A. Popovich

B. R. Hunt

C. Mira

E. Mosekilde

E. Ott

H. Fujisaka

J. C. Alexander

J. C. Sommerer

J. F. Heagy

J. Milnor

N. F. Rulkov

N. Platt

P. Ashwin

R. H. Abraham

S. V. Gonchenko

V. L. Maistrenko

Y.-C. Lai

Yu. L. Maistrenko

When two identical chaotic oscillators interact, one or more intervals of coupling parameters generally exist in which the synchronized state is weakly stable, and its basin of attraction is riddled with holes that are repelled from it. The paper discusses the role of the absorbing area for the emergence of local vs global riddling and for controlling the dynamics, once synchronization breaks down

Maistrenko, Yu.L.

Maistrenko, V.L.

Popovich, A.

Mosekilde, Erik

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Role of the Absorbing Area in Chaotic SynchronizationMaistrenko, Yu.L.; Maistrenko, V.L.; Popovich, A.; Mosekilde, ErikPublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.80.1638Publication date:1998Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Maistrenko, Y. L., Maistrenko, V. L., Popovich, A., & Mosekilde, E. (1998). Role of the Absorbing Area in ChaoticSynchronization. Physical Review Letters, 80(8), 1638-1641. DOI: 10.1103/PhysRevLett.80.1638VOLUME 80, NUMBER 8 P HY S I CA L REV I EW LE T T ER S 23 FEBRUARY 1998Role of the Absorbing Area in Chaotic SynchronizationYu. L. Maistrenko,1 V.L. Maistrenko,1 A. Popovich,1 and E. Mosekilde21Institute of Mathematics, National Academy of Sciences, Kiev, Ukraine 2526012Department of Physics, Technical University of Denmark, 2800 Lyngby, Denmark(Received 14 July 1997)When two identical chaotic oscillators interact, one or more intervals of coupling parameters generallyexist in which the synchronized state is weakly stable and its basin of attraction is riddled withholes that are repelled from it. Here, we discuss the role of the absorbing area for the emergenceof local vs global riddling and for controlling the dynamics, once the synchronization breaks down.[S0031-9007(98)05306-X]PACS numbers: 05.45.+bFujisaka and Yamada [1] have shown how two identicalchaotic oscillators under variation of the coupling parame-ter can attain a state of chaotic synchronization in whichthe motion of the coupled system is restricted to an invari-ant subspace of total phase space. For two coupled one-dimensional maps, for instance, the synchronized motionis one dimensional and occurs along the main diagonal inphase space. A variety of applications of this phenome-non are presently being investigated, particularly in con-nection with electronic communication and monitoring ofdynamical systems [2].An interesting problem concerns the starting conditionsfor which synchronization can be achieved. Recentstudies of chaotic dynamics restricted to low-dimensionalmanifolds have led to the discovery of a variety of newphenomena, including on-off intermittency [3] and riddledbasins of attraction [4,5].If the transverse Lyapunov exponent l' is negative, thesynchronous state, on the average, attracts points from itstwo-dimensional neighborhood [4]. In general, however,there will be intervals of the coupling parameter for whichl' , 0 while, at the same time, certain orbits embeddedin the chaotic state are transversely unstable. The basinof attraction may then be riddled with tongues in whichthe trajectories are repelled from the low-dimensionalmanifold [6,7]. The synchronized state will attract a set ofpoints of positive Lebesgue measure. Arbitrarily close toany such point, however, one can find a positive measureset of points that are repelled from this state, and the basinof attraction is locally a fat fractal.For the synchronized state to be an attractor in thetopological sense, all orbits on the chaotic set must betransversely stable. The riddling bifurcation (also referredto as the bubbling transition [5]) in which the first orbitembedded in the chaotic attractor loses its transversestability has recently been investigated in detail by Laiet al. [8]. They suggest that the bifurcation takes placeas two repellers located symmetrically on either sideof the invariant subspace approach the chaotic attractorand collide with a saddle embedded in this attractor.Alternatively, a point cycle embedded in the synchronizedstate may lose its transverse stability in a period-doublingbifurcation [9], and both types of local bifurcation may beeither soft or hard.However, destabilization of orbits embedded in thechaotic state is not sufficient for directly observableriddling to occur. This will depend on the global dy-namics of the system. Having left the locally repellingregions in the neighborhood of the invariant manifold, thetrajectories may wander around in phase space withoutever approaching another attractor (or escaping to infin-ity). Sooner or later, most of them will return to theneighborhood of the invariant manifold. Some may againbe mapped into the repelling tongues while others will beattracted by the manifold. This gives rise to a dynamicswith temporal bursting (attractor bubbling) and very longtransients. However, at the end, almost all initial condi-tions close to the invariant manifold will end up in thesynchronized state.Denoting the synchronized chaotic state by A, its dneighborhood by UdsAd, and its basin of attraction byB sAd, we may then distinguish between the following twotypes of riddling: (i) BsAd is globally riddled if A attractsa positive Lebesgue measure set of points from UdsAd, butnot the full measure. The remaining points, i.e., the pointsof UdsAd not belonging to B sAd, go to some other state.In this case, A is referred to as a weak Milnor attractor[10]. (ii) B sAd is locally riddled [7] if A attracts almost allpoints in UdsAd; i.e., mesfB sAd > UdsAdg ­ mesUdsAd.A is then said to be a Milnor attractor, and BsAd con-tains no observable holes belonging to the basin of an-other attractor. mesUdsAd denotes the Lebesgue measureof UdsAd.The purpose of the present Letter is to show how theemergence of locally riddled basins of attraction dependson the existence of a so-called absorbing area AsAdinside the basin of attraction B sAd. If such an areaexists, the nonlinearities of the system restrain trajectoriesstarted within AsAd from coming into contact with thebasin boundary. Variation of a parameter that causesthe absorbing area to increase in size (or the basin ofattraction to shrink) may then lead to a crisis where the1638 0031-9007y98y80(8)y1638(4)$15.00 © 1998 The American Physical SocietyVOLUME 80, NUMBER 8 P HY S I CA L REV I EW LE T T ER S 23 FEBRUARY 1998two boundaries touch one another. After this crisis, thebasin of attraction will be globally riddled.We shall also show that the existence of an absorbingarea within the basin of attraction is essential for whathappens in the blowout bifurcation, where l' becomespositive. In the presence of an absorbing area, theblowout bifurcation will be gradual. The symmetry ofthe synchronized state will break, and the attractor willspread into the two-dimensional phase space with amore or less pronounced peak of the invariant densitynear the invariant manifold. This corresponds to a so-called nonhysteric blowout bifurcation [5]. Without therestraining borders of the absorbing area, the blowoutbifurcation is similar to an explosion, leading to the abruptcollapse of the synchronized state (hysteric blowout).Let us consider a system of two coupled logistic mapsof the formFa,« :(xy)!(fasxd 1 «sy 2 xdfasyd 1 «sx 2 yd), (1)where fa : x ! axs1 2 xd, x [ R1, a [ f0, 4g is theone-dimensional logistic map, and « [ f22, 2g is thecoupling parameter.In order to delineate the regions in parameter space inwhich the synchronized state is asymptotically stable, wehave determined the regions of transverse stability for themain low-periodic cycles. For an N-periodic synchronousstate gN ­ hx1, x2, . . . , xN j, the criterion for transversestability isNYn­1jf 0asxnd 2 2«j , 1 . (2)Considering, for instance, the fixed point Psx0, x0d withx0 ­ 1 2 1ya, the condition (2) gives the stability region2sa 2 1dy2 , « , 2sa 2 3dy2. The slanting line de-noted fixed point in the middle of Fig. 1 represents thelower edge of this zone. The upper boundary falls out-side the range of coupling strengths considered in the fig-ure. Hence, the fixed point is transversely stable to theright of the fixed point line.The period-2 cycle arises at a ­ 3 and hereafter alter-nates between x1,2 ­ fa 1 1 6psa 1 1d sa 2 3dgy2a.The condition for transverse stability, in this case, givestwo intervals for « of which the interval to the left of« ­ 2 12 is determined by212 f1 1psa 1 1d sa 2 3d 1 1g , «, 212 f1 1psa 1 1d sa 2 3d 2 1g . (3)The borderlines of this interval are represented by thecurves denoted period 2 in Fig. 1. Similarly, the curvesdenoted period 4, period 6, and period 8 delineate theregions of transverse stability for these cycles. Withthe values of a considered in the figure, period-3 cycleshave not yet appeared in the individual map. Hence,synchronized behavior with this periodicity cannot occur.FIG. 1. Riddling bifurcation curves delineating the parameterregion of absolute stability.Cycles of higher periodicity are not likely to be first tobecome unstable [11].The logistic map is known to exhibit a positive mea-sure Cantor set of parameter values a [ f3, 4g for whicha chaotic attractor exists. In particular, this is the casefor a ­ a0 ­ 3.678 573 510 428 32 . . . , where the fixedpoint x0 ­ 1 2 1ya undergoes its first homoclinic bifur-cation and for a ­ a1 ­ 3.592 572 184 106 97 . . . , wherethe period-2 cycle g2 ­ hx1, x2j undergoes the first homo-clinic bifurcation. At a ­ a0, fasxd displays a one-bandchaotic attractor G ­ G0, which consists of two subbandshaving contact in x0. Similarly, for a ­ a1, fasxd pro-duces a two-band chaotic attractor G ­ G1, which con-sists of four subbands having pairwise contact in theperiod-2 points x1 and x2.G0 has an absolutely continuous invariant measure,and almost all trajectories on the synchronized chaoticset A0 ­ hx ­ y [ G0j will exhibit the same Lyapunovexponent. Figure 2 shows the variation of the transverseLyapunov exponentl' ­ limN!‘1NNXn­1ln j f 0asxnd 2 2«j , (4)with « for a ­ a0. Here, hxn ­ fna sxdj‘n­1 is a typicalitinerary of fasxd. The transverse Lyapunov exponentis negative from « > 21.54 to « > 20.48, and, inthis interval, the synchronous state is at least weaklyattracting in the Milnor sense [10]. The points wherel' changes sign are referred to as blowout bifurcations.The region of strong stability is restricted to the intervalbetween « > 21.31, where the period-6 orbit becomestransversely unstable in a subcritical period-doublingbifurcation, and « > 21.24, where the period-2 orbit1639VOLUME 80, NUMBER 8 P HY S I CA L REV I EW LE T T ER S 23 FEBRUARY 1998FIG. 2. Transverse Lyapunov exponent as a function of thecoupling parameter.becomes transversely unstable in a supercritical perioddoubling.An investigation [9] of the form of the basin ofattraction for different values of the coupling parametershows that to the left of the range of strong stability wehave an interval of global riddling with a dense set ofrepelling tongues leading to divergent orbits. Immediatelyto the right of this range, on the other hand, whilethe basin of attraction exhibits a fractal boundary, onecannot observe tongues emanating from the synchronizedstate and leading the trajectories to diverge. Here, theglobal dynamics of Fa,« is such that points that arerepelled from the synchronized state never reach the basinboundary. The trajectories are restricted to move withinthe absorbing area, and this area falls completely withinthe basin boundary. Hence, we observe the phenomenonof local riddling.Global analyses of noninvertible maps use the methodof critical curves [12]. Similar to the manner in which theiterates of the critical point(s) for a one-dimensional maprestrict the regions of phase space in which the stationarytrajectory can move, the successive images of the criticalcurves restrain the dynamics of our two-dimensional mapFa,«. Together, these images define the absorbing areawith the properties (i) that once a trajectory has enteredthis area, it can never leave again, and (ii) all points fromthe neighborhood will be mapped into the absorbing areain a finite number of iterations.The critical curves L0 are defined as the locus ofjDFj ­ 0, with DF denoting the Jacobian matrix of Fa,«.For our two-coupled logistic maps, we haveL0 ­‰sx, yd [ R2 : y ­2sa 2 «dx 2 sa 2 2«d4ax 2 2sa 2 «d¾.(5)The boundary ›A of the absorbing area may be con-structed following the procedure described by Abrahamet al. [13]: If L ­ A > L0, then for a suitable inte-ger m,›A ­m[k­1Fka,«sLd . (6)Figure 3 shows the results of such a construction fora ­ a0 and « ­ 21.234. Bounded by the curve sectionsLm, m ­ 1, 2, . . . , 8, the absorbing area clearly falls in-side the basin boundary ›B , as represented by the fractalcurve. As long as no other attractor exists inside the ab-sorbing area, only local riddling will arise. As « is furtherincreased, a crisis occurs (approximately at « ­ 21.2) inwhich the border of the absorbing area comes into contactwith the basin boundary. Hereafter, the riddling is globaluntil, for « > 20.48, the blowout bifurcation occurs, andthe synchronized attractor instantaneously disappears.When the riddling bifurcation is supercritical, as inthe present case, yet another invariant region comes intoplay. Immediately after the transverse period-doublingat « ­ 21.237, almost all trajectories repelled from thesynchronized state will be confined to a so-called mixedabsorbing area [14] by the unstable manifold of theperiod-4 saddle cycle. This mixed absorbing area isshown crosshatched in Fig. 3.For a ­ a1, one can find an interval of « in which theorder of bifurcations is different. The blowout bifurca-tion here occurs while an absorbing area still exists in-side the basin of attraction. In the whole interval fromthe riddling bifurcation to the blowout bifurcation, wethen observe a locally riddled basin of attraction. At thesame time, the blowout bifurcation takes a very differ-ent form. Instead of a sudden explosion, the bifurcationmanifests itself as a gradual transition from the synchro-nized chaotic attractor A1 ­ hx ­ y [ G1j to an approxi-mately two-dimensional attractor A01, which is boundedby the absorbing area. This is demonstrated in Fig. 4 for« ­ 21.03.FIG. 3. The absorbing area falls inside the basin boundary.Note also the so-called mixed absorbing area inside AsAd.1640VOLUME 80, NUMBER 8 P HY S I CA L REV I EW LE T T ER S 23 FEBRUARY 1998FIG. 4. The blowout bifurcation (at « > 21.04) leads to anattractor in two-dimensional phase space.Just after the blowout, the probability for the systemto move in the neighborhood of the synchronized stateis large. From time to time, significant bursts may beobserved in which the trajectory moves far out in phasespace without, however, coming into contact with thebasin boundary. This is a typical example of on-offintermittency [3].We conclude that, besides the riddling bifurcation (at«sad) and the blowout bifurcation (at «sbd), the contactbifurcation (or crisis at «scd) in which the boundary of theabsorbing area touches the basin boundary is importantin accounting for the phenomena observed in connectionwith the breakdown of chaotic synchronization. If «scdoccurs before «sbd, the system will display local riddlingbetween «sad and «scd, and global riddling between «scd and«sbd. In this case, the blowout bifurcation takes the formof an abrupt explosion. On the other hand, if «scd occursafter «sbd, there is no global riddling, and the blowoutbifurcation is gradual.A more complicated scenario can arise if other at-tracting states exist inside the absorbing area [9]. Theproblem of the existence of such attractors is extremelycomplicated from the view of the mathematical theory ofdynamical systems. It is associated with the possible ex-istence of parameter intervals with nonremovable homo-clinic tangencies [15].We thank T. Kapitaniak, L. Gardini, G.-I. Bischi,and C. Mira for a number of illuminating discussions.J. Laugesen is acknowledged for assistance with thenumerical computations.[1] H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32(1983).[2] N. F. Rulkov, Chaos 6, 262 (1996).[3] N. Platt, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett.70, 279 (1993).[4] J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Int. J.Bifurcation Chaos Appl. Sci. Eng. 2, 795 (1992).[5] J. C. Sommerer and E. Ott, Nature (London) 365, 136(1993); E. Ott and J. C. Sommerer, Phys. Lett. A 188,39 (1994).[6] J. F. Heagy, T. L. Carroll, and L.M. Pecora, Phys. Rev.Lett. 73, 3528 (1994).[7] P. Ashwin, J. Buescu, and I. Stewart, Nonlinearity 9, 703(1996).[8] Y.-C. Lai, C. Grebogi, J. A. Yorke, and S. C. Venkatara-mani, Phys. Rev. Lett. 77, 55 (1996).[9] Yu. L. Maistrenko, V. L. Maistrenko, A. Popovich, andE. Mosekilde (to be published).[10] J. Milnor, Commun. Math. Phys. 99, 177 (1985).[11] B. R. Hunt and E. Ott, Phys. Rev. Lett. 76, 2254 (1996).[12] C. Mira, L. Gardini, A. Barugola, and J.-C. Cathala,Chaotic Dynamics in Two-Dimensional NoninvertibleMaps (World Scientific, Singapore, 1996).[13] R. H. Abraham, L. Gardini, and C. Mira, Chaos inDiscrete Dynamical Systems (Springer-Verlag, Berlin,1997).[14] C. Mira, C. Rauzy, Yu. Maistrenko, and I. Sushko, Int. J.Bifurcation Chaos Appl. Sci. Eng. 6, 2299 (1996).[15] S. V. Gonchenko, L. P. Shil’nikov, and D.V. Turaev,Chaos 6, 15 (1996).1641

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