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the _ ratchet _ effect , i. e. the possibility to obtain the directed transport by using zero - mean perturbations only , has induced notable scrutiny over the last decade [ 1 ] . initially , most studies have been focused on noisy overdamped models have been inspired by a molecular motors realm [ 1 ] . then the ratchet s approach has been applied to a broad class of physical systems in which inertia effects are essential [ 2 ] . the examples are josepshon junctions [ 3 ] , cold atoms systems [ 4 ] , and mechanical engines [ 5 ] . recently it has been shown that the ratchet idea can be viewed as a part of a general symmetry - breaking approach [ 6 ] . this approach is based on the analysis of all relevant symmetries which have to be broken in order to fulfill necessary conditions for a dc - output appearance . the formalization of symmetry analysis for one particle s dynamics has been addressed in ref.[7 ] .    in the present paper we aim at collective rectification effects which arise in a set of coupled single ratchet units . although various examples of interacting ratchet systems have been proposed already in the context of molecular motors [ 8 ] , here the emphasis is put on the weak - noise underdamped limit . we show that a coupling scheme determines a set of certain symmetries for the ratchet s collective and hence defines necessary conditions for a dc - current generation . dynamical mechanisms of a current rectification are connected with a coherence between units , which depends not only on a coupling scheme , but also on a strength of interactions . let us consider a set consisting of @xmath0 identical dynamical units , @xmath1 , that are linearly and symmetrically coupled . the coupling scheme is described by some graph which can be encoded in the symmetrical @xmath2 binary matrix @xmath3 , @xmath4 . the equations of motion are the following : @xmath5 where @xmath6 is the double periodic force function , @xmath7 is linear over all arguments coupling function , and @xmath8 is the strength of interactions . the stochastic terms @xmath9 are mutually independent delta - correlated gaussian white noises , @xmath10 , where @xmath11 is the noise intensity . we assume also that the force function , @xmath12 , is the same for all the units , but the phases , @xmath13 and @xmath14 , can be different for different units . finally , we are interested in the mean dc - current , @xmath15    following the symmetry analysis ideology [ 6 - 7 ] , in order to determine necessary conditions for a dc - output appearance , we have to check whether there exist symmetry transformations which allow to generate out for each trajectory of the system ( 1 ) another one with a reversal velocity . the presence of the white noise , @xmath9 , does not change the symmetry properties of the system . moreover , the coupling to a heat bath leads to an effective exploration of the whole phase space and produce an averaging in a case of several coexisting attractors [ 6,9 ] . for the one - particle case , @xmath16 , a transformation of interest has to involve a change of the sign of @xmath17 ( and thus change of the current direction ) . it allows also some shifts in time and space domains [ 6 - 7 ] : @xmath18    in the case of several coupled ratchets a symmetry operation should be performed in the global coordinate space @xmath19 and can also involve a permutation between different units . the corresponding symmetry operation can be described as the linear transformation , @xmath20 where @xmath21 is a @xmath2 binary matrix with only one non - zero element at each row and line , and @xmath22 is the vector of shifts . the matrix @xmath23 encodes permutation between units . the properties of the permutation matrix @xmath21 ( and the existence of such a matrix at all ) are strongly depend on the structure of connections among units . below , using simple examples , we will illustrate this statement . _ breaking symmetries by connections . _ let us first consider the case of two particles ( rotators ) in the standing - wave potential with the modulated amplitude [ 10 ] , coupled by the linear spring : @xmath24 where @xmath25 .    in the uncoupled case , @xmath26 , both the systems posses symmetries of the type ( 3 ) : @xmath27 with @xmath28 for the first rotator , eq.(5 ) . the symmetry transformations , eqs.(7 - 8 ) , are independent for each rotators . the whole system , eqs.(5 - 6 ) , is symmetric with respect to the transformation @xmath29 for any choice of @xmath30 and @xmath31 . so , the mean dc - output for the uncoupled case is zero ( line(1 ) in fig.1a ) .    in the case of coupled particles , @xmath32 , abovementioned transformations , , eqs.(7 - 8 ) , should been conjugated and independent spatial and temporal shifts are forbidden now . for @xmath33 and @xmath34 , the connection breaks both the symmetries and we can expect on nonzero current appearance ( see also ref . [ 11 ] for another example of an overdamped dimer with an additive driving force ) . for the set of parameters @xmath35 , @xmath36 , @xmath37 , @xmath38 and @xmath39 , the symmetry violation is realized by the asymmetrical limit cycle with the negative winding number ( line(2 ) in fig.1a ) . so , in this case the connection between the units destroy all the symmetries and leads to the dc - current generation . _ restoring symmetries by connections . _ let us now consider the dimer identical to the previous one , eq.(5 - 6 ) , but spatially shifted by @xmath40 ( half of the period ) : @xmath41 both the systems , eqs.(5 - 6 ) and eqs.(9 - 10 ) , can be transformed from one to another by the simple coordinate shift . thus , the system in eqs.(9 - 10 ) produces the same mean current , as the system in eqs.(6 - 7 ) .    as the next step we couple both the systems by additional connections @xmath42 ( see the inset in fig.1a ) . the generalized symmetry transformation , eq.(4 ) , can be identified with the following permutation matrix : @xmath43 and the shift vector @xmath44 .    here , despite to the previous case , the introduction of additional connections leads to the appearance of the new symmetry and we may expect on the nullification of the dc - output ( line(3 ) in fig.1a ) . for the above set of parameters we found that the symmetry is realized in the phase space by two symmetry - coupled chaotic attractors with opposite mean velocities ( see fig.1b ) it is easy to check , that any other type of connection with two additional links does not restore the system s symmetry . while the presence or the absence of the dc - output is clearly connected to the absence / presence of the symmetry transformation , the dc - current value is determined by dynamical mechanisms . from the previous studies [ 12 ] it is known that global properties of a collective dynamics are defined by a coherence between units . thus , we can expect that an efficiency of the current rectification is closely related to a degree of synchronization within a ratchet s collective .    as a natural example we consider the model of @xmath0 globally coupled array of underdamped josepshon junctions ( jj ) , subjected to an ac - current @xmath45 [ 13 ] . the equation for the superconducting phase difference @xmath46 across the single junction is @xmath47 where @xmath48 is the instantaneous mean array current . as the driving force we used the two - harmonics combination , @xmath49 , which ensures that all the relevant symmetries are broken [ 7 ] . the numerically obtained dependence of the array dc - output on the strength of interaction @xmath8 is shown in fig.2(a ) . for the set of parameters @xmath50 , the dependence demonstrates the presence of two successive current reversals , at @xmath51 and @xmath52 . in order to understand dynamical mechanisms of these events , we introduced the mean decoherence , which we define as @xmath53 where @xmath54 means the averaging over the array and over the one period of ac - driving . the first result from the comparison of both the dependencies ( fig.2(b))is that the current reversal at @xmath55 is connected with the transition to the regime of the complete synchronization , @xmath56 . lets now track briefly a relation between the current reversal and the synchronization . initially , in the uncoupled limit , @xmath26 , all jj s work independently , each one as the single one - dimensional rectificator [ 1,6 ] . the asymmetry of the ac - force @xmath45 is realized through the chaotic attractor with the positive mean current ( see fig.3a ) [ 14 ] . the local phase space of each units , @xmath57 , has identical structure , but the relative phases of different units are randomly distributed ( depending on the initial conditions ) . after the introduction of a nonzero coupling , @xmath32 , some coherence between the units occurs . from a point of view of the dynamics of the single junction , this leads to changes of an attractor structure in the phase space @xmath58 ( see fig.3b ) . this causes changes in a projection of the attractor s invariant density on the velocity subspace @xmath59 and , as a result , leads to the current reversal at @xmath60 . the further increase of the interaction strength up to @xmath55 results in the complete synchronization and to a shrinking of the global attractor in @xmath61 to the hyperplane @xmath58,@xmath62 . the global attractor has now dimension equal to @xmath63 . the attractor of the single unit now is the same as in the uncoupled limit but all the units have the same phase . the mean dc - output returns to its value at the limit @xmath26 .    here some analogy with a current reversal in an one - dimensional deterministic ratchet [ 14 ] can be drawn . in the one - dimensional case current reversals have been identified with tangent bifurcations from chaotic to regular ( limit cycles ) attractors [ 14 ] . in our case the current reversal at @xmath55 corresponds to the transition `` hyperchaos - chaos '' [ 15 ] , connected with the shrinking of a the system attractor in a global phase space @xmath61 . it is interesting to note , that from a point of view of the local attractor of the single junction , this transition corresponds to a crisis , connected with _ expanding _ of the attractor in three - dimensional subspace @xmath64 ( compare fig.3a and fig.3b ) . the presence of a weak noise ( which is equivalent to a weak coupling with a heat bath ) can strongly suppress correlations between units and may lead to a delayed synchronization transition [ 16 ] . due to a strong conjugation between the synchronization and the current rectification process , we can manage a noise - induced current reversal for a fixed strength of interaction ( fig.2(b ) ) . thus , the presence of a thermostat allows to control the dc - output by changing a temperature of the system . finally , we have presented the symmetry approach to the problem of the collective current rectification by a set of coupled dynamical units . this idea can be used for a more general problem such as an obtaining of a non - zero value of some relevant mean ensemble characteristic , @xmath65 . this characteristic can correspond , for example , to a mean magnetization of a spin lattice with a complex geometry [ 17 ] . the proposed collective ratchet s ideology may also be relevant in a context of a cooperative dynamics of neural networks , as an approach to a visual processing with directional selectivity [ 18 ] . relations between symmetries of a collective ratchet and its coupling scheme on the one side , and relations between a topology of interactions and synchronization properties [ 19 ] , on the other one , may open an interesting perspective . it has been found that in the thermodynamic limit , @xmath66 , the synchronization is impossible for nearest - 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we consider a set of coupled underdamped ac - driven dynamical units exposed to a heat bath . 
 the coupling scheme defines the absence / presence of certain symmetries , which in turn cause a nonzero / zero value of a mean dc - output . 
 we discuss dynamical mechanisms of a dc - current appearance and identify current reversals with synchronization / desynchronization transitions in the collective ratchet s dynamics . 
 driven underdamped systems ; synchronization ; ratchet effect    05.45.ac , 05.60.cd