CYCLING CHAOS IN ONE-DIMENSIONAL COUPLED ITERATED MAPS

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1 Letters Iteratioal Joural of Bifurcatio ad Chaos, Vol., No. 8 () c World Scietific Publishig Compay CYCLING CHAOS IN ONE-DIMENSIONAL COUPLED ITERATED MAPS ANTONIO PALACIOS Departmet of Mathematics, Sa Diego State Uiversity, Sa Diego, CA 98-77, USA Received July, ; Revised August 3, Cyclig behavior ivolvig steady-states ad periodic solutios is kow to be a geeric feature of cotiuous dyamical systems with symmetry. Usig Chua s circuit equatios ad Lorez equatios, Dellitz et al. [995] showed that cyclig chaos, i which solutio trajectories cycle aroud symmetrically related chaotic sets, ca also be foud geerically i coupled cell systems of differetial equatios with symmetry. I this work, we use umerical simulatios to demostrate that cyclig chaos also occurs i discrete dyamical systems modeled by oedimesioal maps. Usig the cubic map f(x, λ) =λx x 3 ad the stadard logistic map, we show that coupled iterated maps ca exhibit cycles coectig fixed poits with fixed poits ad periodic orbits with periodic orbits, where the period ca be arbitrarily high. As i the case of coupled cell systems of differetial equatios, we show that cyclig behavior ca also be a feature of the global dyamics of coupled iterated maps, which exists idepedetly of the iteral dyamics of each map. Keywords: Coupled cell systems; cyclig chaos; symmetry.. Itroductio Symmetry i oliear dyamical systems ca force certai subspaces of the phase-space to be ivariat uder the goverig equatios. I cotiuous systems, where the goverig equatios typically cosist of systems of differetial equatios, it is well-kow that the presece of ivariat subspaces ca lead to the existece of solutio trajectories that coect, via saddle-sik coectios, equilibria ad or/periodic solutios that lie o the ivariat subspaces. As time evolves, a typical trajectory stays for icreasigly loger periods ear each solutio before it makes a rapid excursio to the ext solutio. See, for example, [Field, 98] ad [Guckeheimer & Holmes, 988]. Sice saddlesik coectios are robust, these cycles called heterocliic cycles are robust uder perturbatios that preserve the symmetry of the system. Whe a heterocliic cycle is also asymptotically stable, it serves as a model for a certai kid of itermittecy, sice earby trajectories move quickly betwee solutios (equilibria ad periodic solutios) ad stay for a relatively log time ear each solutio. I systems of differetial equatios with O() symmetry, Armbruster et al. [988] showed that heterocliic cycles betwee steady-states ca occur stably, ad Melboure et al. [989] provided a method for fidig cycles that ivolve steady-states as well as periodic solutios. Buoo et al. [] showed that cycles coectig steady-states ad periodic solutios are also foud stably i systems of coupled idetical cells with Dihedral D symmetry. I priciple, more complex heterocliic cycles ca also occur robustly if the ivariat subspaces cotai more complicated solutios. For istace, replacig the equilibria i the 859

2 86 A. Palacios Guckeheimer Holmes cycle with chaotic attractors ca lead to what Dellitz et al. [995] call cyclig chaos. This ca be doe as follows. First, Dioe et al. [996] showed that the Guckeheimer Holmes system ca be iterpreted, with a appropriate choice of couplig fuctio, as a etwork of idetical coupled cells modeled by a system of differetial equatios of the form dx i = f(x i )+ α ij h(x i,x j ), () dt j i where X i =(x i,..., x ik ) R k deotes the state variables of cell i, f govers the iteral dyamics of each cell, h is the couplig fuctio betwee two cells, the summatio is take over those cells j that are coupled to cell i, adα ij is a matrix of couplig stregths. Dellitz et al. [995] the made the critical observatio that, uder certai coditios, cyclig behavior is a feature of the global dyamics that ca persist idepedetly of the iteral dyamics of each cell. It follows that if the iteral dyamics of the Guckeheimer Holmes system is replaced by a dyamical system set to produce a chaotic attractor, the the ew coupled system would produce cyclig chaos. Dellitz et al. demostrated this coclusio usig first Chua s circuit ad the Lorez equatios. Although the existece ad stability of heterocliic cycles i cotiuous dyamical systems with symmetry has bee extesively explored, little is kow, however, about the existece of these cycles i discrete systems. I this work, we explore umerically the existece of heterocliic cycles i coupled cell maps with iteral cell dyamics modeled by oe-dimesioal iterative maps. We demostrate that discrete systems ca also exhibit cycles coectig fixed poits with fixed poits ad periodic orbits with periodic orbits, where the period ca be arbitrarily high. More importatly, we show that cyclig behavior ca persist, idepedetly of the iteral dyamics of each cell, as a feature of the global dyamics. A detailed aalysis of existece ad stability of these cycles is i preparatio [Palacios, ].. Coupled Iterated Cell Maps We cosider here similar coupled cell systems to those used by Dellitz et al. [995] for showig the existece of cyclig chaos, except that ow we assume the iteral dyamics of each cell is govered by oe-dimesioal maps of the form x i+ = f(x i,λ) () where x i R deotes the state variable of cell i ad λ R is a parameter. We model a itercoected etwork of N idetical cells, by a system of oe-dimesioal maps of the form x i+ = f(x i,λ)+ j i α ij h(x i,x j ), (3) where h is the couplig fuctio betwee those cells j that are coupled to cell i, i N. Followig Dellitz et al. [995], we use the term local symmetries to refer to the symmetries of idividual cells. Thus, Σ is a group of local symmetries if, for all σ Σ, we have f(σx) =σf(x). Similarly, we use the term global symmetries to refer to the symmetries that are iduced by the patter of couplig. We also cosider wreath product couplig, so that the local symmetries of idividual cells are also symmetries of the etwork equatios (3). That is, if σ Σthe h(x i,σx j )=σh(x i,x j ), h(σx i,x j )=h(x i,x j ). For illustrative purposes, i this work, we cosider systems of oly three idetical cells coupled i a directed rig as is show schematically i the followig diagram. Similar ideas ca be applied to systems with larger umber of cells. Assumig the couplig amog cells is also idetical, α ij = α, we write the etwork equatios i the form + = f(,λ)+αh(y, ) y + = f(y,λ)+αh(z,y ) (4) z + = f(z,λ)+αh(,z ), so that the global symmetries of the etwork are described by the cyclic group Z 3, which is geerated by the permutatio (x, y, z) (y, z, x).

3 Cyclig Chaos i Oe-Dimesioal Coupled Iterated Maps 86 Fig.. cells. z x Z 3-symmetric directed rig with three idetical 3. Results of Simulatios As a first example, we assume the iteral dyamics of each cell is govered by a cubic map with local Z symmetry through f(x, λ) =λx x 3, λ >. (5) For differet values of the bifurcatio parameter λ, this map exhibits period-doublig cascades leadig y to chaotic attractors. See the bifurcatio diagram of Fig.. The bifurcatios i (5) are remiiscet of those foud i the logistic map [May, 976], except that ow local Z -symmetry forces two otrivial fixed poits (oe with x>adoewithx<) to bifurcate from the trivial solutio x =atλ =. Each fixed poit, i tur, udergoes a period-doublig cascade leadig to a pair of chaotic attractors. Local Z symmetry agai forces the cascades to occur at the same parameter values for each fixed poit [Dellitz et al., 995]. For λ<λ c =3 3/, the attractors are cofied to opposite sides of the x = axis ad each attractor has its ow basi of attractio. At λ = λ c, the basis of attractio collide ad the two attractors merge ito a sigle oe. Rogers ad Whitley [983] provided a more comprehesive aalysis of a similar map f(x, λ) = (λx x 3 ), λ>. Next we form the itercoected etwork equatios (4) with couplig fuctio h(x i,x j )= x j / x i (6) Fig.. Bifurcatio diagram for a cell with iteral dyamics f(x, λ) =λx x 3.

4 86 A. Palacios y z (a) y z (b) Fig. 3. Two types of cycles coectig fixed poits of f(x, λ) =λx x 3 at λ =.5. (a) Type I: With couplig stregth α =, a active cell always shows the same of two cojugate solutios. (b) Type II: With a differet couplig stregth α =.795, each active cell switches betwee two cojugate solutios.

5 Cyclig Chaos i Oe-Dimesioal Coupled Iterated Maps 863 so that the three-cells coupled system possesses global Z 3 -symmetry ad local Z -symmetry. Usig a appropriate value of the couplig stregth α ad a fixed value of λ i the iterval λ λ c, the coupled cell system (4) exhibits trajectories that cycle aroud the orbits geerated by the iteral dyamics of each cell at each value of λ. The wide rage of behavior i the cubic map (5), which cotrols the iteral dyamics of idividual cells, leads to cycles coectig fixed poits with fixed poits, periodic orbits with periodic orbits, ad chaotic attractors with chaotic attractors. A typical trajectory i these cycles produces a patter i which, at ay give time, oe of the cells is active (i a fixed poit, periodic orbit or chaotic orbit) while the remaiig two are quiescet. Depedig o the couplig stregth, two types of cyclig patters are observed. With relatively weak couplig (i absolute value), we fid a cycle (type I) i which whe a cell becomes active it always selects the same of the two cojugate orbits of Fig., either x > or x <. The actual orbit that is selected depeds o the iitial coditios of the active cell. A example of such cycle coectig fixed poits of () is illustrated i Fig. 3. All subsequet figures related to the first example were created usig the iitial coditios (x,y,z )=(.,.3,.). With stroger couplig (i absolute value), we ow fid a differet type of cycle (type II) i which each cell switches itermittetly betwee cojugate orbits as is also show i Fig. 3. As λ icreases towards λ c, both cycles of Fig. 3 persist as a feature of the global dyamics of the coupled map system. The orbits visited by the cycles, however, chage as the iteral dyamics of each map chages accordig to the bifurcatio diagram of Fig.. For istace, at λ = the fixed poits of the cycle of Fig. 3 udergo a period-doublig bifurcatio that leads to two cojugate stable period-two orbits. Near this bifurcatio poit, we agai fid two types of cycles: oe coectig the same of two cojugate period-two orbits ad oe where switchig betwee cojugate period-two orbits occurs itermittetly. A example of the former type of cycle is show i Fig. 4. As the period-doublig cascade i the iteral y z Fig. 4. Type I heterocliic cycle coectig period-two orbits of f(x, λ) =λx x 3 at λ =. ad couplig stregth α =.7.

6 864 A. Palacios y z (a) y z (b) Fig. 5. (a) Type II heterocliic cycle coectig period-four orbits of f(x, λ) =λx x 3 at λ =.3 ad couplig stregth α =.9. (b) Magificatio of cycle show i (a).

7 Cyclig Chaos i Oe-Dimesioal Coupled Iterated Maps 865 y z Fig. 6. Cyclig chaos coectig chaotic attractors of the coupled cell system (4) with f(x, λ) =λx x 3,atλ =.44 ad couplig stregth α =.7. dyamics of each cell cotiues, cycles ivolvig periodic orbits of higher period also occur robustly. A represetative example with radom switchig betwee cojugate period-four orbits is show i Fig. 5. The cyclig behavior illustrated thus far is also foud at values of the bifurcatio parameter (withi λ λ c ) where the attractor is a chaotic orbit. For istace, at λ =.44 calculatios of Lyapuov expoets (ot show for brevity) cofirm the existece of a asymmetric chaotic attractor fillig parts of the iterval [, ]. Local Z symmetry i the iteral dyamics of each cell forces the existece of a cojugate chaotic attractor withi [, ]. Depedig o the couplig stregth, we also fid at λ =.44 two types of cycles coectig chaotic orbits. With α =.7, for istace, we ca observe i Fig. 6 cyclig chaos that is qualitatively similar to the oe foud by Dellitz et al. [995] usig Chua s cotiuous circuit model. That is, a active cell switches radomly betwee two cojugate chaotic orbits. With a differet value of couplig stregth α =.9, the cyclig chaos persists but active cells ca o loger switch betwee cojugate chaotic orbits. This latter type of cycle (ot show for brevity) does ot seem to appear i [Dellitz et al., 995]. At λ = λ c, the basis of attractio of the two chaotic attractors collide ad the two attractors merge ito oe. For values of λ slightly greater tha λ c, Maeville [99] showed that the iteral dyamics of each map, i the ucoupled system, produces a itermittet orbit that switches betwee the remats of the two attractors. Whe thecellsarecoupled,weobtaiathirdtypeofcyclig chaos (see Fig. 7) i which switchig betwee the two remats of the attractors occurs durig the iterval of activity of each cell. I our secod ad last example, we assume the iteral dyamics of each cell to be govered by the stadard logistic map f(x, λ) =λx( x), λ >, (7) so that the local symmetry is ow determied by the idetity group. We use a couplig fuctio of the form h(x i,x j )= x j /m x i (8) where m is a positive iteger. As i our first

8 866 A. Palacios y z (a) y z (b) Fig. 7. Itermittet cyclig chaos i the coupled cell system (4). (a) Near λ c, with couplig stregth α =., switchig betwee remats of two chaotic attractors occurs frequetly. (b) Away from λ c, with couplig stregth α =., cyclig chaos persists ad the mergece of the two attractors is more uiform over [, ].

9 Cyclig Chaos i Oe-Dimesioal Coupled Iterated Maps y z (a) y z (b) Fig. 8. Cyclig chaos i a coupled cell system with iteral cell dyamics govered by the logistic map f(x, λ) = λx( x). Iitial coditios: (x,y,z )=(.6,.3,.3). (a) Parameters: m =, λ =3.8 adα = 3.7. (b) Parameters: m =, λ =4adα = 3.97.

10 868 A. Palacios example, differet combiatios of parameter values (for m, λ, ad couplig stregth α) yield cyclig behavior betwee orbits of the iteral dyamics of each cell. The wide variety of orbits i the logistic map, which ow govers the iteral dyamics of idividual cells, also leads to a full rage of cycles coectig fixed poits with fixed poits, periodic orbits with periodic orbits, ad chaotic attractors with chaotic attractors. However, due to the lack of cojugate orbits, there are ow oly cycles of type I. For brevity purposes, we illustrate oly, through Fig. 8, cyclig chaos obtaied at two differet values of λ where the logistic map is kow to yield chaotic orbits. Observe that whe λ =3.8, cell oscillatios are cofied to a subiterval of the iterval [, ], while whe λ = 4, the oscillatios fill the etire uit iterval [, ]. This is cosistet with the chaotic attractors of the logistic map. Ackowledgmets I would like to thak Prof. Marty Golubitsky for may stimulatig coversatios ad for his metorship i the area of bifurcatio theory i systems with symmetry. Buoo, P. L., Golubitsky, M. & Palacios, A. [] Heterocliic cycles i rigs of coupled cells, Physica D43, Dellitz, M., Field, M., Golubitsky, M., Ma, J. & Hohma, A. [995] Cyclig chaos, It. J. Bifurcatio ad Chaos 5(4), Dioe, B., Golubitsky, M. & Stewart, I. [996] Coupled cells with iteral symmetry. Part I: Wreath products, Noliearity 9, Field, M. J. [98] Equivariat dyamical systems, Tras. Amer. Math. Soc. 59(), Guckeheimer, J. & Holmes, P. [988] Structurally stable heterocliic cycles, Math.Proc.Camb.Phil.Soc. 3, Maeville, P. [99] Dissipative Structures ad Weak Turbulece (Academic Press). May, R. [976] Simple mathematical models with very complicated dyamics, Nature 6, Melboure, I., Chossat, P. & Golubitsky, M. [989] Heterocliic cycles ivolvig periodic solutios i mode iteractios with O() symmetry, Proc. R. Soc. Ediburgh A3, Palacios, A. [] Heterocliic cycles i coupled systems of differece equatios, J. Diff. Eq. Appl., to appear. Rogers, T. & Whitley, D. C. [983] Chaos i the cubic mappig, Math. Model. 4, 9 5. Refereces Armbruster, D., Guckeheimer, J. & Holmes, P. [988] Heterocliic cycles ad modulated travelig waves i systems with O() symmetry, Physica D9, 57 8.