Intermittent stickiness synchronization
IIntermittent stickiness synchronization
Rafael M. da Silva , ∗ Cesar Manchein , † and Marcus W. Beims , ‡ Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, PR, Brazil Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil and Max-Planck Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany (Dated: March 21, 2019)This work uses the statistical properties of Finite-Time Lyapunov exponents (FTLEs) to in-vestigate the
Intermittent Stickiness Synchronization (ISS) observed in the mixed phase space ofhigh-dimensional Hamiltonian systems. Full Stickiness Synchronization (SS) occurs when all FTLEsfrom a chaotic trajectory tend to zero for arbitrary long time-windows. This behavior is a conse-quence of the sticky motion close to regular structures which live in the high-dimensional phasespace and affects all unstable directions proportionally by the same amount, generating a kind of collective motion . Partial SS occurs when at least one FTLE approaches to zero. Thus, distinctdegrees of partial SS may occur, depending on the values of nonlinearity and coupling parameters,on the dimension of the phase space and on the number of positive FTLEs. Through filtering pro-cedures used to precisely characterize the sticky motion, we are able to compute the algebraic decayexponents of the ISS and to obtain remarkable evidence about the existence of a universal behaviorrelated to the decay of time correlations encoded in such exponents. In addition, we show thateven though the probability to find full SS is small compared to partial SSs, the full SS may appearfor very long times due to the slow algebraic decay of time correlations in mixed phase space. Inthis sense, observations of very late intermittence between chaotic motion and full SSs become rareevents.
PACS numbers: 05.45.Ac,05.45.PqKeywords: Stickiness, finite-time Lyapunov spectrum, Poincaré recurrences, synchronization.
I. INTRODUCTION
Synchronization in high-dimensional Hamiltonian sys-tems has been defined as measure synchronization inRefs. [1, 2]. In these works the authors use models con-sisting of two coupled maps. By starting two distinct ini-tial conditions from the uncoupled system, which lead toa regular dynamics, they observe what happens to themby adding a small coupling between the maps. A kind ofsynchronized (collective) motion appears named measuresynchronization . As the synchronization of dissipativechaotic systems [3, 4], synchronization in generic Hamil-tonian systems is also an interesting issue since such sys-tems present a mixed phase-space dynamics which con-tains a rich variety of behaviors. However, it is impor-tant to mention that the synchronization phenomenonobserved in dissipative systems is not possible in Hamilto-nian system due to the Liouville’s theorem that preventsthe full collapse of the orbits to an invariant manifold,since volume must be preserved in phase space.For symplectic two-dimensional maps the chaotic com-ponent is clearly separated from the regular motion [5, 6].However in higher dimensions the chaotic trajectory mayvisits ergodically the whole phase space but, until thishappens, it may suffer a dynamical trapping (or stickymotion) [7, 8] close to quasi-regular structures. The effectof the sticky motion on the chaotic trajectory can be clas- ∗ rmarques@fisica.ufpr.br † [email protected] ‡ mbeims@fisica.ufpr.br sified in distinct regimes [9, 10], defined by the spectrumof Finite-Time Local Lyapunov Exponents (FTLLEs).When all FTLLEs are positive, the regime is chaotic,when all are zero, we have an ordered regime. In be-tween we have semiordered regimes. Separating the dy-namics in distinct regimes, like a filtering procedure,not only a substantial increase in the characterizationof the sticky motion was achieved [11], but allowed tofind a synchronized-like state, leading to the IntermittentStickiness Synchronization (ISS) discussed in the presentwork. Essentially the ISS is characterized by the inter-mittent behaviour between the chaotic motion and a kindof transient measure synchronization generated by sticki-ness. Such synchronized-like states due to stickiness werealso detected some years ago and classified as commonmotion [12]. A somehow similar analysis allowed to syn-chronize drive and slave coupled standard maps [13]. Inthis case, since the coupling between the two maps isunidirectional, once the synchronized state is reached, itdoes not change along the simulations. Such behaviorchanges when the coupling interaction between the mapsis bidirectional, as considered in the present work, whichuses global (all-to-all) interactions.Since events with long times are associated to times forwhich the trajectory was trapped to the nonhyperboliccomponents of the phase space [8, 14–18], in the presentwork the ISS decay is qualitatively described using thetime decay of the ordered regime in the case of coupledmaps. To mention, other alternatives approaches usingFTLEs [19–25] can be used to characterize the phasespace of conservative systems, with recent applicationsusing large deviation techniques [26–28] and the cumu- a r X i v : . [ n li n . C D ] M a r lants [12, 29] of the FTLEs distribution.Recently, a methodology that uses time-series of lo-cal Lyapunov exponents to define the above mentionedregimes of ordered, semiordered, and totally chaotic mo-tion was proposed, making it possible to improve thecharacterization of stickiness in Hamiltonian with few de-grees of freedom [11] and non-Hamiltonian [30] systems.The present work uses such filtering procedure [11] tocheck precisely the algebraic decay exponents of the ISSin higher-dimensional Hamiltonian systems. This investi-gation is motivated by the few amount of numerical stud-ies related to weakly chaotic properties and consequentlythe long time correlations observed in higher-dimensionalmixed phase spaces of Hamiltonian systems (at least forsmall and moderate number of homogeneously coupledtwo-dimensional maps). We find that only the full SSobeys a power-law decay, while all other partial SSs decayexponentially. Thus, sticky effects from the semiorderedregimes are almost irrelevant for long time ISS decay. Ad-ditionally, the algebraic decay exponent of full SS seemsto be independent of the (i) number of coupled maps(at least for a moderate number of them), and (ii) thecoupling intensities used here. Although it is still underdebate whether such an exponent persists in the weak-coupling regime, our investigation corroborates with theclaim suggested in [27] that predicts the existence of generic decay exponent for time correlations χ ∼ . forHamiltonian systems with few degrees of freedom whichis smaller than what says the conjecture proposed in [16]to predict the existence of an universal decay of Poincarérecurrences γ ≈ . − . (see also [31] for earlier work).The corresponding relationship between these algebraicexponents is given by the well-known equation χ = γ − .The plan of this paper is presented as follows. Sec-tion II presents the coupled maps model used for the simulations. In Sec. III the precise definition of ordered,semiordered, and chaotic regimes is given, which leadsto the definition of the synchronized-like state, togetherwith some numerical examples. While in Sec. IV the ISSdecay is discussed qualitatively, Sec. V summarizes ourconclusions. II. THE COUPLED-MAPS MODEL
Consider the time-discrete composition T ◦ M of inde-pendent one-step iteration of N symplectic -dimensionalmaps M = ( M , . . . , M N ) and a symplectic coupling T = ( T , . . . , T N ) . This constitutes a N -dimensionalHamiltonian system. For our numerical investigation weuse the -dimensional Standard Map (SM): M i (cid:32) p i x i (cid:33) = (cid:32) p i + K i sin(2 πx i ) mod 1 x i + p i + K i sin(2 πx i ) mod 1 (cid:33) , (1)and for the coupling T i (cid:32) p i x i (cid:33) = (cid:32) p i + (cid:80) Nj =1 ξ i,j sin[2 π ( x i − x j )] x i (cid:33) , (2)with ξ i,j = ξ j,i = ξ √ N − (all-to-all coupling). This sys-tem is a typical Hamiltonian benchmark tool with mixedphase space presenting all essential features expected incomplex systems. It was studied in Refs. [15, 32] usingthe Recurrence Time Statistic (RTS) and used in Ref. [11]to propose the classification of Lyapunov regimes for im-proving stickiness characterization. In all numerical sim-ulations we used nonlinearity parameters correspondingto a mixed phase space, namely . ≤ K ≤ . (seeFig. 1 which is discussed next). III. DEFINITION OF REGIMES ANDSTICKINESS SYNCHRONIZATION
The numerical technique uses the FTLLEs spectrum { λ ( ω ) i =1 ...N } computed along a chaotic trajectory during awindow of size ω , where λ ( ω )1 > λ ( ω )2 , . . . , λ ( ω ) N > , andexplores temporal properties in the time series of { λ ( ω ) i } [11]. The window size ω has to be sufficiently small toguarantee a good resolution of the temporal variation ofthe λ ( ω ) i ’s, but sufficiently large in order to have a reliableestimation (see Refs. [19, 24, 25]). The sharp transitionstowards λ ( ω ) i ≈ observed earlier motivates the classifi-cation in regimes of motion [9, 10]. For a system with N degrees of freedom, the trajectory is in a regime oftype S M if it has M local Lyapunov exponents λ ( ω ) i > ε i ,where ε i (cid:28) λ ( ∞ ) i are small thresholds. Thereby, S and S N are ordered and chaotic regimes respectively. Regimes S i with < i < N are called semiordered . For the com-putation of the FTLLEs we use the traditional Benettin’salgorithm [33, 34], which includes the Gram–Schmidt re-orthonormalization procedure. On average, the FTLLEsare in decreasing order. However, inversions of the order( λ ( ω ) i +1 > λ ( ω ) i ) may happen for some times t and we havechosen to impose the order of λ ( ω ) i for all t . A. The uncoupled case: N = 1 To get a better understanding of the involved complex-ity in the dynamics and the behavior of the FTLLEs,Fig. 1 displays the phase-space dynamics for a chaotictrajectory for one uncoupled ( ξ = 0 ) SM together withthe corresponding positive FTLLE λ ( ω )1 (see color bar)for ω = 100 . In this case the Lyapunov spectrum hasonly two Lyapunov exponents which, asymptotically, one Figure 1. (Color online) Phase-space dynamics for the uncoupled case ( ξ = 0 ) using (a)-(d) K = 0 . and (e)-(h) K = 0 . . Inpanels (a) and (e) we used initial conditions, each one iterated × times. In (b) and (f) the colors (see the color bar)represent λ ( ω )1 , for ω = 100 , computed through a trajectory of × iterations. Panels (c), (d) and (g), (h) are magnificationsof the cases K = 0 . and K = 0 . , respectively. is positive and the other one negative. Thus, only tworegimes are observed: (i) the ordered one, if λ ( ω )1 < ε and, (ii) the chaotic one, if λ ( ω )1 > ε . While the upperrow in Fig. 1 presents the K = 0 . case, the lower rowshows results for K = 0 . . For both cases, the phasespace has a large regular island located in the center,surrounded by higher order resonances. In Fig. 1(a) weobserve a -order resonance and in Fig. 1(e) a -order res-onance [better seen in Figs. 1(b) and 1(f), respectively].It is well known that additional higher-order resonances Table I. Values of K i used to couple the standard maps andthe thresholds ε i .Value of K i N = 2 N = 3 N = 4 N = 5 K K K - 0.60 0.63 0.63 K - - 0.62 0.62 K - - - 0.61Threshold ε ε ε - 0.05 0.06 0.07 ε - - 0.04 0.06 ε - - - 0.04 (not visible in the scale of these Figures) live aroundthe island. These islands lead to the dynamical trap-ping which can be stronger, or not, depending on thetopological structure of the islands. Such dependencybecomes better visible when the positive FTLLE λ ( ω )1 iscalculated for the trajectories. This is shown in colors inFigs. 1(b) and 1(f) with some magnifications (see blackboxes) shown respectively in Figs. 1(c), 1(d) and 1(g),1(h). We observe that, when approaching the island bor-ders, the FTLLE decreases, as specified by the color barsin Figs. 1(d), 1(h). A very complex behavior is evidentand only motions very close to the regular islands havesmaller FTLLEs. This suggests that these motions closeto the regular islands will belong to the ordered motion. B. The coupled case: N = 2 A nice visualization of the regimes becomes clear whentwo coupled SM are analyzed in phase space, as shownin Fig. 2. Different colors represent points x t ∈ S M be-longing to regimes S (blue circles), S (red points), and S (green points). These points were computed start-ing a single trajectory in the chaotic sea of the coupledsystem and iterating it times. Table I presents thevalues of K i used in the simulations and the thresholds ε i used to define the regimes of motion. Blue circles arethe points in phase space ( x , p ) and ( x , p ) for which λ ( ω ) i < ε i , for i = 1 , . The red color indicates points forwhich λ ( ω )1 > ε and λ ( ω )2 < ε . Green points are used ifboth FTLLEs are larger than the respective thresholds ε i . We observe in Fig. 2 that by increasing the value ofthe coupling strength ξ , the trajectory penetrates the reg-ular domain from the uncoupled case where there is theisland’s hierarchy, inside which only S and S regimesoccur. Thus, by increasing the coupling force between themaps, the number of points which induce sticky motionincrease. C. The stickiness synchronization
From above results it is easy to verify that for the or-dered regime the position of coupled maps tend to bevery close to the almost regular domains and to eachother. This can be checked more precisely by determin-ing, for N = 2 for example, the distance | x − x | asa function of time. This is shown by the gray color inFig. 3 for two distinct time windows. At a given time,the distance | x − x | suddenly approaches zero, leadingto an approximated synchronization of the positions ofthe coupled SMs. Since these positions are not exactlyequal, we say to have a synchronized-like state. In both Figure 2. (Color online) Phase-space dynamics projected in ( x , p ) [(a), (c) and (e)] and in ( x , p ) [(b), (d) and (f)] fordifferent values of ξ . Figure 3. (Color online) Time series of the spectrum ofFTLLEs { λ ( ω =100) i } , with i = 1 , . . . , N for the map (1)–(2)with N = 2 and ξ = 10 − . In (a) and (b) the thresholds ε = 0 . and ε = 0 . are represented by black dash-dottedand black dotted lines, respectively. The gray line indicatesthe distance | x − x | and shows the synchronization of themaps and in the intervals of time for which the regime S occurs. cases the synchronization occurs only for a finite-timewindow. Surprisingly these time windows match withthose times for which the ordered regime S is present.This can be checked in the same picture, where we plotsimultaneously the two positive FTLLEs λ ( ω ) i as a func-tion of time. Thus, synchronization of the position ofthe maps coincides with the full synchronization of theFTLLEs. For the regime S we observe in Fig. 3(b) thatthe distance | x − x | is more away from zero than thisdistance measured in the regime S , leading to a kind of“weaker” synchronization. In this case we say to have a partial synchronization, since only one FTLLE tends tozero. For S there is no synchronization.The relation between the position synchronization ofthe coupled maps shown above allows us to use the con-cept of stickiness synchronization . We have checked thisrelation for all S regimes along a chaotic trajectory oflength t = 10 . Since all regimes S M with M < N are transient, and there is an intermittent transition be-tween these regimes, we say to have the ISS. Our resultsfor higher-dimensions can also be interpreted as the syn-chronization of FTLLEs. It is in fact a consequence ofthe synchronization of local expansion/contraction ratesalong all unstable/stable direction manifolds.
IV. QUALITATIVE DESCRIPTION OF ISS
Numerical techniques used to characterize the stickymotion can now be used to describe the qualitative be-havior of the ISS decay in time. For this we use theconsecutive time τ M spent by a trajectory in the sameregime S M [11]. During a trajectory of length t L we col-lected a series of τ M and important results are obtainedby analyzing the cumulative distribution of τ M , definedas: P cum ( τ M ) ≡ ∞ (cid:88) τ (cid:48) M = τ M P ( τ (cid:48) M ) . (3)Applying this alternative procedure to obtain the P cum ( τ M ) , we are able to estimate the decay exponentfor the recurrence times. This technique is much moreappropriated to estimate such an exponent when com-pared to the former one, based on the cumulative dis-tribution of Poincaré recurrence times [or (RTS)], sinceit remains unclear how to estimate the time scale overwhich a higher-dimensional system reaches its asymptoticregime under the process of weak Arnold diffusion [5]. Asthis technique is based on a filtering process, we can se-lect the events related to longer correlations and thenobtain decay curves with several decades. In addition,there is no more dependence on the choice of recurrenceset to obtain the RTS [35]. A. The uncoupled case: N = 1 To apply the filtering method we have to specify thethreshold ε and the time window ω . Figures 4(a) and4(b) compare the cumulative distribution P cum ( τ M ) forthe regime S , obtained using ω = 100 and two values of ε , with the RTS P cum ( τ ) , both quantities calculated foruncoupled SMs with two different values of K , specifiedin Fig. 4. For the determination of the RTS (plotted ingray) we have: (i) used a recurrence region in the chaoticcomponent of the phase space delimited by < x < and . < p < − . , and (ii) collected the lapses of time τ spent outside of the recurrence region by a trajectorystarted inside of such predefined box. Straight lines inFig. 4 are consequences of the sticky motion. We real-ize that while the usual RTS presents some oscillationsas a function of τ , leading to difficulties in the precisedecay exponent, the filtering method tends to decreasesuch oscillations, mainly if the threshold ε = 0 . is used.These results show that, for practical implementations, Figure 4. (Color online) Comparison between the filteringmethod and the RTS (gray curves) for the uncoupled case( ξ = 0 ) in (a) for K = 0 . and in (b) for K = 0 . . Theblue and red curves are the cumulative distribution P cum ( τ ) of consecutive times τ in the regime S (normalized for con-venience of scale), obtained for values of τ M , using twodifferent values of threshold ε . The RTS P cum ( τ ) was obtainedfor recurrences. In (c) and (d) we compare P cum ( τ ) fordifferent values of ω . the thresholds can be defined as ε i ≈ . (cid:104) λ ( ω ) i (cid:105) , where (cid:104) . . . (cid:105) denotes the average over t , where t = 1 , . . . , t L .It is important to define how sensitive these results arein relation to the time window ω used to calculate theFTLLEs. For this, we compare P cum ( τ ) obtained using ω = 100 and ε = 0 . [blue curves in Figs. 4(a) and4(b)] with P cum ( τ ) for ω = 50 and ω = 200 , keepingthe threshold ε = 0 . . Figures 4(c) and 4(d) show thiscomparison for the cases K = 0 . and K = 0 . , re-spectively, and the results demonstrate that even thoughthe choice of ω may affect quantitatively the cumulativedistributions P cum ( τ M ) , our conclusions about the alge-braic decay obtained for the regime S are not changedby oscillations around the chosen value ω = 100 [11]. B. The coupled cases: N = 2 , , , We start determining the cumulative distribution P cum ( τ M ) for the N = 2 case for which we have regimes S , S and S . This is shown in Fig. 5(a) for coupling ξ = 10 − and using values of K i and ε i according tothe Table I. It shows that the only power-law decay oc-curs for the S regime. Thus, while full SS occurs for S regimes with a power-law decay of the P cum ( τ M ) , allother regimes have a chaotic component leading to an Figure 5. (Color online) (a) The cumulative distribution P cum ( τ M ) of times τ M for the regime S M for the map (1)–(2)with N = 2 and ξ = 10 − , obtained using × values of τ M . The values of K i and ε i used are indicated in the Table I.(b) Comparison between our method and the analysis basedon RTS for the case N = 2 and ξ = 10 − . The result obtainedcombining the curves M + M (normalized for convenienceof scale) is equivalent to cumulative distribution P cum ( τ ) , ob-tained for recurrences. In (c) we compare P cum ( τ ) fordifferent values of ω . exponential decay. This indicates that only full FTLLEssynchronized states tend to occur for consecutive verylong times, even though with small probability.Looking at the distributions P cum ( τ M ) in Fig. 5(a), weobserve for the semiordered regime M = 1 an exponentialtail after an initial power-law decay with scaling β ≈ . [15]. When the full SS takes place ( M = 0 ), P cum ( τ ) ∝ τ − γ , with γ = 1 . . As shown in Fig. 5(b), this scaling iscompatible with the result obtained using RTS. However,the cumulative distribution P cum ( τ ) provides a bettercharacterization of algebraic decay (over several orders ofmagnitudes), which is essential when dealing with high-dimensional systems (which may contain different pre-asymptotic regimes) and for an accurate estimation ofthe stickiness exponent γ . In Fig. 5(c) we show that P cum ( τ ) for the coupled case remains (qualitatively) thesame for different values of ω around ω = 100 .Another very interesting quantity to be studied is theresidence time P ( S M ) in each regime as a function of thecoupling strength, defined by P ( S M ) = 1 β t L (cid:88) t =0 δ t ∈ S M , (4)where β = t L /ω is the factor of normalization. In Eq.(4), δ t ∈ S M = 1 if in time t the trajectory is in the regime Figure 6. (Color online) Residence time in each regime S M using ω = 100 and (a) N = 2 , (b) N = 3 , (c) N = 4 and(d) N = 5 . For each value of ξ , P ( S M ) was computed usinga trajectory with length t L = 10 . The values of K i and ε i used in each case can be found in Table I. S M and δ t ∈ S M = 0 otherwise. The P ( S M ) is shown inFig. 6(a) for N = 2 , in Fig. 6(b) for N = 3 , in Fig. 6(c)for N = 4 and in Fig. 6(d) for N = 5 . For smallercouplings ( ξ (cid:46) × − ) the residence time decreaseswith M , namely P ( S N ) > P ( S N − ) > . . . > P ( S M ) >. . . > P ( S ) > P ( S ) . This means that the probability tofind the ordered regimes ( M = 0 ) is much smaller whencompared to semiordered regimes ( M = 1 ) and so on. Forlarger couplings ξ > − , the probability to find orderand semiordered regimes has roughly the same values andtend to decrease until zero. Only the probabilities of fullychaotic regimes S N remain for larger values ξ .From Figs. 5 and 6 we conclude that even though theprobability to find the full SS is small compared to thepartial SS, it can occur for very long times due to thepower-decay found for P cum ( τ ) . In addition we mentionthat, in distinction to usual synchronization found in dis-sipative systems, here the ISS tends to decrease for largercoupling between the maps [36]. C. Characterizing the decay of synchronization:the ordered regime
The next step is to precisely quantify the ISS decay fordistinct values of coupling ξ between N = 2 , , and SMs. For this we used only the regime S , which is di-rectly related with the full synchronization between thepositions x i . As demonstrated in Figs. 4(a), 4(b) and5(b), the decay of P cum ( τ ) provides an amazing char-acterization of the sticky motion and allows obtainingaccurately the exponent γ , so that the RTS analysis be-comes needless. The results of this study are shown inFig. 7(a) for N = 2 and in Fig. 7(b) for N = 3 , usingdistinct values of ξ , as specified in the Figure. The blackdotted line is the average over all couplings of each caseand fitting this curve we obtain a power-law decay withwell defined exponent γ ≈ . , observed for 6 decades.For ξ = 10 − , long-term trappings tend to disappear. Itis worth to mention that the disappearance of the long-term sticky motion manifest itself in the increasing lack Figure 7. (Color online) The cumulative distribution P cum ( τ ) of consecutive times τ in the regime S using different valuesof coupling ξ for (a) N = 2 with × values of τ M and(b) N = 3 with × values of τ M . The K i and ε i used ineach case are indicated in the Table I. Figure 8. (Color online) The cumulative distribution P cum ( τ ) of consecutive times τ in the regime S using different valuesof coupling ξ for (a) N = 4 and (b) N = 5 , both cases col-lecting × values of τ M . The K i and ε i used in each caseare indicated in the Table I. of data for S as the coupling increases.To finish we would like to present results for N = 4 and N = 5 . Figure 8 displays the cumulative distribu-tion P cum ( τ ) for the regime S and for distinct couplingvalues, specified in the Figure. We observe that for valuesof ξ ≤ − the exponent approaches γ ≈ . for almost decades in Fig. 8(a), and γ ≈ . in Fig. 8(b), valuesobtained fitting the black dotted line that is the aver-age over all couplings. The amount of long-term stickymotion decreases for ξ > − in both cases. Again,this manifests itself in the increasing lack of data for S .However, since we still have at least three decades ofpower-law behavior, it can still be characterized as stickymotion leading to the full SS. V. CONCLUSIONS
This work analysis qualitatively the Intermittent Stick-iness Synchronization (ISS) decay in high-dimensionalgeneric Hamiltonian systems. Such synchronization isgenerated by the regular structures on the chaotic tra-jectory, and can also be interpreted as the synchroniza-tion of FTLLEs. It is a synchronization of local ex-pansion/contraction rates along all unstable/stable di-rection manifolds. We connect the intermittent mo-tion between ordered, semiordered and chaotic dynamicalregimes with, respectively, the full, partial, and absenceof synchronization generated by stickiness. By using thecumulative distribution of the consecutive times τ M spentin each regime S M , we demonstrate the ability of the re-cent proposed filtering procedure [11] to precisely char-acterize the ISS decay generated by the sticky motion.We also show that even though the residence time in thefull SS state is small compared to the residence times inthe partial SS states, it may occur for consecutive verylong times due to the power-decay of the P cum ( τ ) .In addition, our numerical results demonstrate that thealgebraic decay exponent tends to γ ≈ . for higher-dimensional systems. This is in completely agreementwith the estimated decay exponent of time correlations χ ≈ . (both exponents are related by the well-knowrelationship χ = γ − ) obtained in [27] for N = 2 , symplectic maps interacting through a nearest-neighborcoupling scheme. The estimated decay exponents inthese two works were obtained through three differentapproaches and are somehow smaller than recent esti-mates [16] (such observations suggest an universality con-jecture, at least for a moderate number of coupled Hamil-tonian maps).Future investigations can analyze a possible relationbetween the ISS observed here and the hydrodynamicmodes found in many body systems [37]. They presentslow, long-wavelength behavior in the tangent space dy-namics. Besides, the properties of the covariant Lya-punov vectors [38–41] at the full SS might also be promis- ing from the theoretical point of view and applications. ACKNOWLEDGMENTS
This study was financed in part by the Coordenaçãode Aperfeiçoamento de Pessoal de Nível Superior - Brasil(CAPES) - Finance Code 001. C. M. and M. W. B. thankCNPq (Brazil) for financial support. The authors alsoacknowledge computational support from Prof. C. M. deCarvalho at LFTC-DFis-UFPR (Brazil). [1] A. Hampton and D. H. Zanette, Phys. Rev. Lett. ,2179 (1999).[2] U. E. Vincent, New J. Phys. , 209 (2005).[3] S. Boccaletti, J. Kurths, G. Osipov, D. Valladares, andC. Zhou, Phys. Rep. , 1 (2002).[4] L. M. Pecora and T. L. Carroll, Chaos , 097611 (2015).[5] A. J. Lichtenberg and M. A. Lieberman,Regular and Chaotic Dynamics (Springer-Verlag, NewYork, 1992).[6] G. M. Zaslavsky, Hamiltonian Chaos & Fractional Dynamics(Oxford University Press, New York, 2008).[7] B. V. Chirikov and D. L. Shepelyansky, Physica D ,395 (1984).[8] R. Artuso, Physica D , 68 (1999).[9] G. Contopoulos, L. Galgani, and A. Giorgilli, Phys. Rev.A , 1183 (1978).[10] A. Malagoli, G. Paladin, and A. Vulpiani, Phys. Rev. A , 1550 (1986).[11] R. M. da Silva, C. Manchein, M. W. Beims, and E. G.Altmann, Phys. Rev. E , 062907 (2015).[12] C. Manchein, M. W. Beims, and J. M. Rost, Chaos ,033137 (2012).[13] S. Mahata, S. Das, and N. Gupte, Phys. Rev. E ,062212 (2016).[14] G. Cristadoro and R. Ketzmerick, Phys. Rev. Lett. ,184101 (2008).[15] E. G. Altmann and H. Kantz, Europhys. Lett. , 10008(2007).[16] D. L. Shepelyansky, Phys. Rev. E , 055202 (2010).[17] S. Lange, M. Richter, F. Onken, A. Bäcker, and R. Ket-zmerick, Chaos , 24 (2014).[18] S. Lange, A. Bäcker, and R. Ketzmerick, Eur. Phys. Lett , 116 (2016).[19] H. Kantz and P. Grassberger, Phys. Lett. A , 437(1987).[20] M. Falcioni, U. Marini Betollo Marconi, and A. Vulpiani,Phys. Rev. A , 2263 (1991). [21] T. Konishi and K. Kaneko, J. Phys. A: Math. Gen. ,6283 (1992).[22] K. Kaneko and T. Konishi, Physica D , 146 (1994).[23] S. Tomsovic and A. Lakshminarayan, Phys. Rev. E ,036207 (2007).[24] J. D. Szezech, S. R. Lopes, and R. L. Viana, Phys. Lett.A , 394 (2005).[25] M. Harle and U. Feudel, Chaos, Solitons & Fractals ,130 (2007).[26] R. Artuso and C. Manchein, Phys. Rev. E , 036210(2009).[27] T. M. de Oliveira, R. Artuso, and C. Manchein (2018),submitted to publication.[28] T. Laffargue, K.-D. N. T. Lam, J. Kurchan, andJ. Tailleur, J. Phys. A , 254002 (2013).[29] C. Manchein, M. W. Beims, and J. M. Rost, Physica A , 186 (2014).[30] R. M. da Silva, M. W. Beims, and C. Manchein, Phys.Rev. E , 022921 (2015).[31] M. Ding, T. Bountis, and E. Ott, Phys. Lett. A , 395(1990).[32] E. G. Altmann, Ph.D. thesis, Max Planck Institut fürPhysik Komplexer Systeme (2007).[33] G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn,Meccanica , 09 (1980).[34] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,Physica D , 285 (1985).[35] M. Sala, R. Artuso, and C. Manchein, Phys. Rev. E , 94 (2016).[36] J. F. Heagy, L. M. Pecora, and T. L. Carroll, Phys. Rev.Lett. , 4185 (1995).[37] C. Dellago, H. A. Posch, and W. G. Hoover, Phys. Rev. E , 1485 (1996).[38] F. Ginelli, P. Poggi, A. Turchi, H. Chaté, R. Livi, andA. Politi, Phys. Rev. Lett. , 130601 (2007).[39] A. Norwood and et al., J. Phys. A (2013).[40] M. W. Beims and J. A. C. Gallas, Sci. Rep. (2016).[41] M. W. Beims and J. A. C. Gallas, Chaos20