aR)dR},    (7)

related to the condition aM=-1.

The rest of parameter space is occupied by the region

D={p:μL>(1aL)dL,  μR<(1aR)dR,           μR<μL+(1+aL)dL(1+aR)dR},    (8)

which is marked by blue line in Figure 1. First, we notice that for certain parameter values asymptotic orbits of f are located only on the two adjacent branches. Namely,

1. If <dR then only IL and IM are relevant for asymptotic dynamics of f. Hence, the surfaces ξL, ηM, and b1 (defined from the condition =dR) confine the region

D1={p:(1aL)dL<μL<dRdLaL,             μR<μL+(1+aL)dL(1+aR)dR}    (9)

(outlined red in Figure 1). The related bifurcation structure—skew tent map (STM for short) structure—is described in Section 3.

2. Similarly, the condition r>dL holds in the region

D2={p:dLdRaR<μR<(1aR)dR,            μR<μL+(1+aL)dL(1+aR)dR},    (10)

(outlined yellow in Figure 1). This region, also having the STM structure, is confined by ξR, ηM, and the boundary b2 corresponding to the condition r=dL. For pD2 only IM and IR are relevant for asymptotic dynamics of f (see Section 3).

Finally, the conditions >dR and r<dL define the region

D0={p:μR<dL-dRaR,  μL>dR-dLaL},    (11)

delineated green in Figure 1). For parameter values belonging to D0, orbits of f are located on either IL and IR, or involve all three partitions. Two basic bifurcation structures observed in the region D0 are period adding (PA for short) structure and fin structure, described in Section 4.

Figure 2A presents an example of the typical bifurcation structure related to regular dynamics in the (μL,μR) parameter plane of the map f (the colors correspond to periods of cycles as indicated in the color bar). In Figure 2B, on the contrary, colored zones correspond to chaoticity regions related to chaotic attractors with different number of intervals (colors correspond to the number of intervals), while periodicity regions are painted gray.

FIGURE 2
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Figure 2. Bifurcation structure of the (μL,μR)-parameter plane of f. (A) Regular dynamics: colors correspond to periods of related cycles, while gray hatched areas are associated with diverging orbits. (B) Chaotic dynamics: periodicity regions are shown in gray, while chaoticity regions related to chaotic attractors with different number of intervals are shown with different colors. Parameters: aL=0.5,aR=1.3,dL=0,dR=0.3.

In the following sections we characterize different bifurcation structures observed in the parameter space of the map (1).

3. Skew Tent Map Structure

Let pD1 or pD2, then asymptotic dynamics of the map (1) is associated with two adjacent partitions only. In such a case the map f is locally topologically conjugate to the famous skew tent map, which was extensively studied already three decades ago (see e.g., [710]). Below we summarize main facts concerning this bifurcation structure, which we refer to as the skew tent map (STM) structure.

Let us consider the skew tent map family gα,β,μ

gα,β,μ:xgα,β,μ(x)={gA(x)=αx+μ,x0,gB(x)=βx+μ,x>0,    (12)

with αβ < 0. All maps from this family with μ > 0 (μ < 0) are topologically equivalent to gα,β,1 (gα,β,−1). Moreover, gα,β,−1 is topologically equivalent to gβ,α,1. Hence, to describe the STM structure it is enough to fix μ = 1. Non-trivial dynamics in this case is observed for α > 0, β < 0. Below, for shortness, we drop the indices denoting the parameters and refer to the skew tent map simply as g.

Any point x < 0 is associated with the symbol A, while a point x > 0 corresponds to the symbol B. The critical point is denoted as c := g(0) and its images as ci:=gi(c).

The stability region of the fixed point OB is confined by the DB1 boundary θB={(α,β):β=1} and the DFB boundary ηB={(α,β):β=-1}.

The only possible stable cycles of g are the basic n-cycles OBAn-1, n ⩾ 2. Each periodicity region PBAn-1 is confined by the BCB boundary and the DFB boundary

ξABAn-2={(α,β):β=-1-αn-1(1-α)αn-2},ηBAn-1={(α,β):β=-1αn-1}    (13)

(see Figure 3 where the boundaries ξABA and ηBA2 are marked). The BCB boundary ξAB is particular, because it coincides with the DFB boundary ηB. For every n ⩾ 3 the boundary ξABAn-2 corresponds to the fold BCB leading to the appearance of the basic cycle OBAn-1 and its complementary cycle OB2An-2 which is necessarily unstable. The boundary ηBAn-1, n ⩾ 3, is related to the DFB of the cycle OBAn-1, which leads to the appearance of 2n-cyclic chaotic intervals Q2n. The DFB of the fixed point OB and the 2-cycle OBA are particular as described below.

FIGURE 3
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Figure 3. Bifurcation structure of (α, β) parameter plane of the skew tent map.

The chaoticity region C2n, n ⩾ 3, associated with Q2n, is the area restricted between ξABAn-2, ηBAn-1, and γBAn-1c2n. The latter one is related to the first homoclinic bifurcation of the cycle OBAn-1. This bifurcation occurs when

c2n=xBAn-1 with c2n=gBAn-1BAn-2B(c).    (14)

The condition (14) holds for

γBAn-1c2n={(α,β):α2(n-1)β3-β+α=0}.    (15)

Crossing the bifurcation boundary γBAn-1c2n leads to the transition Q2nQn, which is also called the merging bifurcation. The existence region for n-cyclic chaotic intervals Qn is Cn confined by ξABAn-2, boundary γBAn-1c2n, and the boundary ζBAn-2Bcn corresponding to the first homoclinic bifurcation of the cycle OB2An-2. This homoclinic bifurcation occurs when

cn=gBAn-1(c)=xBAn-2B,    (16)

which gives

ζBAn-2Bcn={(α,β):αn-1β2+β-α=0}.    (17)

Crossing the boundary ζBAn-2Bcn leads to the transition QnQ, which is also called the expansion bifurcation. For more detailed description of merging and expansion bifurcations we refer to Avrutin et al. [18]. In Figure 3 the boundaries γBA2c6 and ζBABc3 are marked.

Cases n = 1 and n = 2 differ from the others. The DFB of the fixed point OB leads either to an attracting 2-cycle or to 2m-cyclic chaotic intervals, where m ⩾ 1 depends on particular parameter values. As for the 2-cycle OAB, when it loses stability due to DFB, there can appear 2m-cyclic chaotic intervals Q2m, where m ⩾ 2 depends on α, β. Every region C2m is adjacent to C2m-1 from one side and to C2m+1 from the other side. The boundary separating C2m and C2m+1 is associated with the first homoclinic bifurcation of the 2m-cycle whose symbolic sequence is the m-th harmonic of B. Recall that to obtain the k-harmonic ρk of B one has to apply concatenation operator iteratively as follows:

ρ0:=B,ρ0:=A,  ρk=ρk-1ρk-1,k1,    (18)

where ρk-1 differs from ρk−1 only by the last symbol. For example, ρ1=BA, ρ2=BABB, ρ3=BABBBABA, and so on. Notice that according to the rule (18) it is possible to define harmonics of an arbitrary cycle Oσ provided that xσ is the rightmost point of the cycle. For instance, in the condition (14) for the first homoclinic bifurcation of OBAn-1 the symbolic sequence BAn-1BAn-2B is the first harmonic of BAn-1.

The condition for the first homoclinic bifurcation of the 2m-cycle Oρm is

gρm+1(c)=xρm,    (19)

that holds for

γρmc2m+1={(α,β):α2δmβ2δm+1+(αβ)(-1)m+1-1=0},    (20)

where δm=(2m-(-1)m)/3, m ⩾ 1. As mentioned above this bifurcation leads to the transition Q2m+1Q2m, The case m = 0 (i.e., the merging bifurcation γρ0c2) is related to the first homoclinic bifurcation of the fixed point OB leading to the transition Q2Q. For m → ∞ the curves γρmc2m+1 in the (α, β)-plane crowd the point (α, β) = (1, −1) (see Figure 3).

The region related to diverging orbits (hatched in Figure 3) is confined by the boundary related to the first homoclinic bifurcation of the fixed point OA, which occurs when c1=gB(c)=xA:

χAc1={(α,β):β=α1-α}.    (21)

At this bifurcation the absorbing interval J = [c1, c] collides with a boundary of its basin of attraction (given by OA and its preimage) and for β<α1 - α the interval J is no longer absorbing causing a typical orbit to diverge. Such bifurcation is also called a final bifurcation.

For pD1 or pD2 the original map f is locally topologically conjugate to g with β=aM, α=aL or α=aR, respectively, and μ = 1 in the neighborhood of dL or dR, respectively. Thus, to get the boundaries of STM-regions in the parameter space of f it is enough to substitute L or R, respectively, instead of A and M instead of B. In Figure 4 the STM-regions in the (μL,μR) parameter plane associated with attracting cycles are colored with moderate gray, while for chaoticity regions only their boundaries are shown (blue lines). The hatched region in D2 is related to diverging orbits.

FIGURE 4
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Figure 4. Bifurcation structure of the (μL,μR)-parameter plane of map f: periodicity regions of the STM structure, as well as the PA and the fin structures are shown moderate gray, light-gray, and dark-gray, respectively. For chaoticity regions of the STM structure their boundaries are shown with blue lines. For PA regions of the first complexity level their boundaries are plotted red, while green color corresponds to the second complexity level. The parameters are as in Figure 2.

4. Bifurcation Structures Involving Both Border Points: Regular Dynamics

Let us briefly recall how other two periodicity structures are formed in the parameter space of f for pD0.

4.1. Period Adding Structure

Consider a cycle Oσ of the map f with aL>0, aR>0 such that all its points belong only to ILIR. Consequently, σ consists only of the symbols L and R. Then the related periodicity region Pσ is a part of the so-called period adding structure (PA structure). Such structures are also known as Arnold tongues or mode-locking tongues and often appear in the parameter space of a certain class of circle maps, 1D discontinuous maps, two- and higher-dimensional maps near a Neimark-Sacker bifurcation boundary, etc. (see, for example [3, 2427]). We recall that the regions belonging to PA structure are located in the parameter space according to a specific order based on Farey summation rule. Namely, consider two cycles Oσ1 of period n1 and Oσ2 of period n2 with σi containing mi symbols L. Suppose also that the fractions m1/n1 and m2/n2, called also rotation numbers, are Farey neighbors, that is, |m1n2m2n1| = 1. Then in the parameter space between the two regions related to Oσ1 and Oσ2 there is a region related to a cycle Oσ3 of periodicity n1 + n2 with σ3 having m1 + m2 symbols L, and thus, the rotation number of Oσ3 is (m1 + m2)/(n1 + n2). Notice that the cycles Oσi, i = 1, 2, 3, are not necessarily attracting.

In Figure 4 the regions composing the PA structure are shown light-gray.

If 0<aL,aR<1 then any PA-cycle is globally attracting inside the region of its existence. Indeed, in this case for pD0 there always exists an invariant absorbing interval J = [r, ℓ], and hence, there are no divergent orbits. Moreover, we consider an arbitrary PA-cycle Oσ and immediately get that its multiplier is νσ=aLmaRn-m(0,1), where m is the number of L's in σ.

However, if one of the slopes is greater than one, while the other is between zero and one, the inequality νσ < 1 is satisfied only if n<m(1-logaRaL), so that not all PA-cycles are attracting. There can also be divergent orbits since there is an unstable fixed point on the branch whose slope is greater than one. Furthermore, the interval J is no longer absorbing after the final bifurcation corresponding to the first homoclinic bifurcation of this unstable fixed point. The related bifurcation boundary confines the region in the parameter space in which a typical orbit diverges. For instance, for 0<aL<1, aR>1 (as in Figure 4) the final bifurcation χR occurs when xR=, and thus, for any (1-aR)(aLdL+μL)<μR<dL-aRdR all orbits go to infinity.

To simplify calculation of analytic expressions for PA-region boundaries we use the so-called map replacement technique [28]. At first we group symbolic sequences of PA-cycles into infinite number of families complying with their complexity level (as in [3]). For example, two families related to basic cycles form the first complexity level:

Σ1,1={LRn1}n1=1,Σ2,1={RLn1}n1=1.    (22)

Notice that both families contain the common sequence LRRL.

To construct the rest of symbolic sequence families we introduce the following symbolic replacements:

κmL:={LLRmRRLRm,κmR:={LLRLmRRLm.    (23)

Application of a symbolic replacement κmL to a symbolic sequence σ means that each symbol L in σ is replaced by LRm and each symbol R by RLRm. Similarly, application of κmR means replacing all L's by LRLm and all R's by RLm.

Symbolic sequences of higher complexity levels are obtained by iterative application of κmL and κmR. More precisely, applying κn2L and κn2R to Σ1,1 we get the families

Σ1,2={LRn2(RLRn2)n1}n1,n2=1,Σ2,2={LRLn2(RLn2)n1}n1,n2=1.    (24)

And then applying κn2L and κn2R to Σ2,1 we obtain

Σ3,2={RLRn2(LRn2)n1}n1,n2=1,Σ4,2={RLn2(LRLn2)n1}n1,n2=1.    (25)

The four families Σi,2, i=1,4¯, compose the second complexity level. Notice that the families Σ1,2 and Σ3,2 contain the common sequence LRRLRRLRLR with n1 = n2 = 1, while Σ2,2 and Σ4,2 share the sequence LRLRLRLLRL with n1 = n2 = 1.

In short the applied procedure can be written as

Σ1,2=κn2L(Σ1,1), Σ2,2=κn2R(Σ1,1),Σ3,2=κn2L(Σ2,1), Σ4,2=κn2R(Σ2,1).    (26)

Further, applying the replacements (23) with m = n3 to four families of the second complexity level, we obtain 23 families Σj,3, j=1,23¯, of the third complexity level, and so on. In this way all symbolic sequences of PA-cycles are obtained.

Now let us identify the PA-regions in the parameter space of the map f. Each such region has two boundaries corresponding to BCBs of the related cycle. For the basic cycles OLRn1, n1 ⩾ 1, these boundaries are obtained directly from the relevant BCB conditions giving

         ξLRn1={pD0:μL=Φ1,1(aL,aR,μR,dL,n1)},ξRLRn1-1={pD0:μL=Ψ1,1(aL,aR,μR,dR,n1)},    (27)

where

Φ1,1(aL,aR,μR,d,n1)=-ψ(aR,n1)μR+φ(aR,aL,n1)d,    (28)
Ψ1,1(aL,aR,μR,d,n1)=-(aL+ψ(aR,n1-1))μR                                                + aRφ(aR,aL,n1)d    (29)

with

φ(a,b,n)=1-anban,ψ(a,n)=1-an(1-a)an.    (30)

For ORLn1, n ⩾ 1, their boundaries are obtained from (27) by interchanging L and R. So that, the basic PA-regions are defined as

PLRn1={pD0:Ψ1,1(aL,aR,μR,dR,n1)<μL                          <Φ1,1(aL,aR,μR,dL,n1)},    (31)
PRLn1={pD0:Ψ1,1(aR,aL,μL,dL,n1)>μR                          >Φ1,1(aR,aL,μL,dR,n1)}.    (32)

Further, using the map replacement technique we obtain periodicity regions associated with cycles whose symbolic sequences belong to the families Σ1,2 and Σ2,2, given in (24). The trick is as follows. Any symbolic sequence τ ∈ Σ1,2 can be written as τ=κn2L(σ) with certain n2 ⩾ 1 and σ ∈ Σ1,1 with particular n1 ⩾ 1. Then to get the BCBs ξτdL and ξτdR we take the expressions (27) for ξLRn1 and ξRLRn1-1 and substitute the parameters aL, μL, aR, μR by aLRn2, μLRn2, aRLRn2-1, μRLRn2-1:

ξκn2L(LRn1)={pD0:μLRn2=Φ1,1(aLRn2,aRLRn21,                                                           μRLRn21,dL,n1)},ξκn2L(RLRn11)={pD0:μLRn2=Ψ1,1(aLRn2,aRLRn21,                                                               μRLRn21,dR,n1)}.    (33)

Recall that aLRn2, μLRn2 and aRLRn2-1, μRLRn2-1 denote, respectively, the coefficients of the following composite (auxiliary) functions

fLRn2=fLfRfRn2,  fRLRn2-1=fRfLfRfRn2-1.

Schematically, we summarize the applied procedure as follows:

ξLRn1LLRn2RRLRn2-1ξκn2L(LRn1),ξRLRn1-1LLRn2RRLRn2-1ξκn2L(RLRn1-1).    (34)

Clearly, the coefficients aLRn2, μLRn2, aRLRn2-1, μRLRn2-1 depend on aL, μL, aR, μR, and hence, the equalities in (33) can be solved with respect to μL, implying the PA-region related to Oκn2L(LRn1) to be

Pκn2L(LRn1)={pD0:Ψ1,2(aL,aR,μR,dR,n1,n2)                               <μL<Φ1,2(aL,aR,μR,dL,n1,n2)}    (35)

where Ψ1,2, Φ1,2 are certain functions (see [16] where expressions for Ψ1,2, Φ1,2 are given in explicit form).

Similarly, any symbolic sequence τ ∈ Σ2,2 can be written as τ=κn2R(σ) with n2 ⩾ 1 and σ ∈ Σ1,1. To get the expressions for the boundaries of Pτ we perform analogous trick replacing in (27) the parameters aL, μL, aR, μR with the coefficients aLRLn2-1, μLRLn2-1, aRLn2, μRLn2 of the related (auxiliary) composite functions. The resulting new equalities can be resolved with respect to μL, and the related PA-region is

Pκn2R(LRn1)={pD0:Ψ2,2(aL,aR,μR,dR,n1,n2)<μL                                                     <Φ2,2(aL,aR,μR,dL,n1,n2)}.    (36)

with certain Ψ2,2, Φ2,2.

As for the cycles whose symbolic sequences belong to the families Σ3,2 and Σ4,2 (25), boundaries of the related periodicity regions are obtained by interchanging the indices L and R and changing the inequality signs to the opposite ones in (36) and (35), respectively.

Similarly, having expressions for the boundaries of periodicity regions of the (k − 1)-th complexity level and using this replacement technique, we can compute the related expressions for periodicity regions of the k-th complexity level (for more detail see [16]).

In Figure 4 for the regions of the first complexity level their boundaries are colored red, while for those of the second complexity level they are green.

4.2. Fin Structure

In Figure 2 or in Figure 4 one can notice that some PA-regions have other (smaller) periodicity regions attached to them, which resemble to certain extent fish fins. Every such fin-like region is associated with a stable kn-cycle, where n is the period of the related PA-cycle and k ⩾ 1. To explain briefly how these regions appear, let us consider the n-th iterate of f.

For the map fn every point of the PA-cycle, say Oσ, of the period n is a fixed point. Suppose that the parameter point pPσ moves outside the region crossing transversally, for instance, the boundary ξσdL (see Figure 5). Near the BCB the map fn is locally topologically conjugate in the neighborhood of dL to a skew tent map gα,β,μ (12) with α=aLlaRn-l, β=aLl-1aMaRn-l (l is the number of symbols L in σ) and μ < 0 before, μ = 0 at, and μ > 0 after the bifurcation. In other words, the skew tent map is used as the BCB normal form, and the result of such a bifurcation is completely defined by the values of α and β (see, e.g., [5, 10, 11]). In particular, for certain combination of aL, aM and aR, a stable k-cycle for fn appears after the BCB. In terms of the original map f, after the BCB of a stable n-cycle Oσ there occurs a stable kn-cycle, say, Oτ. Crossing the boundary ξσdR is treated likewise with the only difference that β=aLlaMaRn-l-1 (for details see [16]).

FIGURE 5
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Figure 5. (A) The left and the right 1 · n-, 2 · n-, and 3 · n-fins of the PA-region PLRn-1 shown schematically in the (μL,μR)-parameter plane. (B–D) Plots of fn at the corresponding parameter points marked in (A).

We refer to Oτ/Pτ as the k · n-fin cycle/region, and to Oσ/Pσ as the trunk cycle/region. For clarity sake, the region Pτ is called the left/right k · n-fin if it is attached to the boundary of Pσ that is related to collision of Oσ with the left/right border point dL/dR. Furthermore, the complexity level of the trunk region defines the complexity level of its fins.

Let us consider PLRn-1, n ⩾ 2, of the first complexity level. Its left k · n-fin regions, k ⩾ 1, correspond to cycles whose symbolic sequences are

σknL:=(LRn-1)k-1MRn-1,    (37)

while symbolic sequences of the cycles associated with right k · n-fin regions are

σknR:=(RLRn-2)k-1MLRn-2.    (38)

As for the trunk region PRLn-1 its k · n-fins correspond to cycles whose symbolic sequences are obtained from (37) and (38) by interchanging symbols L and R. Whether a trunk region has fins on both sides or none of them is present depends on the parameters.

Every fin region Pσkns, s{L,R}, k ⩾ 2, obviously has a common BCB boundary with the related trunk region. At most Pσkns has three more bifurcation boundaries that correspond to two other BCB's and the DFB of the associated cycle. Analytic expressions for the BCB boundaries are obtained by using the skew tent map as border collision normal form. The DFB boundary is given by condition aLlaMaRkn-l-1=-1, where l is the number of symbols L in σkns. For left fins PσknL the expressions are (see [16])

ξLRn-1={pD0:μL=Φ1,1(aL,aR,μR,dL,n-1)},    (39)
ησknL={pD0:aMaRk(n-1)aLk-1=-1},    (40)
ξσknLdL={pD0:aMaRn-1=aLaRn-1((aLaRn-1)-k+1-1)aLaRn-1-1}    (41)
ξσknLdR={pD0:aRn1aM(aRn1aL)k11aRn1aL1((aRn2aL+aRn21aR1)μR+aRn2μL)+                  aR(n1)kaMaL(k1)dR+(aRn2aM+aRn21aR1)μR+aRn2μM=dR},    (42)

where Φ1,1 is introduced in (28). Notice that the symbolic sequence of the 1 · n-fin cycle is σnLMRn-1, and hence, the related periodicity region is confined by only three boundaries: ξLRn-1 (39), ηMRn-1 (40), and ξRMRn-2 (42).

Similarly, the bifurcation boundaries confining the right k · n-fin PσknR, contiguous to PLRn-1, are

ξRLRn-2={pD0:μL=Ψ1,1(aL,aR,μR,dR,n-1)}    (43)
ησknR={pD0:aMaRk(n-1)-1aLk=-1},    (44)
ξσknRdR={pD0:aLaMaRn-2=aLaRn-1((aLaRn-1)-k+1-1)aLaRn-1-1},    (45)
ξσknRdL={pD0:aRn2aMaL(aRn1aL)k11aRn1aL1(aRn11aR1μR+aRn1μL)+             aR(n1)k1aMaLkdL+aMaRn21aR1μR+aMaRn2μL+μM=dL},    (46)

where Ψ1,1 is given in (29). Again the symbolic sequence of the 1 · n-fin cycle is σnRMLRn-2, and consequently, the periodicity region PσnR is confined by ξLRn-1 (39), ηMRn-1 (40), and ξRMRn-2 (42).

In Figure 5 the region PLRn-1 and its fins are shown schematically.

The boundaries of fins contiguous to trunk regions PRLn-1 can be obtained by interchanging the indices L and R in (39–45). The expressions for the boundaries of fin regions attached to trunk regions of higher complexity levels can be obtained using the map replacement technique mentioned in Section 4.1 (see [16] for details).

5. Expansion of the Absorbing Interval from Two Adjacent Branches Over the Third One

The present section describes the phenomena that occur when the parameter point p crosses the boundary b1/b2 entering D0. As mentioned above, for pD1/D2 the invariant absorbing interval J (if it exists) involves only one border point dL/dR and is located in two partitions only. At the boundary b1/b2 the interval J collides with the second border point and expands into the third partition. This implies modification of bifurcation boundaries confining periodicity and chaoticity regions.

5.1. Prolongation of the Skew Tent Map Structure

Recall that for pD1 asymptotic orbits of the map f are located in ILIM only. If additionally either 0aL1, or aL>1 and fM()>xL, then there exists invariant absorbing interval J=[fM(dL),dL]ILIM. Consequently, all bifurcation boundaries in D1 engage only the left border point dL and the critical point ℓ (including its images). Similarly, for pD2 the bifurcation conditions are related to the right border point dR and the critical point r only. Conversely, for pD0 the absorbing interval J (if existent) extends over all three partitions. Hence, a bifurcation may be associated with any border or critical point.

Let us consider first chaoticity regions C2n, Cn existing after the DFB ηMLn-1 of the attracting cycle OMLn-1, n ⩾ 3 (generic case). We describe how their boundaries change when the parameter point p enters D0 crossing the border b1 (see, for instance, the regions C6 and C3 shown Figure 6A). As described in Section 3, for pD1 the DFB of OMLn-1 leads to appearance of 2n-cyclic chaotic intervals Q2n, and thus, the corresponding region C2n is confined by ηMLn-1 and γMLn-12n. However, the part of C2n located in D0 is confined by the other two boundaries. The first one is the fold BCB boundary ξLRLn-1MLn-2, associated with the collision of the fin cycle ORLn-1MLn-1 with dL. The second boundary is an extension of γMLn-12n to D0, which we refer to as its D0-prolongation. This boundary is associated as well with a homoclinic bifurcation of OMLn-1, but its condition differs from the one for γMLn-12n, as we show below.

FIGURE 6
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Figure 6. Bifurcation structures associated with D0-prolongations of the boundaries of the chaoticity regions belonging to the STM structure in D1. Parameters: (A) aL=0.5, aR=1.3, dL=0, dR=0.3. (B) aL=0.9, aR=1.1, dL=0, dR=1.

Figure 7A is a schematic representation of 2n-cyclic chaotic intervals Q2n={Ji}i=02n-1 for pC2nD1. It can be shown that the intervals Ji, i=0,2n-2¯, are bounded by ℓi+1 and ℓi+2n+1 and the interval J2n − 1 by ℓ2n and ℓ. Hence, for pD1 the homoclinic bifurcation corresponding to the merging Q2nQn can be defined by the following condition:

f2n()fMLn-1MLn-2M()=xMLn-1    (47)

(cf. (14)). For pb1, the condition =dR holds, and hence ℓ1 = r, ℓi+1 = ri, i ⩾ 1. Therefore, the intervals Ji, i=0,2n-2¯, constituting Q2n are bounded by ℓi+1 = ri and ℓi+2n+1 = ri+2n, while the interval J2n − 1 is given by [r2n-1,dR]. Hence, the condition (47) becomes

f2n()fMLn-1MLn-2M(dR)fLn-1MLn-2M(r)              f2n-1(r)=xMLn-1.    (48)
FIGURE 7
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Figure 7. Schematic representation of function f and 2n-cyclic chaotic intervals Q2n before the merging bifurcation of Q2n for (A) pD1; (B) pD0.

As soon as the point p enters the region D0 by crossing b1, the absorbing interval J spreads over all three partitions IL, IM, and IR. For Q2n it means that J2n-1=[r2n-1,]IMIR with dRJ2n-1 (see Figure 7B). By definition f(J2n − 1) = J0. If p is close to b1, the value of ℓ is greater than the value of dR, but it is still close to dR, and thus,

f(dR)<f()<f(r2n-1)r<1<r2n.

It entails that J0=[f(dR),f(r2n-1)]=[r,r2n], while ℓ1J0. As a result, only the right boundary of J2n − 1 is given by ℓ, and its left boundary, as well as the boundaries of the intervals Ji, i=0,2n-2¯, are given by the images of r (of the same rank as for pb1). Thus, for pD0 the condition for the homoclinic bifurcation leading to the transition Q2nQn coincides with (48). In such a way the D0-prolongation of γMLn-12n is the merging bifurcation boundary

γMLn-1r2n-1={pD0:  fLn-1MLn-2M(r)=xMLn-1}.    (49)

Similarly, we deduce that the chaoticity region Cn consists of two parts, one of which is located in D1 and the other one in D0 (see Figure 6A for a sample of C3). The part of Cn located in D1 is confined by the merging bifurcation boundary γMLn-12n and the expansion bifurcation boundary ζMLn-2Mn. The part of Cn located in D0 is bounded by ξLRLn-1MLn-2, γMLn-1r2n-1 (49) and the D0-prolongation of ζMLn-2Mn, which is associated with the expansion bifurcation as well and leads to the transition QnQ. To obtain the condition related to the mentioned D0-prolongation we recall that for pb1 there holds =dR. There follows that

fn()fMLn-1(dR)fLn-1(r)fn-1(r)=xMLn-2M.

Hence, for pD0 the expansion QnQ occurs at

ζMLn-2Mrn-1={pD0:fLn-1(r)=xMLn-2M}.    (50)

Now we turn to chaotic attractors existing after the DFB of OM or OML. Although these cases are particular they are treated likewise. The DFB of a stable 2-cycle or a fixed point implies appearance of intervals Q2m where m ⩾ 1 depends on parameters. Then there exists a cascade of merging bifurcations ending by the parameter region for that the whole absorbing interval represents a one-piece chaotic attractor. Each boundary γρm2m+1 for pD1 has its D0-prolongation

γρmr2m+1-1={pD0:  fωm+1(r)=xρm}    (51)

related to the transition Q2m+1Q2m. Here ρm is the m-th harmonic of M (cf. (18) with A=L and B=M), and ωm+1=s2s2m+1 such that Ms2s2m+1=ρm+1 is (m + 1)-th harmonic of M.

In Figure 6B we observe the D0-prolongations γMLM2(ML)2r15 and γMLM2r7, while γMLr3 and γMr1 are visible in Figure 6A.

To summarize, we state

Proposition 1 ([17]). Consider a merging bifurcation boundary γσ2nD1 related to the transition Q2nQn, where |σ| = n, n ⩾ 1, xσ is the rightmost point of the cycle Oσ. Then the D0-prolongation γσr2n-1 of γσ2n is given by the homoclinic bifurcation condition fω(r) = xσ where ω is obtained from the first harmonic of σ by dropping the first symbol M.

An expansion bifurcation boundary ζσnD1, n ⩾ 3, and its D0-prolongation ζσrn-1 are related in a similar way.

Further, let us consider an arbitrary chaoticity region C2m, m ⩾ 2, which extends from D1 to D0. As it was shown above, when pD1 the boundaries of intervals Ji, i=0,2m-1¯, constituting the chaotic attractor Q2m are defined by the images of ℓ only (see Figure 7A). When pD0 but still close to b1, the rightmost interval J2m-1 is given by [r2m-1,]. The intervals Ji, i=0,2m-2¯, are confined by the images of r, in particular, J0=[r,r2m]. Moreover, 1<r2m, and therefore ℓ1J0 (see Figure 7B). Then, if p moves away from b1, the values ℓ and ℓ1 increase, and eventually 1=r2m. Consequently, when p moves further away from b1 the leftmost interval of Q2m becomes J0 = [r, ℓ1].

Hence, additionally to γρmr2m+1-1 (which is related to r) another homoclinic bifurcation of the 2m-cycle Oρm related to ℓ may occur. This homoclinic bifurcation defines an additional boundary of C2m and occurs when the leftmost point of Oρm coincides with ℓ1, that is, at the boundary

γρm1={pD0:  fR()=xρm},    (52)

where ρm=s2s2mM with Ms2s2m=ρm (thus, xρm is the leftmost point of Oρm). In Figure 6B the boundaries γLM31 and γLM3(LM)21 are shown.

We remark that the homoclinic bifurcation of type (52) can happen only if the corresponding cycle has at least one point in IL. This is not true for the fixed point OM, and thus, the region C2 does not have this additional bifurcation boundary.

In such a way we can formulate

Proposition 2 ([17]). Suppose that for pD0 (in a certain neighborhood of b1) there exists a D0-prolongation γρmr2m+1-1 of γρm2m+1D1, m ⩾ 1, associated with the transition Q2m+1Q2m. Then there exists a neighborhood U(γρmr2m+1-1) such that for pU(γρmr2m+1-1) there exists a boundary γρm1, where ρm is such that xρm is the leftmost point of the cycle Oρm. The boundary γρm1 is associated with the same transition Q2m+1Q2m and related to a homoclinic bifurcation of the cycle Oρm colliding with an image of the critical point ℓ.

As for chaoticity regions located in D2, to derive expressions for their D0-prolongations it is enough to swap L's and R's, as well as change r to ℓ in (49–51). Similarly, for pD0 in a certain neighborhood of b2, a chaoticity region C2m, m ⩾ 1, has an additional boundary related to a homoclinic bifurcation, whose expression can be derived from (52) by swapping L's and R's and replacing ℓ1 by r1.

5.2. Periodicity Regions for aR0: Stabilization of U-Sequence Cycles

In the previous section we described how the boundaries of chaoticity regions forming the STM structure change when the parameter point p enters D0 by crossing b1 or b2. Every merging or expansion bifurcation boundary existent in D1/D2 has its D0-prolongation. So does every region related to cyclic chaotic intervals. However, the dynamics of the map f for the parameter values pD0 in the neighborhood of b1/b2 is not limited to D0-prolongations of the STM regions. For instance, in Figure 8 between the right 2 · 2-fin PRLML of PRL and the left 1 · 3-fin PMRL of PRL2 one observes several tongues related to n-cycles with small values of n, e.g., with n = 5, 6, 7, 10. At the first site the origin of these tongues is unclear. In this section we explain the reason why such regions appear in the neighborhood of b1. All stated below can be generalized in the obvious way for b2.

FIGURE 8
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Figure 8. Stabilization of U-sequence cycles. Panels (B,C) are the enlargements of the areas marked in (A) by blue and magenta lines, respectively. The parameters are aL=0.57, aR=0.2, dL=0, dR=1.

To discover the origin of the tongues that are not D0-prolongations of the STM structure, recall that for the skew tent map (12) even if the dynamics converges to a chaotic attractor, there exist also many periodic solutions that are unstable. These cycles appear due to BCBs or DFBs and according to a certain universal order with decreasing β. Namely, the symbolic sequences of these cycles form the so-called U-sequence where “U” stands for “universal,” since this order is common for a wide class of unimodal maps [19, 20]. Notice that usually U-sequence is described in terms of kneading sequences associated with superstable cycles (in the symbolic representation of which the symbol C related to the extremum of the function is omitted). The skew tent map obviously cannot have superstable cycles, and by using the term “U-sequence” we mean the order of appearance of the cycles in terms of their complete symbolic representation.

Another peculiarity of the skew tent map is that some blocks of the U-sequence appear simultaneously. For instance, crossing the DFB boundary ηBA (or ηB when α > 1) with decreasing β leads to immediate appearance of cycles with symbolic sequences being harmonics of B, defined in (18). Similar statement is true for a basic cycle OBAn-1, n ⩾ 3, that is, all its harmonics appear at the same parameter values (at the boundary ηBAn-1).

In general, the U-sequence is constructed according to a certain iterative procedure and includes also the basic cycles. Detailed description of this construction procedure, as well as explanation of its peculiarities related to the skew tent map, is beyond the scopes of the current paper. Here we only explain how an unstable cycle whose symbolic sequence belongs to U-sequence may be “stabilized” when p enters D0 crossing b1 or b2. For shortness we refer below to such cycles as U-sequence cycles.

Consider, for example, the region related to the 5-cycle observed in Figure 8A (see also the enlargement in Figure 8B). For pD1 the line ξLMLM2 denotes the fold BCB, at which two unstable 5-cycles are born: the cycle OLMLM2 and its complementary cycle OM2LM2. The condition related to this bifurcation is

f5(dL)=fLMLM2(dL)fM2LM2(dL)=dL.    (53)

In fact, (53) defines two different BCB conditions which are equivalent due to the continuity of f, that is, fL(dL)=fM(dL). This explains why the two cycles (which coincide at the bifurcation moment) appear simultaneously and why their symbolic sequences after the bifurcation differ by a single symbol.

For pξLMLM2b1 the condition (53) can be rewritten taking into account that f(dL)=dR:

f5(dL)=f4(dR)=fMLM2(dR)fRLM2(dR)=dL,    (54)

since fM(dR)=fR(dR). Applying f one more time to (54), and using again the continuity property of f, we get

f5(dR)=fMLM2L(dR)fRLM2L(dR)fMLM2M(dR)                  fRLM2M(dR)=dR(=f(dL)).    (55)

Similarly to (53), the equalities (55) define four BCB conditions that, nevertheless, are equivalent for pξLMLM2b1. Notice that in (55) the BCB conditions depend on dR, in contrast to (53) where they depend on dL. This is similar to how conditions of homoclinic bifurcations change when the parameter point p moves from D1 to D0 (as described in Section 5).

Now, for pD0 not all the BCB conditions (55) remain equivalent. Moreover, the two cycles OLMLM2 and OM2LM2 do not appear simultaneously for pD0. Instead, each of them creates a complementary pair with one of the two new cycles having a single point in IR (and, thus, existing only for pD0). Namely, the cycle OLMLM2 is now complementary to OLRLM2, while OM2LM2 is complementary to OMRLM2. Clearly, the BCB conditions for the two cycles making a pair are equivalent, that is,

 fMLM2L(dR)fRLM2L(dR)=dR,fMLM2M(dR)fRLM2M(dR)=dR.    (56)

However, the related bifurcation is not necessarily fold BCB.

Let us describe in more detail the BCBs related to four mentioned 5-cycles. With decreasing μR (for the other parameter values as in Figure 8) they appear in the following order. First, the two unstable cycles OM2LM2 and OMRLM2 are born due to the fold BCB ξRLM3 (dashed curve). Then there occurs the BCB ξLRLM2 (green curve), at which the point xMRLM2 of OMRLM2 moves from the middle partition IM to the left one IL. Hence, the stable cycle OLRLM2 appears. Finally, the point of OLRLM2 located in the right partition IR moves to the middle partition IM due to the other BCB ξRLM2L creating the unstable cycle OM2LM2. The region PLRLM2 is then confined by ξRLM2L and ξLRLM2. For μR being below ξRLM2L the two unstable cycles OLMLM2 and OM2LM2 continue to exist.

We also remark that the region PLRLM2 has fins similarly to PA-regions. The mechanism of their creation is the same. Indeed, at the BCB boundary of PLRLM2 related to collision of the cycle with dL/dR, the map f5 is locally (in the neighborhood of dL/dR) topologically equivalent to the skew tent map gα,β,1 with α=aL2aM2aR and β=aLaM3aR/β=aLaM3. Crossing transversally this BCB boundary may lead to appearance of a stable 5k-cycle, k ⩾ 1 (cf. Section 4.2). Existence of fins and their number depends on the parameter values.

The scenario due to that the region PLRLM2 occurs corresponds to what we call the “stabilization” of a U-sequence cycle. That is, for pD1 the U-sequence cycle Oσ is unstable (except for the basic cycles OMLn-1, n ⩾ 2, which can be stable). When p crosses b1 entering D0 a new cycle Oσ (among the others) is born. The symbolic sequence σ′ differs from σ by a single symbol. Namely, the symbol M corresponding to the rightmost point of Oσ is replaced by R in σ′. If the slope aR is small enough, the cycle Oσ is stable.

The region related to a 6-cycle observed as well in Figure 8A has the same origin, though it looks somewhat unlike PLRLM2. Let us consider it in more detail and explain why it has different form. For pD1 due to the BCB ξLMLM3 the two unstable 6-cycles OLMLM3 and OM2LM3 appear. Similarly to the case considered above (of the 5-cycle), for pD0 there are two more 6-cycles: OLRLM3 (stable) and OMRLM3 (unstable). However, in contrast to the previous case, the sequence of bifurcations occurring with decreasing μR is different. First, the fold BCB ξRLM3L occurs giving rise to the stable cycle OLRLM3 and the unstable one OLMLM3. Next, the two unstable cycles OMRLM3 and OM2LM3 appear due to the fold BCB ξRLM4. Finally, OLRLM3 (stable) and OMRLM3 (unstable) disappear at ξLRLM3. Notice that the region PLRLM3 (differently from PLRLM2) has one more boundary ηLRLM3 that corresponds to the DFB of the related cycle.

The other regions emerging from b1 and not belonging to PA or fin structures are of the same nature as the two regions described above. Notice that, for a U-sequence cycle OLMσ, the higher is its period, the more of its points are located in IM. Thus, to observe for pD0 the region PLRσ corresponding to the “stabilized” version of OLMσ, the value aR should be small enough. In the limit case aR=0, for every U-sequence cycle there exists the related periodicity region in D0 (see Figure 9), since the multiplier of OLRσ equals zero. Clearly, all these regions have only BCB boundaries and extend to infinite values of μL, μR.

FIGURE 9
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Figure 9. Periodicity regions related to “stabilized” versions of U-sequence cycles in the limit case aR=0.

6. Chaoticity Regions Surrounding the Period Adding and the Fin Structures

In Section 4.1 we have recalled how to obtain boundaries of PA-regions. There are chaoticity regions adjacent to them, and now we discuss how their boundaries can be derived using the skew tent map. As an example, Figure 10 shows the structure of chaoticity regions close to the BCB boundary ξLR of the PA-region PLR. One can observe here two hatched domains. The first one, denoted DstL,LR, is contiguous to the PA-region PLR along the boundary ξLR. For pDstL,LR the map f2 is locally topologically conjugate to the skew tent map. Hence, the bifurcation structure can be described by applying directly the results known for the skew tent map.

FIGURE 10
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Figure 10. Bifurcation structure related to chaoticity regions associated with the PA-region PLR and its left 2 · 2-fin PLRMR. The regions DstL,LR and DstR,2,LR are shown dashed.

In general, in the neighborhood of the boundary ξσdL/ξσdR of every PA-region Pσ associated with an n-cycle Oσ, there exist domains denoted DstL,σ/DstR,σ, respectively, such that for related parameter values the iterate fn is locally (in the neighborhood of dL/dR) topologically conjugate to the skew tent map.

The second hatched domain in Figure 10, denoted DstR,2,LR, is located in the neighborhood of the boundary ξRLRM of the left 2 · 2-fin region PLRMR. In general, let us consider an arbitrary PA-region Pσ, related to the n-cycle Oσ, and its left/right k · n-fin Pτ. In the neighborhood of the BCB boundary ξτdR/ξτdL, respectively, there exists a domain DstR,k,σ/DstL,k,σ such that the restriction of fkn to its absorbing interval is topologically conjugate to the skew tent map.

6.1. Homoclinic Bifurcation Boundaries in DstL,σ

We consider first the trunk region PLR and sub-domain DstL,LR contiguous to it along the boundary ξLR. Clearly, both points of the 2-cycle OLR={xLR,xRL} are fixed points for the second iterate f2, and at pξLR the point xLR collides with dL. The resulting dynamics caused by this collision depends on the slopes of f2 at both sides of dL at the bifurcation moment. In the neighborhood of the border point the map f2 is defined as (see Figure 11)

f2(x){fLR,fL1(dR)<xdL,fMR,dL<xfM1(dR),

where fL-1(dR)=(dR-μL)/aL, fM-1(dR)=(dR-μM)/aM. If the condition

fLR(dL)<fM-1(dR)    (57)

is satisfied, then f2 has a local absorbing interval J=[fMR2(dL),fMR(dL)], on which f2 is topologically conjugate to the skew tent map (12) with α=aLaR and β=aMaR. The parameter region for that the conjugacy takes place is given as

DstL,LR={pD0:  xLR>dL,  fMR(l)<dR}=                {pD0:  μL>μRaR+1aRaLaRdL,     aRaMl+aRμM+μR<dR}.    (58)
FIGURE 11
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Figure 11. The local absorbing interval J of f2 including dL for pDstL,LR.

It is located between two boundaries, namely, ξLR and

bstL,LR={pD0:  fMR()=dR},    (59)

which follows from the equality fLR(dL)=fM-1(dR).

For pDstL,LR expressions for all bifurcation boundaries are derived directly from the related expressions known for the skew tent map (cf. Section 3).

Similarly, for an arbitrary PA-region Pσ related to an n-cycle Oσ, n = |σ|, let σ=Lσ1Rσ2, where the symbolic sequences σ1 and σ2 are such that the point xLσ1Rσ2/xRσ2Lσ1 collides with dL/dR at the BCB boundary ξLσ1Rσ2/ξRσ2Lσ1, respectively. For pPσ consider the map fn, for which every point of Oσ is a fixed point. Close to the boundary ξLσ1Rσ2, two branches of fn joining at the border point dL are

fn(x){fLσ1Rσ2,d<xdL,fMσ1Rσ2,dL<xd¯.

Here d and d̄ are preimages of dR, such that fLσ1(d)=dR and fMσ1(d̄)=dR. As before it can be shown that as long as

fn(dL)fLσ1Rσ2(dL)<d̄    (60)

there exists a local absorbing interval

J=[fMσ1Rσ22(dL),fMσ1Rσ2(dL)]    (61)

such that the restriction fn|J is topologically conjugate to the skew tent map g with the slopes α=aLσ1Rσ2 and β=aMσ1Rσ2. Here aLσ1Rσ2 and aMσ1Rσ2 are the slopes of the composite functions fLσ1Rσ2 and fMσ1Rσ2, respectively. Such a conjugacy persists until the parameter point p crosses the border bstL,σ given by the condition fLσ1Rσ2(dL)=d̄, which is equivalent to fMσ1(fLσ1Rσ2(dL))=fMσ1(d̄)dR, i.e.,

bstL,σ={pD0:fσ1Rσ2Mσ1()=dR}.    (62)

Hence, we can state the following

Proposition 3 ([17]). For pDstL,σ, where the region

DstL,σ={pD0:  xLσ1Rσ2>dL,fσ1Rσ2Mσ1()<dR}    (63)

is confined by ξLσ1Rσ2 and bstL,σ given in (62), the expressions for boundaries of periodicity regions (fins) and chaoticity regions can be obtained from the related expressions known for the skew tent map (62) substituting α by aLσ1Rσ2 and β by aMσ1Rσ2.

For instance, we consider the left k · n-fin Pτ, τ=Mσ1Rσ2(Lσ1Rσ2)k-1, k ⩾ 3, of the PA-region PLσ1Rσ2. This fin region represents the area located between three BCB boundaries: ξLσ1Rσ2, ξτidL (the point xτi of the cycle Oτ collides with dL), ξτjdR (the point xτj of the cycle Oτ collides with dR), and the DFB boundary ητ. We remark that the boundaries ξτidL and ητ emerge from ξLσ1Rσ2, as well as the boundaries between chaoticity regions adjacent to PLσ1Rσ2.

Analytic expressions for all these boundaries are derived by using the skew tent map. Boundaries of the fin region, ξτidL and ητ, are given in (41) and (40), respectively. Let us consider the merging and the expansion bifurcation boundaries confining chaoticity regions Q2kn and Qkn. They can be obtained by applying map replacement technique to expressions for γBAk-1c2k (15) and ζBAk-2Bck (17).

Namely, in (15) we substitute all symbols A with symbolic sequence Lσ1Rσ2, all symbols B with symbolic sequence Mσ1Rσ2, critical point c with fσ1Rσ2(), slope α with aLσ1Rσ2, and β with aMσ1Rσ2. Recall that aLσ1Rσ2 and aMσ1Rσ2 are the slopes of the composite functions fLσ1Rσ2 and fMσ1Rσ2, respectively. Expressing aLσ1Rσ2 and aMσ1Rσ2 in terms of the original parameters aL, aM, aR we obtain the merging bifurcation boundary γMσ1Rσ2(Lσ1Rσ2)k-12kn+n-1 related to transition Q2knQkn. Likewise, by applying the same replacement to (17) one gets the expansion bifurcation boundary ζMσ1Rσ2(Lσ1Rσ2)k-2Mσ1Rσ2kn+n-1. In short this procedure can be represented by the following scheme:

γBAk-1c2kALσ1Rσ2BMσ1Rσ2cfσ1Rσ2()γMσ1Rσ2(Lσ1Rσ2)k-12kn+n-1    (64a)
ζBAk-2BckALσ1Rσ2 BMσ1Rσ2cfσ1Rσ2()ζMσ1Rσ2(Lσ1Rσ2)k-2Mσ1Rσ2kn+n-1.    (64b)

For sake of shortness schemes of such kind are used below to describe similar replacement procedures.

To sum up we can state

Proposition 4 ([17]). Let the parameter point p move along the BCB boundary ξLσ1Rσ2 starting from nk-fin region PMσ1Rσ2(Lσ1Rσ2)k-1, k ⩾ 3. Then p first crosses the DFB boundary ηMσ1Rσ2(Lσ1Rσ2)k-1 (40) leading to 2kn-cyclic chaotic intervals Q2kn, then the merging bifurcation boundary γMσ1Rσ2(Lσ1Rσ2)k-12kn+n-1 (64a) implying the transition Q2knQkn, and finally, the expansion bifurcation boundary ζMσ1Rσ2(Lσ1Rσ2)k-2Mσ1Rσ2kn+n-1 (64b) leading to QknQn.

The case k = 2 is particular. Since a 2n-fin cycle corresponds to the 2-cycle of the skew tent map g, its DFB leads to the appearance of 2m·n-cyclic chaotic intervals, where m ⩾ 2 depends on aL, aR. The cascade of the subsequent merging bifurcations is obtained from (20) applying the same replacement as in (64). In Figure 12 an example of this cascade observed inside the domain DstL,LR is shown, located close to the left 2 · 2-fin region PLRMR of the trunk region PLR. One can see three merging bifurcation boundaries, namely, γMRLR(MR)217, γMRLR9, and γMR5 leading to the transitions Q2·8Q2·4Q2·2Q2, respectively. To get the expression for γMRLR(MR)217 the following replacement

γBAB2c8ALRBMRcfR()γMRLR(MR)217

can be used. In this way we obtain

γMRLR(MR)217={pD0:aLR2aMR6+aLRaMR-1=0}                        ={pD0:aL2aM6aR8+aMaL-1=0}.    (65)

The expressions for the other two boundaries

γMRLR9={pD0:aL2aM2aR4+aLaM-1=0},    (66a)
γMR5={pD0:aM2aR2+aMaL-1=0}    (66b)

can be obtained by the same replacement.

FIGURE 12
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Figure 12. (A) Bifurcation structure of the region DstL,LR. Bifurcation diagram (B) along the arrow shown green in (A) and its enlargement (C).

We have described above bifurcations of cyclic chaotic intervals and the related boundaries in the neighborhood of ξσdL of a PA-region Pσ. To obtain similar bifurcation boundaries in the neighborhood of ξσdR one can swap symbols L and R as well as interchange the critical points ℓ and r in all analytic expressions.

6.2. Homoclinic Bifurcation Boundaries in DstR,K,σ

In this section we describe another parameter domain DstR,k,σ, mentioned above, which is located in the neighborhood of an arbitrary left fin region of a PA-region Pσ. To describe bifurcation structure of DstR,k,σ we again use conjugacy with skew tent map and map replacement technique.

As the first example let us consider the left 2 · 2-fin region PLRMR of the trunk region PLR (see Figure Figures 12A). As explained in Section 4.2 the fin region PLRMR has the BCB boundary ξRLRM related to the collision of the cycle OLRMR with the border dR.

In Figure 13 we show the plot of f4 on the interval including the border point dR for parameter values located in the neighborhood of ξRLRM. The two linear pieces of f4 adjoining at dR are defined as

f4(x){fMLRM,fM1(dL)xdR,fRLRM,dR<xfR1(dL),

where fM-1(dL)=(dL-μM)/aM, fR-1(dL)=(dL-μR)/aR. If the conditions

fRLRM(dR)<fR-1(dL),    (67a)
fRLRM2(dR)>xMLRM    (67b)

are held, then there exists a local absorbing interval

J=[fRLRM2(dR),fRLRM(dR)]    (68)

for f4. The restriction f4|J is topologically conjugate to the skew tent map g with the slopes α=aLaM2aR and β=aLaMaR2.

FIGURE 13
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Figure 13. The local absorbing interval J of f4 including dR for pDstR,2,LR.

Taking into account that fR(fRLRM(dR))fLRMR(r), fRLRM2(dR)=fLRM(fRLRM(r)), and xMLRM=fLRM(xLRM2), the conditions (67) can be written as

fLRMR(r)<dL,    (69a)
fLRMR(r)>xLRM2.    (69b)

The conjugacy between f4|J and g holds in the parameter region DstR,2,LR confined by the BCB boundary ξRLRM, the boundary

bstR,2,LR={pD0:  fLRMR(r)=dL},    (70)

and the boundary

ζLRM2r4={pD0:  fLRMR(r)=xLRM2}.    (71)

Note that for the map f the boundary (71) corresponds to an expansion bifurcation leading from Q4 to Q. Thus, the region DstR,2,LR is defined as

DstR,2,LR={pD0:xRLRM>dR,fLRMR(r)<dL,                 fLRMR(r)>xLRM2}.    (72)

Figure 14A presents the region DstR,2,LR, where one can observe chaoticity regions Q2m, m=2,6¯. The bifurcation boundaries separating these regions are related to transitions Q2mQ2m-1. To get expressions for these boundaries we use map replacement technique. Namely, we take the corresponding formulae (20) known for the skew tent map and apply the following replacement:

γρjc2j+1AMLRM    BRLRMcfLRM(r)γτjr2j+3+3.    (73)

Here the symbolic sequence τj, j ⩾ 0, is derived from the harmonic ρj (18) by substituting MLRM instead of A and RLRM instead of B.

FIGURE 14
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Figure 14. (A) Bifurcation structure of the region DstR,2,LR. Bifurcation diagram (B) along the arrow shown red in (A) and its enlargement (C).

Summing up we can describe DstR,2,LR in the following way. The region DstR,2,LR\PLRMR consists of t + 1 chaoticity regions C4·2j, j=0,t¯, where t ⩾ 1 depends on aL, aR. If the parameter point p crosses the boundary γτjr2j+3+3 between the chaoticity regions C2j+1 and C2j, given by

γτjr2j+3+3={pDstL:  aL2(δj+δj+1)aM2(2δj+δj+1)aR2(δj+2δj+1)                                    +(aMaR)(1)j+11=0},j=0,t1¯,

then the merging bifurcation C2j+1C2j takes place. Here δj=(2j-(-1)j)/3. In Figure 14A the merging bifurcation boundaries γτ3r67, γτ2r35, γτ1r19, and γτ0r11 are shown. See also 1D bifurcation diagrams in Figures 14B,C.

In a similar way we characterize the region DstR,k,σ, related to the boundary ξτdR of the left k · n-fin Pτ, τ=(Lσ1Rσ2)k-1Mσ1Rσ2, of the trunk region PLσ1Rσ2, k ⩾ 2, n ⩾ 2. In order to determine the branches of fkn which are adjacent to one another at the border point, we rewrite the symbolic sequence associated with the fin cycle as follows:

τR_σ2Lσ1(Rσ2Lσ1)k-3Rσ2L_σ1Rσ2Mσ1.    (74)

Here the underlined symbols R_ and L_ are associated with the points xR_τ1, xL_τ2 such that xR_τ1 collides with dR at ξτdR, while xL_τ2 collides with dL at ξτdL. We introduce the abbreviation

ω1:=σ2Lσ1(Rσ2Lσ1)k-3Rσ2,ω2:=σ1Rσ2Mσ1    (75)

and rewrite the symbolic sequence (74) as Rω1Lω2. Then the points xR_τ1 and xL_τ2 become xRω1Lω2 and xLω2Rω1, respectively. Then in the neighborhood of dR the iterate fkn is given by

fkn(x){fMω1Lω2,d˜<xdR,fRω1Lω2,dR<x<d^

Here d~ and d^ are the preimages of dL such that

fMω1(d~)=dL,fRω1(d^)=dL.

The map fkn has in the neighborhood of dR a local absorbing interval

J=[fRω1Lω22(dR),fRω1Lω2(dR)][d~,d^],    (76)

whenever the conditions

fRω1Lω2(dR)<d^,fRω1Lω22(dR)>xMω1Lω2,    (77)

are held. The restriction fkn|J is topologically conjugate to the skew tent map g with

α=aMω1Lω2,β=aRω1Lω2.    (78)

Taking into account that fRω1(fRω1Lω2(dR))fω1Lω2Rω1(r), fRω1Lω22(dR)=fω1Lω2(fω1Lω2R(r)), and xMω1Lω2=fω1Lω2(xω1Lω2M), the conditions (77) can be rewritten as

fω1Lω2Rω1(r)<dL,fω1Lω2R(r)>xω1Lω2M.

Therefore, the region DstR,k,σ in which the asymptotic dynamics of f can be studied by using the skew tent map g with the slopes given in (78), is confined by the BCB boundary ξRω1Lω2, the border

bstR,k,σ={pD0:  fω1Lω2Rω1(r)=dL},    (79)

and the expansion bifurcation boundary

ζω1Lω2Mrnk={fω1Lω2R(r)=xω1Lω2M}.    (80)

Accordingly, the region DstR,k,σ is given by

DstR,k,σ={xRω1Lω2>dR,fω1Lω2Rω1(r)<dL,fω1Lω2R(r)                     >xω1Lω2M}.    (81)

To sum up we state

Proposition 5 ([17]). The region DstR,k,σ\PRω1Lω2 consists of t+1 chaoticity regions C4·2j, j=0,t¯, where t ⩾ 1 depends on aL, aR. Each pair of contiguous chaoticity regions C2j+1 and C2j is separated by the merging bifurcation boundary γτjr2j+1nk+nk-1 which is derived from (20) applying the following replacement

γρjc2j+1AMω1Lω2    BRω1Lω2cfω1Lω2(r)γτjr2j+1nk+nk1,    (82)

where τj is obtained from the harmonic ρj (18) by replacing A with Mω1Lω2 and B with Rω1Lω2.

As before, to describe bifurcation structure of the parameter domain DstL,k,σ, located in the neighborhood of an arbitrary right fin region of a PA-region Pσ, one can interchange L and R as well as the critical points ℓ and r in all analytic expressions.

6.3. Homoclinic Bifurcation Boundaries between DstL,σ and DstR,K,σ

As one can see in Figure 10, between the regions DstL,LR and DstR,2,LR there are more bifurcation boundaries related to some homoclinic bifurcations. In contrast to the previous two sections, where chaotic intervals were associated with only one border point, dL or dR, here we consider the case when chaotic intervals include both border points. This makes impossible direct usage of the results known for the skew tent map.

In Figures 15A,B we show the area between the regions DstL,LR and DstR,2,LR. As one can observe there are several bifurcation boundaries issuing from the curves bstL,LR (59) and bstR,2,LR (70). Right below we briefly describe how these bifurcation boundaries can be derived.

FIGURE 15
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Figure 15. (A) Chaoticity regions located between DstL,LR and DstR,2,LR; In (B) the region outlined red in (A) is shown enlarged. (C) Local absorbing interval of f2 including dL for pDstL,LR immediately after crossing bstL,LR. (D) Local absorbing interval of f4 including dR for pDstR,2,LR immediately after crossing bstR,2,LR.

First, recall that bstL,LR is related to the condition fLR(dL)=fM-1(dR). Until fLR(dL)<fM-1(dR) (that is, inside DstL,LR), the local absorbing interval J of f2 includes only a single kink point dL. Hence, f2|J is defined by only two branches fLR and fMR. When there holds the opposite inequality fLR(dL)>fM-1(dR), the absorbing interval J spreads over another kink point fM-1(dR) (to the right of dL) and becomes

J=[fMR(fM-1(dR)),fLR(dL)]=[r,fR()]fM-1(dR)

(see Figure 15C). Then f2|J consists of three branches: fLR, fMR, and fMM. This transformation is similar to the one described in Section 5.1 for the original map f, where the parameter point p crosses the border b1. Hence, the bifurcation boundaries emerging from bstL,LR are of the same sort as D0-prolongations. Applying the replacement

γρmr2m+11LLR    MMR γτmr2(2m+11)    (83)

to (51), one obtains the related analytic expressions for the mentioned boundaries.

Then we apply the replacement

γρm1LLR    MMR    RMMfR()γτm3    (84)

to the additional boundary γρm1 (52). Here the sequences τm and τm are obtained from the sequences ρm and ρm, respectively, by replacing L with LR and M with MR. Recall, that ρm consists of the symbols L and M and is the m-th harmonic of M [see (18) with A=L and B=M], while ρm is obtained from ρm by shifting the first symbol M to the end of the symbolic sequence. Note that in (83), r must be replaced by the value fMR(fM-1(dR))fR(dR) which coincides with r, and that is why the replacement for r is not necessary.

To illustrate the described procedure let us consider the boundary γMRr2 shown in Figure 15A which corresponds to the transition Q4Q2. For f2 this transition is associated with the merging of 2-cyclic chaotic intervals into a single chaotic interval. Thus, we can apply the replacement (83) to the expression for the D0-prolongation γMr1 [i.e., (51) with m = 0] obtaining

γMRr2={pD0:fLR(r)=xMR}.    (85)

Likewise, we derive the expression for the boundary γMRLRr6, which is associated with the transition Q8Q4, by using the expression for the D0-prolongation γMLr3 [i.e., (51) with m = 1]:

γMRLRr6={pD0:fLR(MR)2(r)=xMRLR}.    (86)

The other boundary γLRMR3 associated with the same transition is derived from the boundary γLM1 (52), m = 1, corresponding to another homoclinic bifurcation of OML. Applying (84) to γLM1 we obtain

γLRMR3={pD0:fRMM()=xLRMR}.    (87)

In Figure 15B one can also see the boundaries γMRLR(MR)2r14 and γLR(MR)33, which correspond to the transition Q16Q8. Setting m = 2 and applying the replacement (83) to (51) and the replacement (84) to (52) we get

γMLM2r7LLR    MMR γMRLR(MR)2r14,γLM31LLR    MMR    RMMfR()γLR(MR)33,

which implies

γMRLR(MR)2r14={pD0:fLR(MR)2(MRLR)2(r)                   ​=xMRLR(MR)2},    (88)
γLR(MR)33={pD0:fRMM()=xLR(MR)3}.    (89)

Analogously we consider the case where the parameter point p leaves the region DstR,2,LR crossing bstR,2,LR. When pDstR,2,LR the condition fRLRM(dR)>fR-1(dL) is satisfied and the local absorbing interval J of f4 in the neighborhood of dR includes only two linear branches. Otherwise, the interval J spreads over the third branch fRMRM and becomes

J=[fRLRM(fR1(dL)),fRLRM(dR)]     [fRM(),fLRM(r)]fR1(dL).

See Figure 15D. Thus, for f4 crossing the border bstR,2,LR is similar to crossing b1 for f. Consequently, the associated boundaries of chaoticity regions for pDstR,2,LR can be derived as well from (51) and (52).

For example, the transition Q16Q8 occurs at the boundaries γMLr3 and γMLRMRLRMr7. We use the expressions for γMLr3 (51) and γLM1 (52) with m = 2 to compute these mentioned boundaries. It is enough to apply the following replacements:

γMLr3LMLRM    MRLRMrfRM()γRLRM2LRM14,γLM1LMLRM    MRLRM    RRMRMfLRM(r)                            γMLRMRLRMr7.    (90)

In this way we obtain

γRLRM2LRM14={pD0:fRMMLRM(RLRM)2()                      =xRLRM2LRM},    (91a)
γMLRMRLRMr7={pD0:fL(RM)3(r)                     =xMLRMRLRM}.    (91b)

Regarding the merging bifurcation boundaries γMR3 and γM1 marked in Figure 15A, it can be shown that γMR3 is the D0-prolongation of γMRr4D2, and γM1 is the D0-prolongation of γMr2D2.

So far all bifurcation boundaries of the regions C2, C4, and C8 related to the transitions Q8Q4Q2Q are fully described. However, in Figure 15B one can see that the region C16 has further boundaries different from those described above. We can suggest that the regions C2m, m ⩾ 4, have much more complicated form and may have other boundaries, whose description falls beyond the scopes of the current paper.

The above example can be generalized. That is, for an arbitrary PA-region Pσ we consider the are located between the regions DstL,σ and DstR,k,σ, k ⩾ 2. The bifurcation boundaries inside this area are derived in a similar way by using the relevant replacements (for details see [17]).

7. Conclusions

In this work we carry on studying asymptotic dynamics of a 1D continuous bimodal PWL map f with the two outermost slopes being positive. Previously we have described certain bifurcation structures related to regular as well as to chaotic dynamics of f. Three basic bifurcation structures have been identified: the skew tent map (STM) structure, the period adding (PA) structure, and the fin structure. In case of STM structure the points of a typical orbit are located on two neighbor branches only, so that f is reduced to the skew tent map. In case of PA structure the orbits belong to the outermost branches only, so that the results known for the discontinuous map defined on two partitions with positive slopes can be used. The fin structure is constituted by periodicity regions that have a common boundary with PA-regions. The analytic expressions for bifurcation boundaries defined by BCBs, DFBs, as well as homoclinic bifurcations (such as merging and expansion) have been obtained.

In the current paper we have summarized the earlier results and described another family of tongues associated with regular dynamics observed in the parameter space of f. They emerge from the boundary b1/b2 that separates the parameter region D1/D2 where f is locally topologically conjugated to the skew tent map (STM structure) from the region D0 where all three branches can be involved in asymptotic dynamics. These tongues are closely related to the cycles whose symbolic sequences form the U-sequence. For parameters belonging to D1/D2 these cycles are located in the two adjacent partitions and are unstable (except for the basic cycles having one point in the middle partition, which can be stable). However, for parameters in D0 every such cycle has a new complementary cycle with a single point on the third branch. If the slope of the third branch is small enough, this new cycle is stable and one observes the related region in the parameter space of f. In the limit case when the third branch is horizontal (the slope is zero), for every U-sequence cycle from D1/D2 there exists the related periodicity region in D0. We refer to this situation as the “stabilization” of a U-sequence cycle.

Author Contributions

The paper is the result of joint research, the contribution of every author is comparable to the others.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

The work of AP and IS has been performed under the auspices of COST Action IS1104 “The EU in the new complex geography of economic systems: models, tools and policy evaluation.” The work of VA was partially supported by the German Research Foundation within the scope of the project “Organizing centers in discontinuous dynamical systems: bifurcations of higher codimension in theory and applications.”

Footnotes

1. ^Except the case where the left/the right branch lies on the main diagonal. In this particular situation every point xIL/xIR is a fixed point and has a stable set consisted of all its preimages, if they exist.

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Keywords: bimodal piecewise linear map, border collision bifurcation, border collision normal form, degenerate bifurcation, homoclinic bifurcation

Citation: Panchuk A, Sushko I and Avrutin V (2017) Bifurcation Structures in a Bimodal Piecewise Linear Map. Front. Appl. Math. Stat. 3:7. doi: 10.3389/fams.2017.00007

Received: 24 June 2016; Accepted: 12 April 2017;
Published: 29 May 2017.

Edited by:

Fabio Lamantia, University of Calabria, Italy

Reviewed by:

Elisabetta Michetti, University of Macerata, Italy
Mauro Sodini, University of Pisa, Italy

Copyright © 2017 Panchuk, Sushko and Avrutin. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Anastasiia Panchuk, anastasiia.panchuk@gmail.com