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delayed dynamical systems ( ddss ) serve to model diverse phenomena in physics @xcite ( prominently in optics ) , but also in other fields @xcite such as engineering , biology , climatology or ecology . more than one decade ago @xcite it was found that delayed systems with one constant delay can be studied like extended systems , and they present not only analogies but equivalent phenomena such as pattern instabilities @xcite and high - dimensional chaos @xcite . chaos in ddss is considered as a particular type of spatio - temporal chaos for which the delay plays the role of the system size . there are a number of studies concerning the tangent dynamics of systems with time delay . previous works mainly focused on the lyapunov exponents ( les ) and properties obtained from them ( dimension , entropy , ... )  @xcite . less is known about the associated tangent space directions , in particular , there are no studies analyzing the structure of lyapunov vectors ( lvs ) apart from the main one ( pointing along the most unstable direction ) . correlations of the main lv were only recently addressed  @xcite taking advantage of the generic mapping of a dds into an extended system , but with contradicting results . when considering lvs other than the main one , one must distinguish among different vector types . the so - called characteristic lvs  @xcite constitute the only intrinsic set of vectors that is univocally defined and is covariant with the dynamics . however , their numerical computation is difficult in high - dimensional systems . we adapt to ddss one of the methods to compute characteristic lvs @xcite . in this paper we report on the generic properties that lvs exhibit in ddss in the large delay limit . our numerical and theoretical results indicate that the lvs ( the main one and the others ) behave in qualitative and quantitative terms like in one - dimensional dissipative systems with spatiotemporal chaos . ddss may exhibit chaos even in the simplest situation in which the main variable is a scalar and its evolution is determined by a delay - differential equation with one constant delay : @xmath0 where @xmath1 . in a lyapunov analysis we are interested in the evolution of infinitesimal perturbations @xmath2 : @xmath3 which is also a delayed equation with @xmath4 , and where @xmath5 and @xmath6 are functions : @xmath7 , @xmath8 . if the dds is chaotic an initial random perturbation becomes after some time aligned with the main lv ( reaching a stationary state in a statistical sense ) , and the average exponential growth rate is the largest le of the system @xmath9 . numerically , this is simply achieved by integrating eqs . ( [ dds ] ) and ( [ tangent ] ) for long enough times . for non - leading lvs , more involved algorithms are needed ( see below ) . we focus our study on systems in which delayed and non - delayed terms are separated : @xmath10 . our results are expected to be generic for this class of models . in our study , we have assumed the non - delayed part is linear that did not revealed relevant differences . ] , @xmath11 , and all nonlinearities appear in the retarded component @xmath12 . many important time - delayed systems , including the mackey - glass ( mg )  @xcite and ikeda models  @xcite , and optical delayed feedback systems  @xcite , can be expressed in this mathematical form . in particular , we have studied numerically different nonlinear functions : the mackey - glass ( mg ) , @xmath13 ,  @xcite ; and the nonlinear function @xmath14 that appears in some optical cryptosystems with delayed feedback  @xcite . almost identical results in qualitative and quantitative terms are obtained for these two systems . therefore , for the sake of brevity , we choose to present the results only for the mg model , also used in @xcite .    in our simulations we have integrated numerically eq . ( [ dds ] ) using a third - order adams - bashforth - moulton predictor - corrector method  @xcite , while linear equations , _ e.g. _  ( [ tangent ] ) , have been integrated using an euler method with the non - delayed part integrated semi - implicitly . the parameters we used for the mg model are @xmath15 , @xmath16 and the time step was @xmath17 . in a numerical integration of a time - delayed system  @xcite , the temporal discretization makes the system finite - dimensional ( with @xmath18 degrees of freedom ) but this has no effect in the results whatsoever since an increase of time resolution ( _ i.e. _  decreasing @xmath19 ) does not modify the largest les ( and their associated vectors ) .    in several previous works @xcite it was found useful to map the dds into an equivalent spatially extended dynamical system of `` size '' @xmath20 that evolves at discrete times @xmath21 as follows . we express the continuous time as @xmath22 where @xmath23 is the `` spatial '' position and @xmath24 is the discrete time . this spatial representation can be applied to both , the state of the system , @xmath25 , and the perturbations , @xmath26 . the benefit of this spatial representation is that one can analyze the dds with the tools available for spatially extended dynamical systems . we shall use the spatial representation of delayed systems here to analyze lvs and analyze the existence of long - range correlations as well as some other dynamical properties . there is some degree of confusion in the literature regarding the definition and computation of lvs . many authors think of lvs as the orthonormal frame of vectors that results as a byproduct of computing the les with the algorithm of benettin et al . these vectors are usually called gram - schmidt or _ backward _ lvs  @xcite and they span the subspaces in tangent space that , at present time @xmath27 , have grown at exponential rates @xmath28 since the remote past . backward lvs @xmath29 depend on the particular scalar product adopted for the gram - schmidt orthogonalization in benettin s algorithm ( though the subspaces they span are genuine ) . this and other undesired properties render the backward lvs unsuited to analyze problems like extensivity or hyperbolicity questions . ruelle and eckmann  @xcite noticed time ago that one can define a different set of lvs , the so - called _ characteristic _ @xcite lvs @xmath30 , that are intrinsic to the dynamics and are univocally defined ( _ i.e. _  independent of how the scalar product is defined ) . each of these vectors is covariant with the dynamics , and the corresponding @xmath31-th le is indeed recovered by either a forward or backward integration of an infinitesimal perturbation initially aligned with the @xmath31-th characteristic lv . actually , the perturbation will be aligned with the lv at all times ( hence the covariance ) . however for backward lvs ( other than the main one , because @xmath32 ) the corresponding les are only recovered integrating backward . until very recently , no efficient numerical algorithms were available to compute characteristic lvs in large extended systems .    in this paper we calculate the set of characteristic lvs by adapting to ddss the method recently introduced by wolfe and samelson  @xcite . in order to find the @xmath31th characteristic lv one has to compute the first @xmath33 forward lvs in addition to the first @xmath31 backward lvs . forward lvs @xmath34 are obtained like backward lvs but integrating the perturbations backward in time ( from the remote future to the present time @xmath27 ) . they have to be calculated using the transpose jacobian matrix so that les are obtained with the usual ordering @xcite : @xmath35 . for our time - delayed system this procedure encompasses to integrate : @xmath36 the diagonal coefficient @xmath37 is identical to that in eq . ( [ tangent ] ) . the off - diagonal term has a curious structure stemming from transposing a jacobian matrix with nonzero elements @xmath20 temporal units below the main diagonal . thus , in eq . ( [ tangentb ] ) , we have @xmath38 . notice that , as the integration runs backward in time ( @xmath39 ) , the minus sign in the left hand side is canceled . as occurs with backward lvs , periodic gram - schmidt orthonormalizations are needed to avoid the collapse of all perturbations along the most unstable direction . note also that , due to the delay , eq . ( [ dds ] ) can not be integrated backward , and hence it is necessary to store the trajectory @xmath40 at every time step in the computer to be used in the computation of  ( [ tangentb ] ) . once both backward lvs and forward lvs have been computed the characteristic lvs are easily calculated following the prescriptions in ref . one delicate point in delayed systems is that one must make sure that both forward and backward lv sets correspond exactly to the same time interval @xmath41 . in this section we study the form of both backward ( gram - schmidt ) and characteristic lvs corresponding to the largest les , their localization properties , and the existence of long - range correlations . lvs are strong localized objects , therefore , like in refs . @xcite , it is very convenient to work with the associated fields obtained after a logarithmic transformation : @xmath42 ( and likewise for backward lvs ) . we will refer to @xmath43 as the @xmath31th _ surface _ due to the similarities of its dynamics with that of the kinetic roughening of fractal surfaces in growth models ( see details below ) . also recall that the norm is irrelevant because a lv only indicates a direction in tangent space , and this implies that the mean height of @xmath43 is irrelevant ( only its profile fluctuations matter ) ; this should be beard in mind for the theoretical analysis below . ( @xmath44 t.u . ) . ( b ) difference of 2nd and 3rd lv - surfaces with respect to the 1st lv - surface . the existence of plateaus evidences the piecewise copy structure of lv surfaces with respect to the 1st one.,width=283 ]    figure  [ vectors](a ) shows a snapshot of the characteristic lv surfaces @xmath45 , @xmath46 , and @xmath47 corresponding to the three largest les t.u . there are 168 positive les . the @xmath48-th le vanishes , with @xmath49 . ] . we find that non - leading lv surfaces are approximately piecewise copies of the leading lv surface @xmath45 , see fig . [ vectors](b ) . this ` replication property ' is a highly nontrivial phenomenon that was originally discovered to occur in chaotic extended dissipative systems  @xcite . we also emphasize that characteristic lvs exhibit a tendency to clusterize , contrary to backward lvs whose localization sites are scattered due to the imposed orthogonality . indeed , as exemplified by the snapshot shown in fig . [ vectors](a ) , the first three characteristic lvs localize ( reach their largest magnitude ) at the same point ( completely overlapping on the leftmost region in this particular snapshot ) . this does not occur all the time but in an intermittent manner . interestingly , these structural features namely , replication and clustering have recently been reported to occur generically for lvs in chaotic spatially extended systems  @xcite . this deepens in the analogy between ddss and systems with extensive chaos in one dimension , which happens to hold even at the level of non - leading lvs . regarding the existence of long - range correlations , we find that lv surfaces have a self - affine spatial structure at long scales , which translates into power - law correlations . a detailed analysis of the spatial structure can be best achieved by computing the fourier transform of the surfaces at discrete times @xmath21 given by @xmath50 , with wavenumbers @xmath51 $ ] , where @xmath20 is the system size in this representation . for a given lv surface @xmath43 , the fourier transform of the two - point correlator @xmath52 , is the so - called structure factor : @xmath53 where the brackets denote average over time @xmath21 and realizations . therefore , the structure factor @xmath54 directly informs about the @xmath31th lv surface correlations at scale @xmath55 . the curves from top to bottom correspond to @xmath56 . the insets show the dependence of the local slopes of different curves . for @xmath57 the asymptotic slope is @xmath58 whereas for @xmath59 the asymptotic slopes are @xmath60 and @xmath61 for backward and characteristic lvs , respectively.,width=283 ]    in fig . [ sf ] it can be seen that the structure factor for the first lv - surface decays asymptotically as @xmath62 , with the so - called roughness exponent @xmath63 ( in agreement with the result in @xcite for the ikeda model ) . figure [ sf ] also reveals the existence of a particular crossover wavenumber @xmath64 for the @xmath31th lv . the exponents observed for @xmath65 are @xmath66 and @xmath67 for characteristic and backward lvs , respectively . these numbers coincide with those previously reported in dissipative systems in one dimension  @xcite . our interpretation of these results is that for backward lvs the @xmath68 dependence seems to be the consequence of residual correlations with a geometric origin in the orthogonality of the basis . the @xmath66 exponent of characteristic lvs indicates they convey more information among distant parts of the system than backward lvs ; which can be understood as a consequence of the covariance of characteristic lvs as they are consistent with both past and future evolutions [ characteristic lvs with @xmath59 are `` saddle solutions '' of ( [ tangent ] ) forced by ( [ dds ] ) , see @xcite ] . as we have seen in the previous section , the surface @xmath45 associated with the main lv exhibits scale - invariant correlations in the @xmath69 limit . this strongly suggests that this lv surface should belong to one of the universality classes of surface growth . the fact that the lv surfaces obey scaling laws and that systems with spatiotemporal chaos could be divided into a few universality classes , according to the scaling of the associated surfaces , has implications in our understanding of chaos in extended systems from a statistical physics point of view . moreover , the generic replication and clustering of characteristic lvs along the main vector direction in extended systems  @xcite ( also observed here for time - delay systems , see previous section ) makes it even more interesting to determine the universality class of the main lv , since this is expected to provide a great deal of information about the structure of space and time correlations of the lv corresponding to leading as well as sub - leading unstable directions .    for a wide class of one - dimensional spatio - temporal chaotic systems ( see  @xcite ) the statistical features of the main lv surface are well captured by the 1@xmath70-kardar - parisi - zhang ( kpz )  @xcite stochastic surface growth equation : @xmath71 where @xmath72 is a white noise . the kpz equation defines itself an important universality class of surface growth . regarding chaotic systems , in many of them the main lv surface belongs to the kpz class . this includes coupled - map lattices , coupled symplectic maps , the complex ginzburg - landau model , lorenz 96 model , among others  @xcite . only lvs of anharmonic hamiltonian lattices are known  @xcite to exhibit correlations that are clearly inconsistent with kpz exponents ( @xmath73 ) . statistical mechanics teaches us that , given the fact that kpz represents a dynamical universality class , one would expect that a large collection of different systems could belong to the kpz universality despite their apparent differences . only symmetries and conservation laws would determine the universality class . that may explain why hamiltonian ( energy conserving ) systems , in contrast to dissipative systems , do not generally belong to the kpz universality . on this basis , ddss were also proposed  @xcite to belong to a different universality class , namely the zhang model , arguing that these systems break the @xmath74 symmetry in the spatial representation . however , this conclusion was in contradiction with an earlier work  @xcite where kpz was postulated . our aim in this section is to clarify this question by means of a theoretical analysis and extensive numerical simulations . our starting point is the evolution equation in tangent space ( [ tangent ] ) , of which the main lv is the asymptotic solution . in the spatial representation ( see sec .  [ model ] ) we perform the change of variables @xmath75 , so that ( [ tangent ] ) is now written as : @xmath76 where @xmath23 is the ` spatial ' position and @xmath24 is the discrete time . we have seen in the previous section that a description in terms of surfaces instead of vectors themselves is more appropriate due to the strong localization of the latter . moreover , to relate eq . ( [ 2 ] ) with one of the stochastic partial differential equations modeling surface growth , we wish to approximate the discrete differences in @xmath21 by a partial derivative . in sum , we have to take two steps :    1 . transform to a surface : @xmath77 . approximate @xmath21 by a continuous variable : @xmath78 . note that these two steps do not commute . the treatment of eq . ( [ 2 ] ) by snchez _ et . @xcite proceeded with step ( ii ) before step ( i ) . in more detail , after step ( ii ) , eq . ( [ 2 ] ) becomes : @xmath79 and now transforming to the surface picture [ step ( i ) ] one has : @xmath80 so that we get an equation for the field @xmath81 . this equation was already analyzed in ref . @xcite and we summarize some of its properties in the following . on the one hand , the random drift term @xmath82 gives rise to an effective diffusion term @xmath83 . on the other hand , the presence of @xmath6 in the denominator induces large fluctuations of those terms proportional to @xmath84 every time that @xmath6 takes values close to zero . usually , the probability density function is algebraic at the lowest order : @xmath85 with @xmath86 to be normalizable ( and @xmath87 in general ) . in turn , @xmath88 is heavy tailed with large fluctuations @xmath89 ) . these reasoning led the authors of ref . @xcite to conclude that the main lv surface in ddss behaves following ( [ h_sanchez ] ) and generically falls into the universality class of the linear zhang model  @xcite : @xmath90 that describes surface growth driven by a heavy - tailed noise @xmath91 with a probability distribution @xmath92 for @xmath93 and the index @xmath94 ( @xmath95 if @xmath96 ) . in this work we propose an alternative analysis of eq . ( [ 2 ] ) , where step ( i ) is carried out before step ( ii ) . this would be more suited because one applies the nonlinear transformation ( i ) prior to the approximation ( ii ) . thus after defining the corresponding lv surface , @xmath97 , eq . ( [ 2 ] ) becomes @xmath98 } , \label{interm}\ ] ] where the choice of sign @xmath99 comes from the absolute value and is irrelevant for the arguments that follow . we approximate the difference @xmath100 by the partial derivative @xmath101 . with no further approximations small enough so that the exponential in eq . ( [ interm ] ) can be expanded and only the lowest order may be retained yields eq . ( [ h_sanchez ] ) . we propose here a different derivation that avoids this assumption and , remarkably , leads to a different result . ] and taking logarithms in eq . ( [ interm ] ) we get : @xmath102 where the term @xmath103 is again a fluctuating noise - like source due to the chaotic character of the trajectories . in the neighborhood of @xmath104 the probability distribution of @xmath6 is ( [ pv ] ) , and thus large values of @xmath105 are exponentially rare : @xmath106 , _ i.e. _  the noise is not heavy tailed . we can obtain a more intuitive equation by making use of the small gradient approximation , @xmath107 . expanding  ( [ logeq ] ) in the form : @xmath108 ,   \label{logeq2}\ ] ] where the prototypical quadratic term @xmath109 of kpz appears . if @xmath5 is fluctuating , the drift term @xmath110 again yields ( at large scales , after a spatial averaging ) an effective diffusion term . in this case , eq . ( [ logeq2 ] ) would lead to the kpz eq . ( [ kpz_eq ] ) . in contrast , if @xmath5 is a constant , like for instance in the case of the mg model , a diffusion term would also eventually appear far from the small gradient limit as a result of the absolute value in @xmath111 in  ( [ logeq ] ) . finally , notice that the noise - like term @xmath112 , which is different from that in  ( [ zhang_eq ] ) , will not lead to rare events and its distribution will be exponentially decaying in general .    at variance with snchez _ et al . _ derivation  @xcite ( _ cf . _  sec . [ zhang_sec ] ) we have arrived here at the lv surface equation  ( [ logeq2 ] ) by expanding on the local slope @xmath113 , which is generally expected to be a reasonable approach when the roughness exponent @xmath114 since the spatial average over a window of extent @xmath115 should scale as @xmath116 .    . the lower panel shows the local derivative . for the largest value of the delay , @xmath117 t.u . , the local derivative exhibits a plateau about @xmath118 , close to the theoretical value for kpz ( @xmath119 ) . the curves are averages over at least 1000 realizations.,width=283 ]      in the field of growing interfaces it is customary to quantify the temporal features by looking at the growth of the surface width at a given time when started from a flat initial perturbation [ @xmath120 . to do so we let random initial perturbations @xmath121 to evolve and measure the average surface width at time @xmath21 as @xmath122 ^ 2 } \right\rangle,\ ] ] where @xmath123 is the mean surface position and brackets denote averaging over different initial realizations . the growth exponent @xmath124 is defined as the exponent of the transient power - law growth : @xmath125 , before the surface fluctuations saturate ( for @xmath126 ) ; while @xmath127 , when saturation is reached ( for @xmath128 ) and the perturbation is virtually aligned with the lv : @xmath129 . the key point now is that @xmath124 equals @xmath130 for kpz , while it is @xmath131 for the zhang model with @xmath132  @xcite . this makes the time exponent @xmath124 an excellent index to distinguish between kpz and zhang behavior ( note that the spatial exponent @xmath133 is the same in both models : @xmath63 ) . figure  [ beta ] shows that @xmath134 grows with an exponent that , for large enough delays , progressively approaches the kpz growth exponent @xmath135 , which strongly supports the kpz asymptotic ( @xmath136 ) scaling for chaotic ddss . in the paper by snchez et al .  @xcite the largest time delay used @xmath137 t.u .  resulted in an estimation @xmath138 . in the light of our simulations with much larger delays we conclude that the value of @xmath20 used in ref . @xcite was insufficient to detect the true asymptotic scaling exponent . in addition to the growth exponent , ref . @xcite also invoked the existence of multiscaling of the main lv surface as an argument supporting its identification with the universality class of the linear zhang model ( [ zhang_eq ] ) . this model , contrary to the kpz equation , leads to surfaces that exhibit multiscaling , _ i.e. _  strongly non - gaussian tails , induced by extreme events . if we compute the @xmath139th - height - height correlation function for points separated a distance @xmath140 , @xmath141 ( where the overline denotes the average over @xmath142 and the brackets denote a realizations average ) one finds that @xmath143 , and multiscaling exists if the roughness exponents @xmath144 depend on the index @xmath139 .    , @xmath145 . multiscaling is observed at small @xmath140 . above a certain characteristic size @xmath146 no multiscaling is observed . note that @xmath147 is insensitive to @xmath20 , and the curves overlap almost perfectly ( except at large @xmath140 due to finite - size effects).,width=283 ]    we have computed the @xmath139th order height - height correlation function @xmath148 , given by eq . ( [ g_q ] ) , for several values of @xmath20 ( _ i.e. _  system sizes ) ; see fig . [ multiscaling ] . the distribution of the surface fluctuations shows clear signs of multiscaling with local roughness exponents @xmath144 that depend on @xmath139 for length scales below a typical scale of about @xmath149 . these non - gaussian features disappear at larger scales ( @xmath150 ) , where the @xmath151 for all @xmath139 . for truly extreme event dominated fluctuations , this length scale is expected to diverge ( however slowly ) with the system size , so that the fluctuation distribution is truly non - gaussian in the thermodynamic limit . this slow divergence was indeed measured for models in the zhang universality class  @xcite . in the case of time - delay systems , this would correspond to having @xmath152 increasing with @xmath20 . in contrast , we find that the characteristic length @xmath146 remains constant even after an increase of the delay of 16 times . again , the simulations by snchez _ et . @xcite were carried out in systems with delays that were too short to obtain conclusive evidence on the dependence of @xmath147 with @xmath20 . certainly , we can not rule out for sure an extremely weak dependence of @xmath147 on @xmath20 , but if it exists it must be sub - logarithmic and well beyond the precision that we can reach in our simulations . we conclude that multiscaling of the lv surface fluctuations in ddss seems to be a short scale phenomenon that has no effect in the thermodynamic limit where the universality class is defined ; in this case described by the kpz equation . in this work we have implemented ( to our knowledge for the first time ) characteristic lvs in ddss . adaptation of the method proposed in @xcite to this kind of systems together with the computer capabilities nowadays available allowed us to reach systems with fairly large delays , which serves to investigate the `` thermodinamic limit '' of these systems . our results for the lvs coincide quantitatively with those obtained in extended dissipative systems  @xcite . in addition we have revisited the question of which universality class the main lv belongs to . after simulations with very large delays we may conclude that the main lv surface falls into the universality class of the kardar - parisi - zhang equation . our theoretical arguments support this conclusion as well .    in sum , dds are equivalent to extended dynamical systems in the sense that infinitesimal perturbations exhibit the same exponents characterizing spatiotemporal correlations . dds have been traditionally considered to be different because of the lack of extensivity of the lyapunov spectrum : the positive exponents approach zero as @xmath153 @xcite and the ( kolmogorov - sinai ) entropy saturates with @xmath20 . however the identification of @xmath20 with a size implies that comparisons should be done in temporal units of @xmath154 , and extensivity is then recovered . as @xmath155 , the redefined les @xmath156 do not decay to zero with @xmath20 , and the lyapunov spectrum converges to a density in the thermodynamic limit . d.p .  acknowledges support by csic under the junta de ampliacin de estudios programme ( jae - doc ) . financial support from the ministerio de ciencia e innovacin ( spain ) under project no . fis2009 - 12964-c05 - 05 is acknowledged .
we compute lyapunov vectors ( lvs ) corresponding to the largest lyapunov exponents in delay - differential equations with large time delay . 
 we find that characteristic lvs , and backward ( gram - schmidt ) lvs , exhibit long - range correlations , identical to those already observed in dissipative extended systems . 
 in addition we give numerical and theoretical support to the hypothesis that the main lv belongs , under a suitable transformation , to the universality class of the kardar - parisi - zhang equation . 
 these facts indicate that in the large delay limit ( an important class of ) delayed equations behave exactly as dissipative systems with spatiotemporal chaos .