BIFURCATION ANALYSIS OF A CIRCUIT-RELATED GENERALIZATION OF THE SHIPMAP

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1 International Journal of Bifurcation and Chaos, Vol. 16, No. 8 (26) c World Scientific Publishing Company BIFURCATION ANALYSIS OF A CIRCUIT-RELATED GENERALIZATION OF THE SHIPMAP FEDERICO BIZZARRI and MARCO STORACE Biophysical and Electronic Engineering Department, University of Genoa, Via Opera Pia 11a, Genova, Italy federico.bizzarri Marco.Storace@unige.it LAURA GARDINI Istituto di Scienze Economiche, Universita di Urbino, Urbino, Italy gardini@econ.uniurb.it Received May 9, 25; Revised June 2, 25 In this paper a bifurcation analysis of a piecewise-affine discrete-time dynamical system is carried out. Such a system derives from a well-known map which has good features from its circuit implementation point of view and good statistical properties in the generation of pseudo-random sequences. The considered map is a generalization of it and the bifurcation parameters take into account some common circuit implementation nonidealities or mismatches. It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of three parameters. In particular, it will be shown that chaotic intervals may coeist, may be cyclical, and may undergo several global bifurcations. All the global bifurcation curves and surfaces will be obtained either analytically or numerically by studying the critical points of the map (i.e. etremum points and discontinuity points) and their iterates. In view of a robust design of the map, this bifurcation analysis should come before a statistical analysis, to find a set of parameters ensuring both robust chaotic dynamics and robust statistical properties. Keywords: Bifurcation analysis; piecewise-affine map; chaos; global bifurcations. 1. Introduction In the last decades, many works concerning engineering applications of chaos have been focused on either the analysis (direct problem) or the design (inverse problem) of nonlinear dynamical systems to be used in telecommunication frameworks. In many cases, the possible dynamics of most of them have been etensively analyzed a priori from a bifurcation point of view, in order to find out the regions of the parameter space ensuring chaotic dynamics. After this preliminary study, both model analysis and system synthesis have been carried out on the basis of the statistical characteristics of the considered systems, used as chaos generators. Such a kind of working procedure is the most direct way to design a system on the basis of a given mathematical model. However, dealing with the physical implementation of engineering systems (typically, electronic circuits), it is often necessary to redesign/analyze the implemented systems to take into account mismatches with respect to the original models. In such cases, each original mathematical model should be redefined (introducing new parameters related with the real implementation) and new analyses should be carried out. First of all, a bifurcation analysis should be performed, to find the regions of the new parameter space allowing the presence of a unique invariant chaotic interval. 2435

2 2436 F. Bizzarri et al. Then, a statistical analysis within such parameter regions should be carried out, to check that the desired (e.g. uniform) statistical distribution of the trajectory points over such an interval is ensured. In this paper we shall propose a bifurcation analysis of a Piecewise-Affine Map (PWAM), defined starting from a known map and introducing new bifurcation parameters to take into account some common circuit implementation nonidealities (not related to a specific electronic implementation). We shall focus our attention on the piecewiseaffine map known as shipmap [Callegari et al., 2], since it has good features from its implementation point of view and good statistical properties in the generation of pseudo-random sequences [Delgado, 1999]. This kind of analysis is a first step towards a robust design of the map and should be the prelude to a statistical analysis, to find regions in the parameter space ensuring both robust chaotic dynamics (i.e. structural stability of the map) and robust statistical properties. The bifurcation analysis will be carried out by varying the parameters within ranges that are chosen just on the basis of the mathematical formulation of the map. So doing, we are guaranteed that the reasonable ranges from a circuit point of viewarecoveredbytheanalysis:therealmismatch ranges, whose eact estimation depends on the specific circuit implementation and on the adopted technology, will be certainly contained within the considered domain of analysis. Our model is ultimately described by a one-dimensional PWAM, with several control-parameters. The bifurcation analysis will be carried out with respect to three parameters whose changing preserves the basic map structure (in particular, the number of branches). It will be shown that several different dynamic situations may arise, which will be completely characterized as a function of the three chosen parameters. The approach followed to detect global bifurcations is based on the analysis (either analytical or numerical) of the critical points of the map (i.e. etremum points and discontinuity points) and their iterates. The obtained results are quite general, since by choosing other bifurcation parameters, or by etending the chosen parameter domain, the bifurcation scenario will not significantly change. In Sec. 2, we shall define the considered PWAM and the three bifurcation parameters. Section 3, will be devoted to sketch a bifurcation scenario through one-dimensional bifurcation diagrams. In Sec. 4, we shall derive analytical epressions for the most significant bifurcation sets. In Sec. 5, some further numerical results will be provided. Finally, some concluding remarks will be discussed in Sec Map Definition Preliminarily, we recall some basic definitions: AsubsetA of the state space is said invariant for a map T if the set A is mapped onto itself (T (A) =A), i.e. if all points of A are images of points of A. AsubsetA of the state space is said trapping set for a map T if the set A is mapped into itself (T (A) A), i.e. if the images of all points of A belong to A. A closed invariant set A is called asymptotically stable or attracting or absorbing if a neighborhood U of A eists such that T (U) U and T n () A as n + for any U. The basin of attraction of an attracting set A is the set of all points that generate trajectories asymptotically converging to A: B(A) ={ : T n () A as n + }. Ofcourse,A B(A). The shipmap :R Ris strictly related to the tent map and can be derived from it through a cutting and flipping technique. The main advantage in doing so is that, for a given parameter set ensuring chaotic dynamics within an invariant interval I, the basin of attraction B(I) ofi is enlarged, while preserving the main statistical properties of the tent map [Callegari et al., 2]. A possible sequence of operations (circuit blocks) providing the shipmap is shown in Fig. 1. Such operations can be performed through highlevel circuit building blocks like the ones proposed in [Callegari et al., 2]. A further step in the synthesis procedure is the low-level hardware implementation of such blocks, that usually yields to nonidealities. In particular, in the shipmap case, the main possible implementation nonidealities concern: slope of the linear branches; breakpoint positions; quote of the sail maimum; positions of the stepwise transitions in the block B6 The parameters introduced by this block scheme can be related also to other architectures. Other parameters could be introduced a posteriori on the basis of circuit simulations to model,

3 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap 2437 Fig. 1. High-level block scheme of the circuit architecture, with input/output characteristic of each block. The function w() is 1 for (p 4 δw) <(p 5 + δw) and elsewhere. e.g. mismatches due to circuit dynamics, delay times, parasitic effects [Callegari et al., 2]. Each block can introduce mismatches with respect to the ideal operation, thus leading to a nonideal version of the shipmap, where, besides the mismatch δw introduced by the block B6, the fully controllable parameters p k are replaced by mismatched parameters P k = p k + δp k. For instance, the block B3 can introduce a slope P 2 slightly different from p 2, whereas the block B4 can introduce a vertical offset P 3 different from p 3.Inthis paper, we shall focus on the nonidealities due to the blocks B3 and B4. In particular, the error δp 3 introduced by the block B4 influences the parameters P 4 (= P 1 P 3 /P 2 )andp 5 (= P 1 +P 3 /P 2 ), which are assumed to be always symmetric with respect to P 1 (i.e. δp 5 = δp 4 ). Also the value M(P 1 )is influenced by δp 3, but such a value depends on the block B6 too, then we shall take into account the effect of a parameter δm = δp 3 + δp 6 on M(P 1 ). The parameter δw can change the structure of the map, by introducing two further branches, as showninfig.2.withthisinmind,weshallconsider the effects of the other three main parameter mismatches (δp 2, δp 4,andδm), to carry out a bifurcation analysis of a four-branches PWAM (a) (b) Fig. 2. Eamples of si-branches maps, obtained for (a) δw =.5 and (b) δw =.5. The other parameters (δp 2, δp 4,and δm) aresettozero.

4 2438 F. Bizzarri et al. The aforesaid effects can be summarized by the following map definition: P 2 ( P 4 ) <P 4 = P 3 + P 6 P 2 P 1 P 4 <P 5 P 2 ( P 5 ) P 5 (1) Henceforth, we shall fi P 1 = p 1 = (i.e. δp 1 = ), p 2 = 2, p 4 =.25, p 5 =.75, and p 3 + p 6 = 1 and we shall carry out our bifurcation analysis with respect to δp 2, δp 4,andδm. For δp 2 = δp 4 = δm =, the shipmap has been proven to have good statistical properties and to ehibit chaotic dynamics [Delgado, 1999]. Summarizing, the map we shall consider is defined as follows: (2 + δp 2 )[ (.25 + δp 4 )] <(.25 + δp 4 ) = 1+δm (2 + δp 2 ) (.25 + δp 4 ) <(.75 δp 4 ) (2 + δp 2 )[ (.75 δp 4 )] (.75 δp 4 ) (2) 3. Bifurcation Scenarios First of all, we define a domain on the parameter space (δp 2,δp 4,δm). The maimum allowed value for δp 4 should be.25, otherwise the map definition (1) loses its validity. For symmetry reasons, we shall assume that the lower limit for δp 4 is.25. We shall also assume that δp 2 and δm range in [ 2, 2] and [ 1, 1], respectively: so doing, we shall admit a maimum error of 1% in both directions for all the parameters p 2, p 4,andp 3 + p 6. The admitted region of the parameter space will be henceforth denoted by Ω p Equilibrium points In the chosen range for the parameter δp 4,theleft branch of the map always intersects the main diagonal in correspondence with the equilibrium point E 1 = ((2 + δp 2 )(1 + 4δp 4 ))/4(3 + δp 2 ). The following inequalities locate the eistence of other equilibria, say E 2, E 3,andE 4, originated by the intersections of the other map branches with the diagonal: the fied point E 2 = ( 2δ m + δp 2 )/2(1+ δp 2 ) eists in the subregion of Ω p where the following conditions on the parameters are satisfied: ( δp 4 δp ) ( 2 1 4δm δm 1 ) 4(δp 2 +1) 2 The boundary of such a subregion of Ω p will be denoted by S 2 and is defined as follows: { S 2 = (δp 2,δp 4,δm) Ω p : δp 4 = δp } 2 1 4δm 4(1 + δp 2 ) (3) the fied point E 3 =(4+2δ m + δp 2 )/2(3 + δp 2 ) eists in the subregion of Ω p where the following conditions on the parameters are satisfied: ( δp 4 δp δm 4(δp 2 +3) ) ( δm 1 2 The boundary of such a subregion of Ω p will be denoted by S 3 and is defined as follows: { S 3 = (δp 2,δp 4,δm) Ω p : δp 4 = 1 4δm + δp } 2 4(3 + δp 2 ) ) (4) the fied point E 4 = (2+δp 2 )(3 4δp 4 )/ 4(1 + δp 2 ) eists in the subregion of Ω p where δp 2 > 1 The boundary δp 2 = 1 of such a subregion of Ω p will be denoted by S 4. Since the absolute values of the map branches slopes are equal to (2 + δp 2 ), all the equilibrium points are unstable for δp 2 > Bifurcation curves Even if we find analytical results in terms of three parameters, for the sake of graphical clarity, we will show two-dimensional sections of the significant sets of points, obtained for δm = 1/3 (seefig.3).the results obtained for other values of δm within the range [ 1, 1] are subsets of these. Figure 3 shows some significant bifurcation curves in the considered plane (δp 2,δp 4 ). For instance, the bifurcation curves obtained as intersections between the plane δm = 1/3 and the surfaces S 2, S 3,andS 4 are represented by dashed black lines in Fig. 3.

5 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap δ p S 3 S 1 S 4 S Z H 1 S 9 L 2 S 2 S 5 H 3 S δ p Fig. 3. Significant bifurcation curves on the plane (δp 2,δp 4 ) for δm = 1/3. Since the bifurcations are determined by the map critical points (relative maimum and minimum points, discontinuity points) and their iterates, for the sake of compactness we shall denote such points as follows (for k 1): C = P 1 ; C k = M k (C) D = P 4 ; D k = M k (D) F = P 5 ; F k = M k (F ) To be more precise, we should consider both the right and the left limits of M at each discontinuity point. However, according to (2), the left limit of M in P 4 (that is ) is equal to the right limit of M in P 5, whereas the right limit of M in P 4 (that is 1 + δm (2 + δp 2 )(.25 δp 4 )) is equal to the left limit of M in P 5. Then, the definitions of M in P 4 and P 5 give us the two limit values at both the discontinuities. In the net subsections, we will illustrate the main bifurcations by resorting to one-dimensional bifurcation diagrams, obtained by varying δp 2 for three fied values of δp 4, and to the maps obtained for some significant values of δp 2. Once the bifurcation scenario will be clearly sketched, we shall find analytical epressions for the surfaces in the threedimensional parameter space corresponding to the bifurcation curves in Fig. 3. The one-dimensional bifurcation diagrams will be obtained by varying δp 2 within ranges (depending on the chosen value of δp 4 ) such that H 4 H 2 S 8 L 1 L 3 S 6 the asymptotic dynamics of the map is bounded and chaotic everywhere. We are interested in values of δp 2 larger than 1, as for 2 <δp 2 < 1 theonly invariant sets of the map are stable fied point(s). For the chosen value of δm, suchfiedpointsare either E 1 (above S 3,inFig.3)orbothE 1 and E 3 (below S 3, in Fig. 3). For different values of δm and for 2 <δp 2 < 1, also E 2 could be a stable fied point, whereas E 4 never eists. For δp 2 = 1 infinite nonattracting period-two cycles eist around such equilibrium points. For δp 2 > 1, the absolute value of the slopes of the map branches is always larger than 1. This implies that (i) no stable cycle, of any period, can eist, and (ii) the bounded asymptotic dynamics of the system can be only chaotic. In other words, S 4 marks the appearance of chaotic dynamics. As we shall see, chaotic intervals may coeist, may be cyclical, and may undergo several global bifurcations. The boundaries of such chaotic intervals will be defined in terms of the critical points C k, D k, and F k. The gray region in Fig. 3 contains parameter sets either unfeasible (according to the definition of Ω p ) or corresponding to unbounded dynamics (see Sec. 4.1 for details) One-dimensional bifurcation diagram for δp 4 =.16 Figure 4 shows the one-dimensional bifurcation diagram obtained by varying δp 2 within the range [ 1,.4], for δp 4 =.16. For low values of δp 2 (i.e. close to 1), the map dynamics are confined within the absorbing interval J 2 =[C 2,C 1 ]. Within such an interval, as long as it is absorbing, the map is equivalent (topologically conjugated) to a tent map, then its behaviors are deeply analyzed in [Maistrenko et al., 1993]. As a consequence, for δp 2 close to 1 there is a very large number of cyclic chaotic intervals, whose number decreases and whose amplitude increases by increasing δp 2.ForthewholemapM, themechanismgoverning the change in the number of cyclic chaotic intervals will be briefly described in Sec. 5 and is based on the bifurcations that orbits homoclinic to either a fied or a periodic point (i.e. a fied point in higher-order iterates of M) undergo [Gardini, 1994; Bizzarri et al., 25]. If we increase δp 2, the number of chaotic intervals decreases. To locate such intervals, we have to focus on the critical points of the map M and on

6 244 F. Bizzarri et al..8.4 asynt H 1 H 2 S 6 S 7 H 3 S 8 S δ p.4 2 Fig. 4. One-dimensional bifurcation diagram for δp 4 =.16. their iterates. For instance, for δp 2 =.85, we can locate on the map the four cyclic chaotic intervals J 2,k (k = 1,...,4) shown in Fig. 5(A1). Of course, if δp 2 changes, also the chaotic intervals change. The critical points play a significant role in the definition of such intervals. In the considered eample, for instance, the first eight iterates of C (i.e. the points C k, k =1,...,8) give the boundaries of the four cyclic chaotic intervals (see also Sec. 5). Figure 5(A2) shows the map M for δp 2 =.75, with the two cyclic chaotic intervals J 2,1 and J 2,2, and the trapping interval I =[F 1,C 1 ] (bottom dark-green line). The equilibrium point E 2 does not eist. Generally speaking, for any triplet (δp 2, δp 4,δm), the map can have one or more absorbing sets belonging to a given trapping interval I. In order to include all the absorbing sets, we shall assume that the domain of the map M for a given parameter triplet coincides with the corresponding trapping interval I. With this caveat in mind, from Fig. 5(A2) it is evident that, since D 1 > F 2, the interval Z = [F 2,D 1 ] = [P 2 P 4, 1+δm+ P 2 (P 4 1/2)] does not belong to the range of the map. The amplitude δp 2 /2+δm of such an interval is not affected by P 4 (and then by δp 4 ), that only influences the vertical position of Z. From a circuit point of view, this domain setting criterion is the minimal choice that allows one to set the input and output ranges so as to ensure bounded dynamics. For some parameter triplets, the trapping interval I includes all the absorbing sets and parts of their basins of attraction, thus making the system robust with respect to noise as well. For other parameter triplets, the trapping interval I is absorbing; in this case, it would be better to enlarge the input and output ranges to obtain robustness with respect to noise. If we further increase δp 2, the region Z within I shrinks and disappears at the vertical line S Z = {(δp 2,δp 4,δm) Ω p : δp 2 =2δm} (5) Moreover, the equilibrium E 2 appears (when we cross the dashed curve S 2 in Fig. 3) within the interval [F 1,F 2 ], whereas the two cyclic chaotic intervals J 2,1 and J 2,2 approach each other. They have a contact and merge together at the value of δp 2 such that C 3 = C 4 (vertical red line H 1 in Fig. 3). At this bifurcation value, there is an homoclinic orbit to the equilibrium E 3,sinceC 3 = E 3.After,the absorbing interval J 2 =[C 2,C 1 ] becomes chaotic, as pointed out in Figs. 4 and 5(A3). The system undergoes another significant bifurcation when we cross the first green curve S 5 in Fig. 3, which marks the locus of points where a second absorbing set, contained within the interval J 1 = [F 1,F 2 ], appears. This happens when F 2 = E 2, i.e. when the repellor E 2 eits the interval J 1. Before this bifurcation, the interval J 1 is not invariant, being J 1 M(J 1 ). On the contrary, after the bifurcation, J 1 becomes invariant and includes two small cyclic chaotic intervals bounded by images of the discontinuity points: J 1,1 J 1,2 =[F 1,D 2 ] [D 1,F 2 ] J 1.Thetwoinvariant chaotic sets (J 1,1 J 1,2 )andj 2 (both belonging to I =[F 1,C 1 ]) are pointed out in Fig. 5(A4). When we cross the red line H 2 in Fig. 3, which is the locus of the points (δp 2,δp 4 ) such that D 1 = D 2, the two cyclic chaotic intervals J 1,1 and J 1,2 have a contact and merge together. At this bifurcation value, there is an homoclinic orbit to the equilibrium E 1,sinceD 1 = E 1. After, the whole absorbing interval J 1 becomes chaotic, as pointed out in Figs. 4 and 6(A5). After the blue curve S 6 in Fig. 3, D 1 becomes smaller than F 1. This means that the left chaotic interval becomes J 1 =[D 1,D 2 ], and that the trapping interval I (that includes also J 2 =[C 2,C 1 ]) becomes [D 1,C 1 ] as shown in Figs. 4 and 6(A6). When we cross the second green curve S 7 in Fig. 3, the chaotic attractor corresponding to

7 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap (A1) δ p 2 = (A2) δ p 2 = Z J 2,1 J 2,2 J 2,3 J 2,4.6.7 J 2,1 J 2, (A3) δ p 2 = 8 (A4) δ p 2 = J J J 1,1 1, Fig. 5. Maps M for δp 4 =.16 and δp 2 =.85 (A1),.75 (A2), 8 (A3), and 4 (A4). The intervals J 1 and J 2 or their cyclic chaotic subintervals are represented by red and purple horizontal segments, respectively. The trapping interval I is the dark-green horizontal segment (not reported in the zoom of map (A1)), and the interval Z is the vertical red segment (eisting in the maps (A1) and (A2), and visible in the map (A2) only). In map (A4), the green trajectory starts from C, the magenta one from D. the absorbing interval [D 1,D 2 ] disappears. This happens when D 2 = E 2, i.e. when the repellor E 2 enters the interval [D 1,D 2 ] (see Figs. 4 and 6(A7)). As a matter of fact, after the bifurcation the interval [D 1,D 2 ] is no longer invariant ([D 1,D 2 ] M([D 1,D 2 ])) and the old chaotic attractor belonging to such an interval becomes a chaotic repellor. The eistence of such a chaotic repellor is proved by the eistence of infinitely many homoclinic orbits of E 2 belonging to the interval [D 1,E 2 ] (see, for instance, the cyan homoclinic orbit in Fig. 6(A7)).

8 2442 F. Bizzarri et al..75 (A5) δ p 2 = (A6) δ p 2 = J 1 J 2 J 1 J (A7) δ p 2 =.1 (A8) δ p 2 =.5.25 J Fig. 6. Maps M for δp 4 =.16 and δp 2 = (A5),.25 (A6),.1 (A7),and.5 (A8). The intervals J 1 and J 2 or their cyclic chaotic subintervals are represented by red and purple horizontal segments, respectively. The green trajectories start from C, the magenta ones from D. The trapping interval I is the dark-green horizontal segment. A further homoclinic bifurcation occurs when C 2 = E 2, i.e. when the repellor E 2 enters the interval [C 2,C 1 ], thus causing the reunion of the chaotic attractor with the chaotic repellor. After the bifurcation value (vertical red line H 3 in Fig. 3), the absorbing chaotic interval [C 2,C 1 ]has an eplosion and is etended to I =[D 1,C 1 ], as showninfigs.4and6(a8). The absorbing interval I changes again after the brown curve S 8 in Fig. 3, where C 1 = D 2.At the right of such a curve, we have I =[D 1,D 2 ], as showninfigs.4and7(a9). Finally, when D 2 overcomes E 4,themapcannot admit bounded dynamics, as pointed out in Fig. 7(A1). This happens in correspondence of the black curve L 1 in Fig. 3.

9 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap 2443 (A9) δ p 2 =.2 (A1) δ p 2 = Fig. 7. Maps M for δp 4 =.16 and δp 2 =.2 (A9)and.72 (A1). The trapping interval I is the dark-green horizontal segment (eisting in the map (A9) only). To conclude this section, we remark again that the boundaries of the chaotic intervals in the bifurcation diagram shown in Fig. 4 are the points C k or D k or F k (for proper values of k), and the same holds also for the net bifurcation diagrams One-dimensional bifurcation diagram for δp 4 =.3 Figure 8 shows the one-dimensional bifurcation diagram obtained by varying δp 2 within the range [ 1, 1.2], for δp 4 =.3. Figure 9(B1) shows the map M for δp 2 =.8, with the two cyclic chaotic intervals J 2,1 and J 2,2 (that constitute the chaotic attractor J 2 = [C 2,C 4 ] [C 3,C 1 ]), inside the trapping interval I = [F 1,C 1 ]. The equilibrium point E 2 does not eist, whereas inside the interval I there is the region Z. If we further increase δp 2, the region Z decreases and disappears at δp 2 =2δm. Thetwo intervals J 2,1 and J 2,2 merge together when we cross the homoclinic bifurcation red curve H 1 in Fig. 3, at which C 3 = E 3. After the bifurcation, the absorbing interval J 2 =[C 2,C 1 ] becomes chaotic, as pointed out in Figs. 8 and 9(B2). The equilibrium E 2 appears within the interval [F 1,F 2 ] when we cross the dashed curve S 2 in Fig. 3. A further homoclinic bifurcation occurs when we cross the vertical red line H 3 in Fig. 3, at which C 2 = E 2. After the bifurcation value, the chaotic dynamics is etended to the trapping interval [F 1,C 1 ], within two cyclic chaotic intervals asynt H 1 H 3 H 2 S 9 S 6 1 δ p 1 2 Fig. 8. One-dimensional bifurcation diagram for δp 4 =.3.

10 2444 F. Bizzarri et al..75 (B1) δ p 2 =.8 (B2) δ p 2 = Z.25 J 2,1 J 2, J 2.75 (B3) δ p 2 =.5 (B4) δ p 2 =.3.25 J A J B Fig. 9. Maps M for δp 4 =.3 and δp 2 =.8 (B1), (B2),.5 (B3), and.3 (B4). The chaotic subintervals (see tet) are represented by red and purple horizontal segments. The trapping interval I is the dark-green horizontal segment, and the interval Z is the vertical red segment (eisting in the map (B1) only). In map (B3), the green trajectory starts from C, the magenta one from D. J A = [F 1,D 2 ] and J B = [D 1,C 1 ], as shown in Figs. 8 and 9(B3). The absorbing interval [F 1,C 1 ] becomes chaotic when we cross the homoclinic curve H 2 (see Figs. 8 and 9(B4)). After the cyan curve S 9 in Fig. 3, F 2 becomes larger than C 1. This means that the absorbing interval becomes [F 1,F 2 ], as shown in Figs. 8 and 1(B5). When we cross the blue curve S 6 in Fig. 3, becomes lower than F 1, and the absorbing D 1

11 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap (B5) δ p 2 =.6 (B6) δ p 2 = (B7) δ p 2 = Fig. 1. Maps M for δp 4 =.3 and δp 2 =.6 (B5),1.2 (B6),and1.4 (B7). The trapping interval I is the dark-green horizontal segment. interval [F 1,F 2 ] etends to [D 1,D 2 ](seefigs.8and 1(B6)). Finally, when D 2 overcomes E 4,themapcannot admit bounded dynamics, as pointed out in Fig. 1(B7). This happens again in correspondence of the black curve L 1 in Fig One-dimensional bifurcation diagram for δp 4 =.24 When we overcome the curve S 3 in the plane (δp 2, δp 4 ), the fied points E 2 and E 3 cannot eist, whereas a local maimum persists. As a consequence, the map is no longer equivalent to a skew

12 2446 F. Bizzarri et al asynt S Z S 9 H δ p.4 2 Fig. 11. One-dimensional bifurcation diagram for δp 4 =.24. tent map. In Fig. 11 (obtained for δp 4 =.24), there is a small number of cyclic chaotic intervals (just three) for δp 2 1. Figure 12(C1) shows the map for δp 2 =.85, where the presence of many cyclic chaotic intervals (bounded by the iterates of the critical points) within the trapping interval [F 1,C 1 ] is evident. The equilibrium points E 2 and E 3 do not eist, whereas there is the region Z, whose preimages cannot be reached by the trajectories and that consequently determines the lace structure of the bifurcation diagram. The borders of the holes in the bifurcation diagram are bounded by iterates of the critical points, as we shall see in Sec. 5. By increasing δp 2, the region Z decreases and disappears when we cross the vertical line S Z in Fig. 3, as usual. From this point on, the bifurcation diagram loses its lace structure, as pointed out in Figs. 11 and 12(C2), where the absorbing set is J 1,1 J 1,2 =[F 1,C 2 ] [C 3,C 1 ]. When we cross the cyan curve S 9 in Fig. 3, F 2 becomes higher than C 1. This means that the trapping interval becomes [F 1,F 2 ], and J 1,1 J 1,2 =[F 1,F 3 ] [F 4,F 2 ], as shown in Figs. 11 and 12(C3). A further homoclinic bifurcation occurs when F 3 = F 4 = E 1, which causes the merging of the two cyclic chaotic intervals J 1,1 and J 1,2. After the bifurcation value (red line H 4 in Fig. 3), the trapping interval [F 1,F 2 ]becomeschaotic,asshown in Figs. 11 and 12(C4)..7 (C1) δ p 2 =.85 (C2) δ p 2 =.65 Z J 1,1 J 1, Fig. 12. Maps M for δp 4 =.24: δp 2 =.85 (C1),.65 (C2),.6 (C3), (C4). The chaotic subintervals (see tet) are represented by purple horizontal segments. The trapping interval I is the dark-green horizontal segment, and the interval Z is the vertical red segment (eisting in the map (C1) only).

13 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap 2447 (C3) δ p 2 =.6 (C4) δ p 2 =.7.25 J 1,1 J 1, Fig. 12. (Continued) Finally, when D 2 overcomes E 4,themapcannot admit bounded dynamics. This happens in correspondence of the black curve L 2 in Fig Analytical Epressions for Some Bifurcation Sets The analytical epressions for the sets of points S 2, S 3, S 4 and S Z have been already derived in the previous section. In this section, we shall derive analytical epressions for the bifurcation sets corresponding to the other curves in Fig Trapping intervals Generally speaking, for a given triplet (δp 2,δp 4, δm), the trapping interval of the map M is I = [I INF,I SUP ], where I INF and I SUP depend on the parameter values. The lower bound is I INF = min{f 1,D 1 }. In the parameter domain Ω p, the condition D 1 (= F 1 ) can be epressed as f 1 (δm, δp 2 ). = 2 4δm + δp 2 4(δp 2 +2) ( δp 4 1 ) 4 (6) where the constraint introduced in Sec. 3 on the parameter δp 4 (i.e. δp 4 <.25) has been taken into account. Henceforth, the feasibility constraints on the bifurcation parameters will be highlighted whenever necessary. Then, the set of points S 6 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 1 (δm, δp 2 )} (7) marks the limit for the points in the parameter space such that the lower bound of I is I INF = D 1 (<F 1 ). The section of S 6 for δm = 1/3 isthe blue curve in Fig. 3. The upper bound of the trapping interval is I SUP =ma{c 1,M(I INF )} Case 1 If I INF = F 1 (i.e. above S 6 ), we can have two cases. (1a) The upper bound is I SUP = C 1 M(I INF )= F 2 as long as the parameter set (δp 2,δp 4,δm) fulfils the following inequality: f 2 (δm, δp 2 ) =. 2 4δm + δp 2 4(δp 2 +2) ( δp 4 1 ) (8) 4 The trapping interval is [F 1,C 1 ] whenever the parameters belong to the region delimited by the planes corresponding to the boundaries of Ω p and by the sets of points S 6 and S 9 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 2 (δm, δp 2 )} (9) The section of S 9 for δm = 1/3 is the cyan curve in Fig. 3. This means that the trapping

14 2448 F. Bizzarri et al..25 δ p S 9 L 2 Then, the set of points L 2 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 4 (δp 2 )} (13) marks the limit for the points in the parameter space such that the trapping interval [F 1,F 2 ] eists. This means that the trapping interval is I =[F 1,F 2 ]inthepalegreenregionpointed out in Fig S 6 S 8 L δ p 2 Fig. 13. Subregions of Ω p where the trapping interval I is [F 1,C 1 ] (yellow), [F 1,F 2 ] (pale green), [D 1,C 1 ] (pale pink), [D 1,D 2 ](palecyan). interval is I = [F 1,C 1 ] in the yellow region pointed out in Fig. 13. We point out that, for different values of δm, the region of eistence of the trapping interval [F 1,C 1 ] could be limited also by the set of points such that C 1 is smaller than E 4 [Bizzarri et al., 25]: f 3 (δm, δp 2 ) =. 2+4δm + δp 2 +4δmδp 2 4(δp 2 +2) ( δp 4 1 ) (1) 4 In Fig. 3, the set of points L 3 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 3 (δm, δp 2 )} (11) is reported for the sake of completeness, even if, for the chosen value of δm, thecurvel 3 lies outside Ω p. (1b) When inequality (8) is not fulfilled, we have I SUP = F 2. The region of eistence of the trapping interval [F 1,F 2 ]isthesetofpointssuch that F 2 is smaller than E 4 : f 4 (δp 2 ). = 2+δp 2 4(δp 2 +2) δp 4 ( 1 ) 4 (12) Case 2 On the other hand, if I INF = D 1,wecanhavetwo cases as well. (2a) The parameter sets such that I SUP = M(I INF ) = D 2 lie in the region defined by the following inequality (and by the parameter ranges): f 5 (δm, δp 2 ). = 6 12δm + δp 2 4δmδp 2 + δp 2 2 4(2 + 3δp 2 + δp 2 2 ( ) δp 4 1 ) (14) 4 Then the trapping interval is [D 1,C 1 ] whenever the parameters belong to the region delimited by the planes corresponding to the boundaries of Ω p and by the sets of points S 6 and S 8 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 5 (δm, δp 2 )} (15) The surface S 8 corresponds to the brown curve in Fig. 3. As in case (1a), for the chosen value of δm, thesetofpointsl 3 is not relevant. This means that the trapping interval is I =[D 1,C 1 ] in the pale pink region pointed out in Fig. 13. (2b) When the inequality (14) is not fulfilled, we have I SUP = D 2. The region of eistence of the trapping interval [D 1,D 2 ]isthesetofpoints such that D 2 is smaller than E 4 : f 6 (δm, δp 2 ) =. 4 4δm 4δmδp 2 + δp 2 2 4(2 + 2δp 2 ) δp 2 sign(δp 2 )δp 4 (16) Then, the set of points L 1 = {(δp 2,δp 4,δm) Ω p : sign(δp 2 )δp 4 = f 6 (δm, δp 2 )} (17)

15 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap 2449 marks the limit for the points in the parameter space such that the trapping interval [D 1,D 2 ] eists. This means that the trapping interval is I =[D 1,D 2 ] in the pale cyan region pointed out in Fig. 13. When I INF = F 1 (i.e. above S 6 in Fig. 3), this occurs for f 7 (δm, δp 2 ). = 2+4δm + δp 2 + δp 2 2 4(2 + 3δp 2 + δp 2 2 ) δp 4 (19) Also in Fig. 13, the unfeasible parameter region (either outside Ω p or at the right of the black curves corresponding to L 1 and L 2 ) is represented in lightgray. For any point of such a region, the generic trajectory is divergent, as infinitely many repelling cycles still eist for M, but neither invariant interval, nor cyclic chaotic intervals, nor other chaotic dynamics can eist..7 asynt C 5 C 7 C 1 C Bifurcation set S 1.6 H b H a The interval [C 2,C 1 ] can contain an absorbing set provided that the following conditions hold: the equilibrium point E 3 eists; C 2 belongs to the third branch of the map, i.e. C 2 P 5, i.e. δp 4 (1/4) δm; if E 2 eists, C 2 is greater than E 2, i.e. sign(δp 2 )δm (1/2)sign(δp 2 ); The latter condition can be epressed in compact form in terms of δm only, by assuming δp 2 < : C C 4 8 C 6 C δ p.8 2 Fig. 14. Zoom of the one-dimensional bifurcation diagram for δp 4 =.16. δm 1 2 (18) Indeed, for δm < (1/2), the graph of the map is below the diagonal. In Fig. 3, the first condition is fulfilled under the dashed curve S 3, the second condition is fulfilled under the horizontal dark-green line S 1, whereas the third condition holds at the left of the vertical red line H δ p 2 = Bifurcation sets S 5 and S 7 The bifurcation sets S 5 and S 7 are the boundaries of a region where two absorbing sets can coeist. For instance, in Fig. 4, between the two green vertical lines S 5 and S 7 it is evident the presence of an absorbing set (either [F 1,F 2 ]or[d 1,D 2 ]) coeisting with J 2 =[C 2,C 1 ]. Such an absorbing set may eist when: the equilibrium point E 2 eists; M(I INF ) >E 2 ;.6.7 Fig. 15. Homoclinic orbit (in magenta) to a 2-cycle (in green) on the map M for δp 4 =.16 and δp

16 245 F. Bizzarri et al. and when I INF = D 1 (i.e. below S 6 in Fig. 3) for f 8 (δm, δp 2 ) Thus,. = 2 4δm 3δp 2 12δmδp 2 +2δp 2 2 4δmδp2 2 + δp3 2 4(2 + 5δp 2 +4δp δp3 2 ) δp 4 (2) S 5 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 7 (δm, δp 2 )} (21) and S 7 = {(δp 2,δp 4,δm) Ω p : δp 4 = f 8 (δm, δp 2 )} (22) 4.4. Bifurcation sets H k In this subsection we only summarize the homoclinic bifurcation conditions derived in Sec. 3. The curves can be derived by epressing in terms of the bifurcation parameters the equality conditions asynt C 1 asynt C 7 C 9 C 13 C 15 C 11 C 5 C 3 δ p 2 δ p 2 (a) (b) asynt C 4 asynt C 6 C 12 C 14 C 1 C 16 C 2 C 8 δ p 2 δ p 2 (c) (d) Fig. 16. Four zooms (at decreasing quotes from (a) to (d)) of the one-dimensional bifurcation diagram for δp 4 =.16 and δp 2 [.915,.95].

17 Bifurcation Analysis of a Circuit-Related Generalization of the Shipmap 2451 stated in the previous section and involving iterates of critical points and equilibria. Of course, also the conditions of eistence of the involved equilibria must be imposed. The bifurcation set H 1 is obtained by imposing that C 3 = E 3 and that E 3 eists: { H 1 = (δp 2,δp 4,δm) Ω p : δp 2 = 2+ 2,δp 4 1 4δm + δp } 2 (23) 4(3 + δp 2 ) The bifurcation set H 2 is obtained by imposing that D 1 = E 1 : H 2 = { (δp 2,δp 4,δm) Ω p : δp 4 = 4+2δp 2 12δm 4δmδp 2 + δp 2 } 2 4(4 + 4δp 2 + δp 2 2 ) (24) The bifurcation set H 3 is obtained by imposing that C 2 = E 2 and that E 2 eists: { H 3 = (δp 2,δp 4,δm) Ω p : δp 2 =,δp 4 δp } 2 1 4δm (25) 4(1 + δp 2 ) Finally, the bifurcation set H 4 is obtained by imposing that F 3 = E 1 : H 4 = { (δp 2,δp 4,δm) Ω p : δp 4 = δp2 2 +2δp } 2 4 4(8 + 6δp 2 + δp 2 2 ) (26) 5. Other Numerical Results In this section, we shall describe the mechanism governing the change in the number of cyclic chaotic intervals, that can be summarized as follows. Let us suppose that for a bifurcation value δp 2 = δp ˆ 2 it is M p ( ) =M r ( ), with p<r. The last equality means that M p ( ) is a fied point for the map M r p, i.e. it is a periodic point of period r p for the map M. As a matter of fact, M r ( ) = M r p [M p ( )]. Figure 14 shows a zoom of Fig. 4. The curves superimposed to the bifurcation diagram represent the kth iterates of the point = C, with k = 1,...,8, as a function of δp 2. At the bifurcation value corresponding to the vertical red line H a in Fig. 14, two pairs of cyclic chaotic intervals merge together. This happens when C 5 = C 7, i.e. when p =5andr = 7. This condition corresponds to the presence of a 2-cycle on the map M. Evenifthe cycle is unstable, it is reached in five map iterations.6 δ p 2 = Fig. 17. Homoclinic orbit (in magenta) to a 4-cycle (in green) on the map M for δp 4 =.16 and δp δ p.92 2 Fig. 18. Zoom of the one-dimensional bifurcation diagram for δp 4 =.24 and δp 2 [.94,.92] together with curves C k (δp 2 ) (magenta curves), D k (δp 2 ) (blue curves), and F k (δp 2 ) (cyan curves) for several values of k.

18 2452 F. Bizzarri et al. starting from C. This means that the map M contains orbits homoclinic to such a cycle [Gardini, 1994], as pointed out in Fig. 15. Figure 16 shows four zooms of Fig. 14 across the vertical red line H b. The curves superimposed to thebifurcationdiagramrepresentthekth iterates of the point C, withk =1,...,16. At the bifurcation value corresponding to the line H b in Fig. 14, four pairs of cyclic chaotic intervals merge together. This happens when C 9 = C 13. This condition corresponds to the presence of an unstable 4-cycle on the map M, that is reached in eight map iterations starting from C. A homoclinic orbit to such a cycle isshowninfig.17. A completely similar mechanism is the basis of the lace structure in the bifurcation diagram shown in Fig. 11. Figure 18 shows a zoom of such a diagram together with kth iterates of the points C (magenta curves), D (blue curves), and F (cyan curves) as a function of δp 2. In order to have a comprehensible figure, we superimposed to the diagram the most significant iterates only: the curves C k (δp 2 ) are drawn for even values of k from 2 up to 18; the curves D k (δp 2 ) are drawn for even values of k from 2upto2andforoddvaluesofk from 17 to 39; the curves F k (δp 2 ) are drawn for even values of k from 6 up to 32, for k = {1, 3} and for odd values of k from 21 to Concluding Remarks The analysis carried out in this paper provides criteria (at least with respect to the considered bifurcation parameters) for the design of circuits implementing the so-called shipmap. More in general, this paper proposes a remodeling of dynamical systems to introduce new bifurcation parameters related to circuit implementations mismatches. This is the essential prerequisite for an effective bifurcation analysis, strictly related to real implementations. To apply such a design philosophy to dynamical systems in the telecommunication framework, the net step would be a statistical analysis of the redefined model, so as to find regions in the parameter space ensuring both good dynamics (i.e. the structural stability of the system) and good statistical features. We point out that the usual parameter configuration makes the system structurally unstable, since the point δp 2 = δp 4 = δm = lies on a bifurcation curve (see [Bizzarri et al., 25]). From the analysis carried out in this paper, one can deduce what are the regions in the parameter space candidate to configure the system so as to generate chaotic sequences interesting from a telecommunication point of view, that are regions where absorbing sets do not coeist and chaotic dynamics arise within a single absorbing interval (i.e. no cyclic chaotic intervals are present). The combination of statistical and bifurcation analysis should yield to the design of structurally stable chaotic sequences generators. Acknowledgments The authors would thank Dr. Linda Carezzano for her preliminary work on the map studied in this paper. This work has been supported by the MIUR, within the FIRB framework, and by the University of Genoa. References Bizzarri, F., Carezzano, L., Storace, M. & Gardini, L. [25] Bifurcation analysis of a circuit-related piecewise-affine map, Proc. ISCAS 5, Kobe, Japan, May, 25. Callegari, S., Rovatti, R. & Setti, G. [2] Robustness of chaos in analog implementations, in Chaotic Electronics in Telecommunications, eds. Kennedy, M. P., Rovatti,R.&Setti,G.(CRCPress,BocaRaton,FL), pp Delgado-Restituto, M. & Rodriguez-Vásquez, A. [2] Piecewise affine Markov maps for chaos generation in chaotic communication, in Proc. ECCTD 99, Stresa, Italy, 29 August 2 September, 2, Rev. Mod. Phys. 57, Gardini, L. [1994] Homoclinic bifurcations in n- dimensional endomorphisms, due to epanding periodic points, Nonlin. Anal. TMA 23, Maistrenko, Yu. L., Maistrenko, V. L. & Chua, L. O. [1993] Cycles of chaotic intervals in a time-delayed Chua s circuit, Int. J. Bifurcation and Chaos 3,