document: the transport properties of nonlinear non - equilibrium dynamical systems are far from well - understood@xcite . 
   consider in particular so - called ratchet systems which are asymmetric periodic potentials where an ensemble of particles experience directed transport@xcite . 
   the origins of the interest in this lie in considerations about extracting useful work from unbiased noisy fluctuations as seems to happen in biological systems@xcite . 
   recently attention has been focused on the behavior of deterministic chaotic ratchets@xcite as well as hamiltonian ratchets@xcite . 
   chaotic systems are defined as those which are sensitively dependent on initial conditions . whether chaotic or not , the behavior of nonlinear systems  including the transition from regular to chaotic behavior  is in general sensitively dependent on the parameters of the system . 
   that is , the phase - space structure is usually relatively complicated , consisting of stability islands embedded in chaotic seas , for examples , or of simultaneously co - existing attractors . 
   this can change significantly as parameters change . 
   for example , stability islands can merge into each other , or break apart , and the chaotic sea itself may get pinched off or otherwise changed , or attractors can change symmetry or bifurcate . 
   this means that the transport properties can change dramatically as well . 
   a few years ago , mateos@xcite considered a specific ratchet model with a periodically forced underdamped particle . 
   he looked at an ensemble of particles , specifically the velocity for the particles , averaged over time and the entire ensemble . 
   he showed that this quantity , which is an intuitively reasonable definition of ` the current ' , could be either positive or negative depending on the amplitude @xmath0 of the periodic forcing for the system . at the same time , there exist ranges in @xmath0 where the trajectory of an individual particle displays chaotic dynamics . 
   mateos conjectured a connection between these two phenomena , specifically that the reversal of current direction was correlated with a bifurcation from chaotic to periodic behavior in the trajectory dynamics . 
   even though it is unlikely that such a result would be universally valid across all chaotic deterministic ratchets , it would still be extremely useful to have general heuristic rules such as this . 
   these organizing principles would allow some handle on characterizing the many different kinds of behavior that are possible in such systems . 
   a later investigation@xcite of the mateos conjecture by barbi and salerno , however , argued that it was not a valid rule even in the specific system considered by mateos . 
   they presented results showing that it was possible to have current reversals in the absence of bifurcations from periodic to chaotic behavior . 
   they proposed an alternative origin for the current reversal , suggesting it was related to the different stability properties of the rotating periodic orbits of the system . 
   these latter results seem fundamentally sensible . however , this paper based its arguments about currents on the behavior of a _ single _ particle as opposed to an ensemble . 
   this implicitly assumes that the dynamics of the system are ergodic . 
   this is not true in general for chaotic systems of the type being considered . in particular , there can be extreme dependence of the result on the statistics of the ensemble being considered . 
   this has been pointed out in earlier studies  @xcite which laid out a detailed methodology for understanding transport properties in such a mixed regular and chaotic system . depending on specific parameter value , the particular system under consideration has multiple coexisting periodic or chaotic attractors or a mixture of both . 
   it is hence appropriate to understand how a probability ensemble might behave in such a system . 
   the details of the dependence on the ensemble are particularly relevant to the issue of the possible experimental validation of these results , since experiments are always conducted , by virtue of finite - precision , over finite time and finite ensembles . 
   it is therefore interesting to probe the results of barbi and salerno with regard to the details of the ensemble used , and more formally , to see how ergodicity alters our considerations about the current , as we do in this paper . 
   we report here on studies on the properties of the current in a chaotic deterministic ratchet , specifically the same system as considered by mateos@xcite and barbi and salerno@xcite . 
   we consider the impact of different kinds of ensembles of particles on the current and show that the current depends significantly on the details of the initial ensemble . 
   we also show that it is important to discard transients in quantifying the current . 
   this is one of the central messages of this paper : broad heuristics are rare in chaotic systems , and hence it is critical to understand the ensemble - dependence in any study of the transport properties of chaotic ratchets . 
   having established this , we then proceed to discuss the connection between the bifurcation diagram for individual particles and the behavior of the current . 
   we find that while we disagree with many of the details of barbi and salerno s results , the broader conclusion still holds . 
   that is , it is indeed possible to have current reversals in the absence of bifurcations from chaos to periodic behavior as well as bifurcations without any accompanying current reversals . 
   the result of our investigation is therefore that the transport properties of a chaotic ratchet are not as simple as the initial conjecture . 
   however , we do find evidence for a generalized version of mateos s conjecture . 
   that is , in general , bifurcations for trajectory dynamics as a function of system parameter seem to be associated with abrupt changes in the current . 
   depending on the specific value of the current , these abrupt changes may lead the net current to reverse direction , but not necessarily so . 
   we start below with a preparatory discussion necessary to understand the details of the connection between bifurcations and current reversal , where we discuss the potential and phase - space for single trajectories for this system , where we also define a bifurcation diagram for this system . in the next section , 
   we discuss the subtleties of establishing a connection between the behavior of individual trajectories and of ensembles . 
   after this , we are able to compare details of specific trajectory bifurcation curves with current curves , and thus justify our broader statements above , after which we conclude . 
   the goal of these studies is to understand the behavior of general chaotic ratchets . 
   the approach taken here is that to discover heuristic rules we must consider specific systems in great detail before generalizing . 
   we choose the same @xmath1-dimensional ratchet considered previously by mateos@xcite , as well as barbi and salerno@xcite . 
   we consider an ensemble of particles moving in an asymmetric periodic potential , driven by a periodic time - dependent external force , where the force has a zero time - average . 
   there is no noise in the system , so it is completely deterministic , although there is damping . 
   the equations of motion for an individual trajectory for such a system are given in dimensionless variables by @xmath2 where the periodic asymmetric potential can be written in the form @xmath3 + \frac{1}{4 } \sin [ 4\pi ( x -x_0 ) ] \bigg ] .\ ] ] in this equation @xmath4 have been introduced for convenience such that one potential minimum exists at the origin with @xmath5 and the term @xmath6 . 
   ( a ) classical phase space for the unperturbed system . for @xmath7 , @xmath8 , 
   two chaotic attractors emerge with @xmath9 ( b ) @xmath10 ( c ) and a period four attractor consisting of the four centers of the circles with @xmath11.,title="fig:",width=302 ]    the phase - space of the undamped undriven ratchet  the system corresponding to the unperturbed potential @xmath12  looks like a series of asymmetric pendula . 
   that is , individual trajectories have one of following possible time - asymptotic behaviors : ( i ) inside the potential wells , trajectories and all their properties oscillate , leading to zero net transport . outside the wells , the trajectories either ( ii ) librate to the right or ( iii ) to the left , with corresponding net transport depending upon initial conditions . 
   there are also ( iv ) trajectories on the separatrices between the oscillating and librating orbits , moving between unstable fixed points in infinite time , as well as the unstable and stable fixed points themselves , all of which constitute a set of negligible measure .    when damping is introduced via the @xmath13-dependent term in eq . 
   [ eq : dyn ] , it makes the stable fixed points the only attractors for the system . 
   when the driving is turned on , the phase - space becomes chaotic with the usual phenomena of intertwining separatrices and resulting homoclinic tangles . 
   the dynamics of individual trajectories in such a system are now very complicated in general and depend sensitively on the choice of parameters and initial conditions . 
   we show snapshots of the development of this kind of chaos in the set of poincar sections fig . 
   ( [ figure1]b , c ) together with a period - four orbit represented by the center of the circles .    a broad characterization of the dynamics of the problem as a function of a parameter ( @xmath14 or @xmath15 ) emerges in a bifurcation diagram 
   . this can be constructed in several different and essentially equivalent ways . 
   the relatively standard form that we use proceeds as follows : first choose the bifurcation parameter ( let us say @xmath0 ) and correspondingly choose fixed values of @xmath16 , and start with a given value for @xmath17 . 
   now iterate an initial condition , recording the value of the particle s position @xmath18 at times @xmath19 from its integrated trajectory ( sometimes we record @xmath20 . 
   this is done stroboscopically at discrete times @xmath21 where @xmath22 and @xmath23 is an integer @xmath24 with @xmath25 the maximum number of observations made . of these , discard observations at times less than some cut - off time @xmath26 and plot the remaining points against @xmath27 . 
   it must be noted that discarding transient behavior is critical to get results which are independent of initial condition , and we shall emphasize this further below in the context of the net transport or current . 
   if the system has a fixed - point attractor then all of the data lie at one particular location @xmath28 . a periodic orbit with period @xmath29 ( that is , with period commensurate with the driving ) shows up with @xmath30 points occupying only @xmath31 different locations of @xmath32 for @xmath27 . 
   all other orbits , including periodic orbits of incommensurate period result in a simply - connected or multiply - connected dense set of points . for the next value @xmath33 , the last computed value of @xmath34 at @xmath35 are used as initial conditions , and previously , results are stored after cutoff and so on until @xmath36 . 
   that is , the bifurcation diagram is generated by sweeping the relevant parameter , in this case @xmath0 , from @xmath27 through some maximum value @xmath37 . 
   this procedure is intended to catch all coexisting attractors of the system with the specified parameter range . 
   note that several initial conditions are effectively used troughout the process , and a bifurcation diagram is not the behavior of a single trajectory . 
   we have made several plots , as a test , with different initial conditions and the diagrams obtained are identical . 
   we show several examples of this kind of bifurcation diagram below , where they are being compared with the corresponding behavior of the current . 
   having broadly understood the wide range of behavior for individual trajectories in this system , we now turn in the next section to a discussion of the non - equilibrium properties of a statistical ensemble of these trajectories , specifically the current for an ensemble . 
   the current @xmath38 for an ensemble in the system is defined in an intuitive manner by mateos@xcite as the time - average of the average velocity over an ensemble of initial conditions . 
   that is , an average over several initial conditions is performed at a given observation time @xmath39 to yield the average velocity over the particles @xmath40 this average velocity is then further time - averaged ; given the discrete time @xmath39 for observation this leads to a second sum @xmath41 where @xmath25 is the number of time - observations made .    for this to be a relevant quantity to compare with bifurcation diagrams , @xmath38 should be independent of the quantities @xmath42 but still strongly dependent on @xmath43 . 
   a further parameter dependence that is being suppressed in the definition above is the shape and location of the ensemble being used . 
   that is , the transport properties of an ensemble in a chaotic system depend in general on the part of the phase - space being sampled . 
   it is therefore important to consider many different initial conditions to generate a current . the first straightforward result we show in fig . 
   ( [ figure2 ] ) is that in the case of chaotic trajectories , a single trajectory easily displays behavior very different from that of many trajectories . 
   however , it turns out that in the regular regime , it is possible to use a single trajectory to get essentially the same result as obtained from many trajectories .    further consider the bifurcation diagram in fig . 
   ( [ figure3 ] ) where we superimpose the different curves resulting from varying the number of points in the initial ensemble . 
   first , the curve is significantly smoother as a function of @xmath0 for larger @xmath44 . even more relevant is the fact that the single trajectory data ( @xmath45 ) may show current reversals that do not exist in the large @xmath44 data . 
   current @xmath38 versus the number of trajectories @xmath44 for @xmath7 ; dashed lines correspond to a regular motion with @xmath46 while solid lines correspond to a chaotic motion with @xmath47 . 
   note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the @xmath44 exceeds a certain threshold , @xmath48.,title="fig:",width=302 ]     current @xmath38 versus @xmath0 for different set of trajectories @xmath44 ; @xmath45 ( circles ) , @xmath49 ( square ) and @xmath50 ( dashed lines ) . note that a single trajectory suffices in the regular regime where all the curves match . in the chaotic regime 
   , as @xmath44 increases , the curves converge towards the dashed one.,title="fig:",width=302 ]    also , note that single - trajectory current values are typically significantly greater than ensemble averages . 
   this arises from the fact that an arbitrarily chosen ensemble has particles with idiosyncratic behaviors which often average out . as our result , with these ensembles we see typical @xmath51 for example , while barbi and salerno report currents about @xmath52 times greater . 
   however , it is not true that only a few trajectories dominate the dynamics completely , else there would not be a saturation of the current as a function of @xmath44 . 
   all this is clear in fig . 
   ( [ figure3 ] ) . 
   we note that the * net * drift of an ensemble can be a lot closer to @xmath53 than the behavior of an individual trajectory 
   .    it should also be clear that there is a dependence of the current on the location of the initial ensemble , this being particularly true for small @xmath44 , of course . 
   the location is defined by its centroid @xmath54 . for @xmath45 , it is trivially true that the initial location matters to the asymptotic value of the time - averaged velocity , given that this is a non - ergodic and chaotic system . 
   further , considering a gaussian ensemble , say , the width of the ensemble also affects the details of the current , and can show , for instance , illusory current reversal , as seen in figs . 
   ( [ current - bifur1],[current - bifur2 ] ) for example . 
   notice also that in fig . 
   ( [ current - bifur1 ] ) , at @xmath55 and @xmath56 , the deviations between the different ensembles is particularly pronounced . 
   these points are close to bifurcation points where some sort of symmetry breaking is clearly occuring , which underlines our emphasis on the relevance of specifying ensemble characteristics in the neighborhood of unstable behavior . 
   however , why these specific bifurcations should stand out among all the bifurcations in the parameter range shown is not entirely clear .    to understand how to incorporate this knowledge into calculations of the current , 
   therefore , consider the fact that if we look at the classical phase space for the hamiltonian or underdamped @xmath57 motion , we see the typical structure of stable islands embedded in a chaotic sea which have quite complicated behavior@xcite . in such a situation , the dynamics always depends on the location of the initial conditions . 
   however , we are not in the hamiltonian situation when the damping is turned on  in this case , the phase - space consists in general of attractors . 
   that is , if transient behavior is discarded , the current is less likely to depend significantly on the location of the initial conditions or on the spread of the initial conditions .    in particular , in the chaotic regime of a non - hamiltonian system , the initial ensemble needs to be chosen larger than a certain threshold to ensure convergence . 
   however , in the regular regime , it is not important to take a large ensemble and a single trajectory can suffice , as long as we take care to discard the transients . 
   that is to say , in the computation of currents , the definition of the current needs to be modified to : @xmath58 where @xmath59 is some empirically obtained cut - off such that we get a converged current ( for instance , in our calculations , we obtained converged results with @xmath60 ) . when this modified form is used , the convergence ( ensemble - independence ) is more rapid as a function of @xmath61 and the width of the intial conditions . 
   armed with this background , we are now finally in a position to compare bifurcation diagrams with the current , as we do in the next section . 
   our results are presented in the set of figures fig . 
   ( [ figure5 ] )  fig . 
   ( [ rev - nobifur ] ) , in each of which we plot both the ensemble current and the bifurcation diagram as a function of the parameter @xmath0 . 
   the main point of these numerical results can be distilled into a series of heuristic statements which we state below ; these are labelled with roman numerals .     for @xmath7 and @xmath8 , we plot current ( upper ) with @xmath62 and bifurcation diagram ( lower ) versus @xmath0 . 
   note that there is a * single * current reversal while there are many bifurcations visible in the same parameter range.,title="fig:",width=302 ]    consider fig . 
   ( [ figure5 ] ) , which shows the parameter range @xmath63 chosen relatively arbitrarily . in this figure , we see several period - doubling bifurcations leading to order - chaos transitions , such as for example in the approximate ranges @xmath64 . however , there is only one instance of current - reversal , at @xmath65 . 
   note , however , that the current is not without structure  it changes fairly dramatically as a function of parameter . 
   this point is made even more clearly in fig . 
   ( [ figure6 ] ) where the current remains consistently below @xmath53 , and hence there are in fact , no current reversals at all . 
   note again , however , that the current has considerable structure , even while remaining negative 
   .     for @xmath66 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 . 
   notice the current stays consistently below @xmath53.,title="fig:",width=302 ]     current and bifurcations versus @xmath0 . in ( a ) and ( b ) 
   we show ensemble dependence , specifically in ( a ) the black curve is for an ensemble of trajectories starting centered at the stable fixed point @xmath67 with a root - mean - square gaussian width of @xmath68 , and the brown curve for trajectories starting from the unstable fixed point @xmath69 and of width @xmath68 . in ( b ) , all ensembles are centered at the stable fixed point , the black line for an ensemble of width @xmath68 , brown a width of @xmath70 and maroon with width @xmath71 . 
   ( c ) is the comparison of the current @xmath38 without transients ( black ) and with transients ( brown ) along with the single - trajectory results in blue ( after barbi and salerno ) . 
   the initial conditions for the ensembles are centered at @xmath67 with a mean root square gaussian of width @xmath68 . 
   ( d ) is the corresponding bifurcation diagram.,title="fig:",width=302 ]    it is possible to find several examples of this at different parameters , leading to the negative conclusion , therefore , that * ( i ) not all bifurcations lead to current reversal*. however , we are searching for positive correlations , and at this point we have not precluded the more restricted statement that all current reversals are associated with bifurcations , which is in fact mateos conjecture . 
   we therefore now move onto comparing our results against the specific details of barbi and salerno s treatment of this conjecture . in particular , we look at their figs . 
   ( 2,3a,3b ) , where they scan the parameter region @xmath72 . 
   the distinction between their results and ours is that we are using _ 
   ensembles _ of particles , and are investigating the convergence of these results as a function of number of particles @xmath44 , the width of the ensemble in phase - space , as well as transience parameters @xmath73 .    our data with larger @xmath44 yields different results in general , as we show in the recomputed versions of these figures , presented here in figs . 
   ( [ current - bifur1],[current - bifur2 ] ) . 
   specifically , ( a ) the single - trajectory results are , not surprisingly , cleaner and can be more easily interpreted as part of transitions in the behavior of the stability properties of the periodic orbits . 
   the ensemble results on the other hand , even when converged , show statistical roughness . 
   ( b ) the ensemble results are consistent with barbi and salerno in general , although disagreeing in several details . 
   for instance , ( c ) the bifurcation at @xmath74 has a much gentler impact on the ensemble current , which has been growing for a while , while the single - trajectory result changes abruptly . note , ( d ) the very interesting fact that the single - trajectory current completely misses the bifurcation - associated spike at @xmath75 . 
   further , ( e ) the barbi and salerno discussion of the behavior of the current in the range @xmath76 is seen to be flawed 
   our results are consistent with theirs , however , the current changes are seen to be consistent with bifurcations despite their statements to the contrary . on the other hand ( f ) , the ensemble current shows a case [ in fig . 
   ( [ current - bifur2 ] ) , at @xmath77 of current reversal that does not seem to be associated with bifurcations . 
   in this spike , the current abruptly drops below @xmath53 and then rises above it again . 
   the single trajectory current completely ignores this particular effect , as can be seen . 
   the bifurcation diagram indicates that in this case the important transitions happen either before or after the spike . 
   all of this adds up to two statements : the first is a reiteration of the fact that there is significant information in the ensemble current that can not be obtained from the single - trajectory current . 
   the second is that the heuristic that arises from this is again a negative conclusion , that * ( ii ) not all current reversals are associated with bifurcations . 
   * where does this leave us in the search for ` positive ' results , that is , useful heuristics ? 
   one possible way of retaining the mateos conjecture is to weaken it , i.e. make it into the statement that * ( iii ) _ most _ current reversals are associated with bifurcations . *     same as fig . 
   ( [ current - bifur1 ] ) except for the range of @xmath0 considered.,title="fig:",width=302 ]     for @xmath78 and @xmath8 , plotted are current ( upper ) and bifurcation diagram ( lower ) versus @xmath0 with @xmath62 . 
   note in particular in this figure that eyeball tests can be misleading . 
   we see reversals without bifurcations in ( a ) whereas the zoomed version ( c ) shows that there are windows of periodic and chaotic regimes . 
   this is further evidence that jumps in the current correspond in general to bifurcation.,title="fig:",width=302 ]     for @xmath7 and @xmath79 , current ( upper ) and bifurcation diagram ( lower ) versus @xmath0.,title="fig:",width=302 ]    however , a * different * rule of thumb , previously not proposed , emerges from our studies . 
   this generalizes mateos conjecture to say that * ( iv ) bifurcations correspond to sudden current changes ( spikes or jumps)*. note that this means these changes in current are not necessarily reversals of direction . 
   if this current jump or spike goes through zero , this coincides with a current reversal , making the mateos conjecture a special case . 
   the physical basis of this argument is the fact that ensembles of particles in chaotic systems _ can _ have net directed transport but the details of this behavior depends relatively sensitively on the system parameters . 
   this parameter dependence is greatly exaggerated at the bifurcation point , when the dynamics of the underlying single - particle system undergoes a transition  a period - doubling transition , for example , or one from chaos to regular behavior . 
   scanning the relevant figures , we see that this is a very useful rule of thumb . for example 
   , it completely captures the behaviour of fig . 
   ( [ figure6 ] ) which can not be understood as either an example of the mateos conjecture , or even a failure thereof . as such 
   , this rule significantly enhances our ability to characterize changes in the behavior of the current as a function of parameter . 
   a further example of where this modified conjecture helps us is in looking at a seeming negation of the mateos conjecture , that is , an example where we seem to see current - reversal without bifurcation , visible in fig . 
   ( [ hidden - bifur ] ) . 
   the current - reversals in that scan of parameter space seem to happen inside the chaotic regime and seemingly independent of bifurcation . however , this turns out to be a ` hidden ' bifurcation  when we zoom in on the chaotic regime , we see hidden periodic windows . 
   this is therefore consistent with our statement that sudden current changes are associated with bifurcations . 
   each of the transitions from periodic behavior to chaos and back provides opportunities for the current to spike . 
   however , in not all such cases can these hidden bifurcations be found . 
   we can see an example of this in fig . 
   ( [ rev - nobifur ] ) . 
   the current is seen to move smoothly across @xmath80 with seemingly no corresponding bifurcations , even when we do a careful zoom on the data , as in fig . 
   ( [ hidden - bifur ] ) . 
   however , arguably , although subjective , this change is close to the bifurcation point . 
   this result , that there are situations where the heuristics simply do not seem to apply , are part of the open questions associated with this problem , of course . 
   we note , however , that we have seen that these broad arguments hold when we vary other parameters as well ( figures not shown here ) .    in conclusion , 
   in this paper we have taken the approach that it is useful to find general rules of thumb ( even if not universally valid ) to understand the complicated behavior of non - equilibrium nonlinear statistical mechanical systems . in the case of chaotic deterministic ratchets 
   , we have shown that it is important to factor out issues of size , location , spread , and transience in computing the ` current ' due to an ensemble before we search for such rules , and that the dependence on ensemble characteristics is most critical near certain bifurcation points . 
   we have then argued that the following heuristic characteristics hold : bifurcations in single - trajectory behavior often corresponds to sudden spikes or jumps in the current for an ensemble in the same system . 
   current reversals are a special case of this 
   . however , not all spikes or jumps correspond to a bifurcation , nor vice versa . 
   the open question is clearly to figure out if the reason for when these rules are violated or are valid can be made more concrete . 
   a.k . gratefully acknowledges t. barsch and kamal p. singh for stimulating discussions , the reimar lst grant and financial support from the alexander von humboldt foundation in bonn . a.k.p . 
   is grateful to carleton college for the ` sit , wallin , and class of 1949 ' sabbatical fellowships , and to the mpipks for hosting him for a sabbatical visit , which led to this collaboration . 
   useful discussions with j .- 
   m . rost on preliminary results are also acknowledged . 
   p. hnggi and bartussek , in nonlinear physics of complex systems , lecture notes in physics vol . 
   476 , edited by j. parisi , s.c . 
   mueller , and w. zimmermann ( springer verlag , berlin , 1996 ) , pp.294 - 308 ; r.d . 
   asturmian , science * 276 * , 917 ( 1997 ) ; f. jlicher , a.  ajdari , and j.  prost , rev .  mod . 
   phys .  * 69 * , 1269 ( 1997 ) ; c. dring , nuovo cimento  d*17 * , 685 ( 1995 )                      s. flach , o. yevtushenko , and y. zolotaryuk , phys 
   .  rev . 
   lett .  * 84 * , 2358 ( 2000 ) ; o. yevtushenko , s. flach , y. zolotaryuk , and a.  a. ovchinnikov , europhys . 
   lett .  * 54 * , 141 ( 2001 ) ; s. denisov et al . 
   e  * 66 * , 041104 ( 2002 )
summary: in 84 , 258 ( 2000 ) , mateos conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes . 
   this is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential . 
   barbi and salerno , in 62 , 1988 ( 2000 ) , have further investigated this claim and argue that , contrary to mateos claim , current reversals can occur also in the absence of bifurcations . 
   barbi and salerno s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system . 
   the behavior of ensembles can be quite different , depending upon their characteristics , which leaves their results open to question . in this paper we present results from studies showing how the current depends on the details of the ensemble 
   used to generate it , as well as conditions for convergent behavior ( that is , independent of the details of the ensemble ) . 
   we are then able to present the converged current as a function of parameters , in the same system as mateos as well as barbi and salerno . 
   we show evidence for current reversal without bifurcation , as well as bifurcation without current reversal . 
   we conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation , rather than current reversals , and show numerical evidence for our claims .