Figure 1

Plot of $x(t)$ (solid line) and $F(t)$ (dot-dashed line) for $T=0.015$. The particle exhibits both kinds of trajectories with a transition from out-of-phase to in-phase, in this case, at around $t=460$; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 2

Plot of $\overline{W}$ with $x(0)$ for $T=0.001$ (a), $T=0.003$ (b), and $T=0.016$ (c); $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$. The insets in (a) and (c) show the stroboscopic (Poincaré) plots in the $(x\text{−}v)$ plane at times ${\tau }_n=n\tau ,n=0,\pm 1,\pm 2,\dots$Reuse & Permissions

Figure 3

The basins of attraction of the in-phase (inp) and out-of-phase (outp) attractors at temperature $T=0.001$ are shown. The bracketed numbers on inp and outp indicate the well number of the periodic potential where the trajectory settles down in the in-phase or out-of-phase states. The region marked “$a$” corresponds to inp($-1$), “$b$” corresponds to outp($-2$), “$c$” corresponds to the continuation of inp($-2$) indicated on the left side, “$d$” corresponds to inp($-3$), and “$e$” corresponds to outp($-3$). The boundaries correspond to the end of the phase indicated above the respective lines. The thick line separating outp(2) and inp(1) indicates that some other phases also chip in between. The phase inp(1) at the top right corner is just the continuation of the phase inp(1) on the left top corner of the figure. For this figure, $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 4

Plot of the particle trajectories $x(t)$ for $T=0.04$. Transitions are seen between the in-phase to the out-of-phase states. Inset is a magnified figure. $F(t)$ (dashed line) is also included for comparison; $x(0)=2.0$, $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 5

A schematic depiction of the two dynamical energy states of the particle trajectories. In the figure, $E3-E1\simeq 0.016$ and $E3-E2\simeq 0.003$.Reuse & Permissions

Figure 6

Plot of $x(t)$ for $T=0.08$. The figure shows interwell transitions as also numerous transitions between the in-phase (lower amplitude) and out-of-phase (higher amplitude) states. At around $t=3700$ and $5400$, the interwell jump also leads to a transition from the out-of-phase to the in-phase state; $x(0)=2.0$, $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 7

Plot of $P(W)$ for different values of $T$ [(a) for $T<0.016$ and (b) for $T>0.016$] for the homogeneous system. For large $T$, e.g., at $T=1.0$, the distribution has a single peak structure; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 8

Plot of $\langle \overline{W}\rangle$ as a function of $T$ for the homogeneous system; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 9

Plot of $\langle \overline{A}\rangle$ (top), $x_0$ (middle), and $-\overline{\varphi }$ (bottom) with $T$ for the homogeneous system; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$. Beyond $T=0.4$, the curves have a monotonic behavior and are not shown. The top figure is an exact reproduction of Fig. 8.Reuse & Permissions

Figure 10

Plot of $\overline{W}$ with $x(0)$ for $T=0.001$ (a), $T=0.012$ (b), $T=0.014$ (c), and $T=0.016$ (d); $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$ for the inhomogeneous system. The inset of (a) shows the two attractors in the $(x\text{−}v)$ plane stroboscopic plots at times ${\tau }_n=n\tau$.Reuse & Permissions

Figure 11

Plot of $\langle \overline{W}\rangle$ (top), the amplitude $x_0$ (middle), and the phase difference $-\mathrm{\varphi ̄}$ (bottom) with $T$ for the inhomogeneous system; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions

Figure 12

Plot of $P(W)$ for different values of $T$ [(a) for low $T$ and (b) for higher $T$] for the inhomogeneous system; $\tau =8$, $F_0=0.2$, and ${\gamma }_0=0.12$.Reuse & Permissions