A higher-dimensional generalization of the Lozi map: Bifurcations and dynamics
aa r X i v : . [ n li n . C D ] M a y A higher-dimensional generalization of the Lozi map: Bifurcations and dynamics
Shakir Bilal ∗† and Ramakrishna Ramaswamy ‡ Department of Physics and Astrophysics, University of Delhi, Delhi, 110 007, India Department of Chemistry, Indian Institute of Technology, New Delhi, 110 016, India
We generalize the two dimensional Lozi map in order to systematically obtain piece–wise con-tinuous maps in three and higher dimensions. Similar to higher-dimensional generalizations of therelated H´enon map, these higher-dimensional Lozi maps support hyperchaotic dynamics. We carryout a bifurcation analysis and investigate the dynamics through both numerical and analyticalmeans. The analysis is extended to a sequence of approximations that smooth the discontinuity inthe Lozi map.
I. INTRODUCTION
The behavior of low-dimensional nonlinear iterativemaps and flows has been extensively studied and charac-terized over the past few decades, particularly with ref-erence to the creation of chaotic dynamics [1–4]. Thevarious scenarios or routes to chaos in such systems areby now fairly well known [4–7]. Similar exploration of theproperties of higher dimensional dynamical systems—forinstance the dynamics of attractors with more than onepositive Lyapunov exponent and the bifurcations throughwhich they have been created—has not been studied inas much detail even in relatively simple systems [3, 8–11].Linear and piecewise-linear mappings are among thesimplest examples of iterative dynamical systems. Theso–called Lozi map [3] is analogous to the quadraticH´enon mapping [2] but has the advantage that more ex-tensive analysis is possible [12]. The mapping itself isonly piecewise continuous, and this introduces some ad-ditional features that need to be understood more clearly[14, 15, 17, 21]. Indeed, specific bifurcation phenomenonsuch as border collision bifurcations can only occur inpiecewise smooth dynamical systems [14, 21].Our interest in the present paper is the generalizationof the Lozi map to higher dimensions. One motivation isto compare this piecewise continuous system to a similarhigh-dimensional H´enon mapping [16]. Of the differentways in which this can be done, we choose to extend themap to d –dimensions by incorporating time-delay feed-back while ensuring that the system remains an endo-morphism in the absence of dissipation. The dissipationis introduced at the k th step, k < d , and this also en-sures that the map is a diffeomorphism. The system cantherefore have k –positive Lyapunov exponents, and weexamine the transition to high-dimensional chaos as afunction of parameters, characterizing the different bi-furcations that can occur. An intermediate “smooth”approximation [13] of the piecewise map is also investi-gated vis-a-vis bifurcations for comparison. ∗ This article was written when the author was at University ofDelhi. He works from home at the time of submission. † email: [email protected] ‡ email: [email protected] In the next Section, the generalized Lozi map is de-scribed and a detailed analysis of the local bifurcations ofthe elementary fixed points is presented. The emergenceof chaotic and hyperchaotic attractors and the global bi-furcations that arise are discussed in Section III. SectionIV is devoted to the analysis of smooth approximationsto the map. The paper concludes with a discussion andsummary in Section V.
II. THE GENERALIZED LOZI MAP
The two dimensional Lozi map [3] is given by x n +1 = 1 − (1 − ν ) y n − a | x n | y n +1 = x n . (1)This map is a modification of the quadratic H´enon map-ping, with the parameters ν and a tuning the dissipationand nonlinearity respectively. Since the map is piecewiselinear, it lends itself to extensive analysis, some of whichhas been recently summarized [11].Rewriting the above as a difference delay equation, onehas x n = 1 − a | x n − | − (1 − ν ) x n − , (2)which suggests a natural generalization to higher dimen-sions, x n = 1 − a | x n − k | − (1 − ν ) x n − d . (3)Here d and k are integers such that k < d , and we take0 ≤ ν ≤
2. The mapping is conservative when ν is 0or 2, and is dissipative otherwise. For ν = 1 the mapreduces to a k –dimensional endomorphism, while for ν =1 the map is a d –dimensional diffeomorphism. In thenext subsection we analyze the implications of differentchoices of d and k for this map. A. The base maps and q –degeneracy The integers d and k are either co-prime or share acommon factor q . When k and d have a common factor q , it is easy to see that all the eigenvalues of the Jacobian FIG. 1: (a) Region of period–1 dynamics for d =3, k =2;see Eq.(9). (b)The organization of a – ν plane into un-bounded dynamics (ES) and bounded dynamics: hyper-chaotic (HC,orange), chaotic (C, green), periodic (P,blue).White regions sandwiched between period one dynamics andhyperchaotic/chaotic/periodic corresponds to quasiperiodicdynamics (QP). The blue regions indicate that supercriti-cal bifurcations are possible on either side of the period oneboundaries (see Fig. 2). (c) The region of period one dynam-ics converges asymptotically (i.e. for d → ∞ ) and is containedwithin the curves L TL , L TR , L BL , L BR . (d) The convergenceto the curves TR,BL,BR is captured by the distance r of thelargest root from the unit circle as a function of dimension d on these curves. L NS indicates the Neimark–Sacker bound-ary, and the other labels L TR , L TL , L BR and L BL are shortfor top left,right, and bottom left,right. are q –fold degenerate, and this leads to a q –fold degen-eracy in the Lyapunov exponents. It therefore sufficesto examine the case of d, k co–prime since these formthe base for all other values of k < d , and it suffices toconsider only base-maps as can be seen by the followingargument. For the n = qm th iterate, the substitution x q · by ξ · gives x qm = 1 − a | x qm − qk ′ | − (1 − ν ) x qm − qd ′ ↓ ξ m = 1 − a | ξ m − k ′ | − (1 − ν ) ξ m − d ′ . (4)The maps Eq. (3) and Eq. (4) differ in that there are q hidden variables within each ξ m . Thus the Jacobiancan be separated into q identical blocks, giving rise to a q –degenerate ( d ′ , k ′ ) map. B. Fixed points: Stability
The delay map, Eq. (3) can be rewritten as a d -dimensional iteration, x n +1 = 1 − a | x kn | − (1 − ν ) x dn x n +1 = x n ... x dn +1 = x d − n (5)The fixed points of the mapping are those for which { x n +1 , . . . , x dn +1 } = { x n , . . . , x dn } . Solving, we find x ∗ = x ∗ = · · · = x d ∗ = x ± with x ± = 12 − ν ± a . (6)of these two fixed points, x − is always unstable. Thematrix elements of the Jacobian of the map in Eq. (3)are given by[ J ] ij = j = i − , d ≥ i ≥ ± a i = 1 , j = k − (1 − ν ) i = 1 , j = d , (7)and it is straightforward to obtain the stability conditionson the fixed point x + (for arbitrary d and k ) from thecharacteristic polynomial P ( λ ), P , ( λ ) = λ d ± aλ d − k + (1 − ν ) . (8)If the number of real roots of the polynomials P , withreal part greater than +1 (or smaller than -1) is σ +1 , ( σ − , ), then according to Feigin’s classification of bordercollision bifurcations in piecewise smooth maps [14], afold bifurcation occurs when σ +1 + σ +2 is odd. If σ − + σ − is odd, on the other hand, a flip bifurcation occurs.The characteristic of the Neimark–Sacker bifurcationin smooth dynamical systems is that a pair of complexeigenvalues cross the unit circle. This theory has beenextended to piecewise smooth maps only recently [15]and although some of the features are common, there aremajor differences [15]. We find from numerical estimationof eigenvalues of P , for the base maps that a Neimark–Sacker bifurcation occurs via a border collision if an oddnumber of pairs of complex eigenvalues cross the unitcircle. In particular, for the base map with d = 3 , k = 2we find that period one region is bounded by the curvesas shown in Fig. 1(a): L fold ( d ) : a = − νL NS ( d = 3)(supercritical) : a = ν (2 − ν ) .L flip : a = − ν . If either of the polynomials produce their largest rootswith absolute values less than one, the fixed point x + is stable and contributes to the period–one region in the a – ν parameter space, unless it hits the border x = 0. Un-like the smooth case (i.e. the H´enon map) studied in [16]the bifurcations in the Lozi map can show supercriticalbifurcations on either side of the period one boundaries(of course limited upto the saddle node curve), as shownin Fig. 1(b). Since these bifurcation clearly show thatorbit of the period one hits the boundary x = 0 at suchbifurcation, we attribute this distinct feature of the gen-eralized version of the Lozi map (3) to border collisionbifurcations of the map. FIG. 2: Bifurcation diagrams as a function of the nonlin-earity a for different base maps (a) d =3, k =1, ν = 0.3 (b) d =4, k =1, ν =0.5, (c) d =3, k =2, ν =0.6, (d) d =4, k =3, ν =0.6.The boundaries of stable period-1 dynamics are indicated bydashed vertical lines; these are also the border-collision bifur-cation points. C. Bifurcations diagrams
Bifurcation diagram in the two dimensional a – ν pa-rameter space for d = 3 , k = 2 is shown in Fig. 1(b). Atypical feature of a – ν parameter space is the existenceof hyperchaotic (HC), chaotic (C), quasiperiodic (QP),and periodic regions. These features are shared by othermembers (with different d and k values) of this general-ized Lozi map. Representative bifurcation diagrams as afunction of a for different ( d, k ) combinations are shownin Fig. 2 and the corresponding orbital characteristics areshown via Lyapunov exponents in Fig. 3. In each case ν isdifferent but fixed. An interesting feature of these bifur-cation diagrams is that for for 0 < ν < a is decreased to the left of the period one boundary.Typically these were found to be a flip bifurcation below a < − ν when d = 3 , k = 2 and supercritical Neimark–Sacker type bifurcations for d = 4 , k = 3. Such featureare typically of these maps even across different combi-nation of dimensionality parameters ( d , k ). We shouldmention that such phenomenon was not found for a sim- ilarly generalized H´enon map [16], and appears to be aresult of border collision bifurcations. It is important tonote that the theory of bifurcations in smooth dynamicalsystems does not explain these features [17]. III. HIGH DIMENSIONAL DYNAMICS
In this Section we examine the bifurcations startingfrom the period-1 fixed point as a function of nonlinearityparameter for different embedding dimensions d and theendomorphism dimension k . FIG. 3: k- largest Lyapunov exponents for the systems corre-sponding to the bifurcation diagrams shown in Fig. 2 . Thedifferent base maps are (a) d =3, k =1, ν =0.3, (b) d =4, k =1, ν =0.5, (c) d =3, k =2, ν =0.6, and (d) d =4, k =3, ν =0.6. Theboundaries of the stable period-1 dynamics are indicated bydashed vertical lines; the left boundary is a border collisionbifurcation point. A. Bounded dynamics
In the limit d → ∞ period–1 motion converges to a re-gion shown in Fig. 1(c): the boundaries are the followingcurves, L T L : a = − νL T R : a = 2 − νL BL : a = − ν (9) L BR : a = ν, in the a − ν parameter space. These curves can be un-derstood from the properties of the characteristic poly-nomials Eq. (8) in the limit of d → ∞ . The distance r between the leading root of the characteristic polynomialEq. (8) and the unit circle on the curve L T L , in the ex-treme case of k = 1 and k = d −
1, are shown in Fig. 1(d): r approaches zero as the dimension d is increased. Sim-ilar behavior of r is also observed on L BR , L BL , T LR and for 1 < k < d − B. Hyperchaos
The map Eq. (3) exhibits at most k positive Lyapunovexponents as nonlinearity parameter a is varied: this isdue to the fact that the nonlinearity in the map occurs atthe k th previous iteration step. The stretching and fold-ing that is responsible for introducing sensitivity to initialconditions in the map [18], occurs in k directions, and thisresults in the maximum k number of possible positiveLyapunov exponents; see Figs. 4(a)–(b). Additionally,for ν = 1, the map is a k –dimensional endomorphism(and is not invertible) with k –fold degenerate Lyapunov FIG. 4: The Lyapunov exponent spectrum ( λ i , i = 1 , ..., k +1) as a function of nonlinearity parameter for d = 5. (a)–(b) demonstrate that for a given dimension d the number ofpositive Lyapunov exponents are governed by the parameter k . We keep ν = 1 . k = 2 in (a) and k = 4 in (b). (c)–(d) demonstrate thebreaking of degeneracy in the Lyapunov spectrum observedin the endomorphism ν = 1 by making the map diffeomorphic ν = 1 . exponents, i.e. all of these LEs are identical and becomepositive at the same value of the nonlinearity parameter a as can be seen in Fig. 4(c). Embedding the k –dimensionalendomorphism in a d –dimensional space does not changethis behavior. However when the map is made diffeomor-phic by enabling the contraction/dissipation parameter( ν = 1 & 0 ≤ ν ≤
2) the degeneracy in the Lyapunovexponents is lifted, although the maximum possible num-ber of positive Lyapunov exponents is still k as can beseen in Fig. 4(d).Route to chaos is observed via the quasiperiodic andalso via finite period–doubling route, as seen in the Lya-punov spectra Fig. 3 and Fig. 4. The period doubling cas-cade terminates after a few doublings, leading to chaos.On the other hand chaos and hyperchaos transition issmooth, since the first k –largest Lyapunov exponents be-have smoothly as they hierarchically become positive atdifferent values of the nonlinear parameter a . This typ-ically means that the map Eq. (3) can be written as ahierarchy of chaotically driven maps at subsequent tran-sitions to higher chaos [20], this is similar to the chaos hyperchaos transition in the generalized H´enon map [16]. IV. SMOOTH APPROXIMATIONS
In this section we analyse a smooth approximation ofthe generalized Lozi map (3). We replace the modulus
FIG. 5: The bifurcation diagram for d = 3 , k = 1 for differentvalues of ǫ . As ǫ is increased from (a)–(d) characteristics ofsmooth bifurcations emerge in the form of period doubling. function | · | in Eq. (3) with a smooth function S ǫ ( · ): x n = 1 − aS ǫ ( x n − k ) − (1 − ν ) x n − d (10) S ǫ ( x n − k ) = ( x n − k / ǫ + ǫ/ if | x n − k | ≤ ǫ | x n − k | if | x n − k | ≥ ǫ , where 0 < ǫ <
1. The function S ǫ ( x n − k ) extends thesmooth approximation applied to the two dimensionalLozi map [13] to our high dimensional generalization ofthe Lozi map and removes the discontinuity in the slopeat x n − k = 0.The fixed points of the new map in Eq. (10) are givenby: x ± = − ν ± a if | x | > ǫ ǫa (cid:16) − (2 − ν ) ± q (2 − ν ) + aǫ − a (cid:17) if | x | ≤ ǫ (11)the first set of these fixed points are similar to those ofLozi map (see Eq. (6)) and lose stability by colliding withone of the borders located at ± ǫ as nonlinearity parame-ter a is varied. The new orbits that appear following thisborder collision bifurcation depend on the delay parame-ters d and k , although the maximum number of positiveLyapunov exponents is still limited to k . Assuming fixedvalues of the dissipation parameter and ǫ , the variationin nonlinearity parameter can take iterations of the mapalso inside the region | x | < ǫ then subsequent bifurca-tions are no longer only due to border collisions: bordersat ± ǫ have well defined first derivatives and once an orbitenters the region | x | < ǫ the dynamics is also governedby the smooth approximation.The case of d = 2 and k = 1 was illustrated in [13],where it was observed that the period-doubling route tochaos is achieved for finite values of ǫ (note that perioddoubling route to chaos is absent for ǫ =0).For d > ǫ = 1 / n ǫ , n ǫ ≥ d =3, k =1 the loss of stability of the fixedpoint as nonlinearity parameter is varied is dependent on n ǫ : border collision bifurcations gives way to period dou-bling bifurcations as n ǫ decreases (Fig. 5(a)–(d)). Similarobservations are made when a Neimark–Sacker type bi-furcation is involved in the Lozi map with d = 3 , k = 2,where border collision bifurcations (Fig. 6(a) ) give wayto smooth bifurcations Fig. 6(d). In the H´enon maps FIG. 6: The bifurcation diagrams for d = 3 , k = 2 for differ-ent values of ǫ . As ǫ is increased from (a)–(d) characteristicsof smooth bifurcations emerge in the form of period doubling.The figures (d)–(a) also indicate emergence of border collisionbifurcation beginning from smooth period doubling bifurca-tions in (d). the chaotic dynamics follows directly after the Neimark–Sacker bifurcation via a crisis–like transition [16]. Thusthe effect of ǫ > V. DISCUSSION AND SUMMARY
In this paper we have introduced a generalized time–delayed Lozi map with nonlinear feedback from k earlier steps and linear feedback from d earlier steps. This sim-ple feedback process governs the dimensionality of themaps. The parameter k ( < d ) determines the number ofpositive Lyapunov exponents. Further more, the familyof maps formed as a result of different combinations ofdimensionality parameters are classified into base mapswhen d and k are co–prime. All other maps reducibleto these base maps exhibit a q -fold degenerate Lyapunovspectrum when d and k share a common factor q . Bi-furcation analysis was performed in a limited region ofthe parameter space. In particular, fixed point dynam-ics loses stability through the fold, flip and the NeimarkSacker bifurcations via border collisions. Analytic formswere determined for these boundaries, and the flip andNS bifurcation curves were found to depend on the di-mension d . With increasing dimension, the region of pe-riod one dynamics was found to converge in the param-eter plane.The dynamics evolves abruptly from regular to chaosdue to piece-wise nature of the map. Subsequent transi-tions from chaos to hyperchaos, however, are smooth asindicated by Lyapunov spectrum: the dimension of theattractor changes smoothly if there are no abrupt tran-sitions in Lyapunov spectrum [16, 20].A smooth approximation of the map enabled the anal-ysis of the bifurcations vis-a-vis further comparing someof the bifurcations to the generalized the H´enon map. Itshowed that some of the bifurcations observed persist onboth the piecewise Lozi and H´enon map. Further explo-ration of a more general unified mapping is a project forfuture work. Another possible project for a future work isthe analysis of conservative limit in these class of maps:orbits in the conservative limit are only possible when d = 2 k ,therefore in the conservative limit hyperchaoticorbits are indeed possible with k − positive Lyapunov ex-ponents. The exploration of the conservative limit of thismap could be a task for future work. Acknowledgement
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