Bifurcations and sudden current change in ensembles of classically chaotic ratchets

Mateos [Phys. Rev. Lett. 84, 258 (2000)] 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 [Phys. Rev. E 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.

Figure 1

(Color online) (a) Classical phase for the unperturbed system. For $\omega =0.67$, $b=0.1$, and $ϵ=1$, two chaotic attractors emerge with (b) $a=0.11$ and (c) $a=0.155$. Circles superposed in (c) represent a period four attractor obtained with $a=0.08125$.

Figure 2

Current $J$ versus the number of trajectories $N$ for $\omega =0.67$, $b=0.1$; dashed lines correspond to regular motion with $a=0.12$ while solid lines correspond to chaotic motion with $a=0.08$. Note that a single trajectory is sufficient for a regular motion while the convergence in the chaotic case is only obtained if the $N$ exceeds the threshold, $N\ge N_{\mathrm{thr}}=100$.

Figure 3

(Color online) Current $J$ versus $a$ for different set of trajectories $N$; $N=1$ (circles), $N=10$ (square), and $N=100$ (dashed lines). Note that a single trajectory suffices in the regular regime where all the curves match. In the chaotic regime, as $N$ increases, the curves converge toward the dashed one.

Figure 4

(Color online) Current $J$ versus $a$ for a single trajectory $(N=1)$, with $n_c=5\times {10}^5$ time steps discarded from a total $M={10}^7$ time steps (blue) showing many transitions between the upper and lower values, compared to Barbi and Salerno’s results (dashed in magenta) showing a single transition. Here $\omega =0.67$ and $b=0.1$.

Figure 5

(Color online) Currents $J$ and corresponding bifurcation diagrams $v$ versus $a$. In (a) $J$ is computed with an ensemble of trajectories of different root-mean-square Gaussian width $W$, starting centered either at the stable fixed point $(0,0)$ or at the unstable fixed point $(-0.375,0)$. The blue (topmost) and the black (lowermost) curves are obtained with trajectories centered at $(0,0)$, for $W=0.25$ and with $W=0.5$, respectively, while the magenta (middle) is obtained with trajectories centered at $(-0.375,0)$ and $W=0.25$. In (b) we compare our converged results from (a) (shown in magenta) obtained with trajectories centered at $(0,0)$ and covering the entire space, $W=1$, to the single trajectory results (shown in blue) of Barbi and Salerno. In (c) the bifurcation diagram (blue) is obtained with increasing $a$ while the bifurcation diagram (red) is obtained with decreasing $a$. Here $\omega =0.67$, $b=0.1$, $N={10}^3$, $n_c=5\times {10}^5$, and $M={10}^7$.

Figure 6

(Color online) Same as Fig. 5 except for the range of $a$ considered.