Current reversals and synchronization in coupled ratchets

Current reversal is an intriguing phenomenon that has been central to recent experimental and theoretical investigations of transport based on ratchet mechanism. By considering a system of two interacting ratchets, we demonstrate how the coupling can be used to control the reversals. In particular, we find that current reversal that exists in a single driven ratchet system can ultimately be eliminated with the presence of a second ratchet. For specific coupling strengths a current-reversal free regime has been detected. Furthermore, in the fully synchronized state characterized by the coupling threshold $k_{th}$, a specific driving amplitude $a_{opt}$ is found for which the transport is optimum.

Figure 1

(Color online) Equipotential contours plot of $V\left(x_1,x_2\right)$ with colors growing from dark-blue (minima) through yellow/gray (intermediate) to white (maxima): (a) no interaction, $k=0$, (b) weak coupling, $k=0.05$, (c) moderate coupling, $k=0.15$, and (d) strong coupling, $k=1.0$.

Figure 2

(Color online) Typical on-off intermittency for $a=0.0809472$ (a) of the error state $\Delta x\left(t\right)=x_1\left(t\right)-x_2\left(t\right)$ for $k=0.065$, (b) of trajectories $x_1\left(t\right)$ (black) and $x_2\left(t\right)$ (red/gray) for $k=0.065$ and $k=0.45$ as indicated with dashed arrows, and (c) of the enlarged portion of (b) for $k=0.065$.

Figure 3

(Color online) (a) Distribution of laminar phases $\Lambda \left(t\right)$ of $\Delta x\left(t\right)$ (black circles), satisfying a $-3/2$ power-law scaling typical of on-off intermittency and (b) average laminar lengths with varying parameter $a$ satisfying the scaling law $⟨l⟩\propto {ϵ}^{-0.5}$ with $ϵ=a-a_c$ and $a_c=0.0809474$ [$x_1$ (black circles) and $x_2$ (green/gray squares)] showing type-I intermittency. Here, $k=0.065$ and the lines (red/gray) are corresponding fits.

Figure 4

Transition to full synchronization as a function of the coupling strength $k$ and for $a=0.0809472<a_c$, indicated by (a) the average error dynamics $\eta$ and (b) the current $J_{-}=J_1-J_2$. In the same coupling range, (c) the current $J_1$ is plotted showing regions of zero current and of current reversals. Here, $k_{th}$ is the coupling threshold for full synchronization.

Figure 5

The bifurcation diagrams for $a=0.0809472$ as a function of the coupling strength $k$. The transition from the intermittent chaotic regime to a period 4 orbit (a) $v_1\left(t\right)={\dot{x}}_1\left(t\right)$ happens at the synchronization threshold $k\ge k_{th}\approx 0.576$ (b) $v_2\left(t\right)-v_1\left(t\right)$.

Figure 6

Bifurcation diagram for large amplitudes of the driving force $a$ for $k=0$, showing several windows of chaos as well as quasiperiodic ones.

Figure 7

(Color online) Current $J_1$ as a function of the driving amplitude $a$ for (a) strong couplings $0.4<k<1$ and (b) weak couplings $0<k<0.07$ as indicated in each panel.

Figure 8

Optimal current $J_{1}opt$ as a function of the driving amplitude $a$ over the coupling interval $0\le k\le 1.0$. One clearly sees the transport suppression for weak amplitudes $\left(a<0.1\right)$ and also for $0.5<a<0.6$, and transport enhancement for other values of $a$. The peak optimal current is at a moderate driving force $a=a_{opt}\approx 0.3$.