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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*.