Inertial effects in the dc+ac driven underdamped Frenkel-Kontorova model: Subharmonic steps, chaos, and hysteresis

The effects of inertial terms on the dynamics of the dc+ac driven Frenkel-Kontorova model were examined. As the mass of particles was varied, the response of the system to the driving forces and appearance of the Shapiro steps were analyzed in detail. Unlike in the overdamped case, the increase of mass led to the appearance of the whole series of subharmonic steps in the staircase of the average velocity as a function of average driving force in any commensurate structure. At certain values of parameters, the subharmonic steps became separated by chaotic windows while the whole structure retained scaling similar to the original staircase. The mass of the particles also determined their sensitivity to the forces governing their dynamics. Depending on their mass, they were found to exhibit three types of dynamics, from dynamical mode-locking with chaotic windows, through to a typical dc response, to essentially a free-particle response. Examination of this dynamics in both the upforce and downforce directions showed that the system may not only exhibit hysteresis, but also that large Shapiro steps may appear in the downforce direction, even in cases for which no dynamical mode-locking occurred in the upforce direction.

Figure 1

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $K=4,{\nu }_0=0.2,F_{\text{ac}}=0.2,\omega =\frac{1}{2}$, and different values of the mass $m=0,0.1$ and 0.2 represented by different colors. The case $m=0$ corresponds to the standard overdamped case. Results in (a) and (b) corresponds to the dc+ac and dc driven dynamics, respectively. Numbers mark the harmonic steps, while $F_{c0}$ marks dynamical dc threshold.

Figure 2

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $m=0.1$ in the region between the first and the second harmonic step in (a), and in the region between the second and the third harmonic step in (b). The rest of parameters are the same as in Fig. 1.

Figure 3

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $\omega =1$, and $m=0$, and 0.1 in the region between the first and the second harmonic step in (a), and in the region between the second and the third harmonic step in (b). The rest of parameters are the same as in Fig. 1.

Figure 4

Fractal dimension $D$ as a function of mass $m$ measured in the region between the first and the second $(F_{1\text{dc}}<F_{\text{dc}}<F_{2\text{dc}})$, and the second and the third $(F_{2\text{dc}}<F_{\text{dc}}<F_{3\text{dc}})$ harmonic steps for $\omega =\frac{1}{2},$ and 1 in (a) and (b), respectively. The rest of the parameters are the same as in Fig. 1.

Figure 5

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ and the corresponding Lyapunov exponents ${\lambda }_i$ for $K=4,F_{\text{ac}}=0.2,{\nu }_0=0.2,\omega =\frac{1}{2}$, and $m=0.041$ and 0.1 in (a) and (b), respectively. On this scale of $y$ axis only the largest Lyapunov exponent is visible. Numbers mark the harmonic steps.

Figure 6

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ and the corresponding Lyapunov exponents ${\lambda }_i$ for $m=0.15$ and 1.5 in (a) and (b), respectively. The rest of parameters are the same as in Fig. 1. On this scale of $y$ axis only the largest Lyapunov exponents are visible. Numbers mark the harmonic steps.

Figure 7

The average velocity as a function of the average driving force $\overline{v}\left(\overline{F}\right)$ for $m=0.2,K=4,F_{\text{ac}}=0.2,{\nu }_0=0.2$, and $\omega =\frac{1}{2}$. Numbers mark harmonic and subharmonic steps. The staircase in (b), (c), and (d) represent the high-resolution views of the selected areas in (a), (b), and (c), respectively.

Figure 8

The Lyapunov exponents ${\lambda }_i$ as a function of the average driving force $\overline{F}$ and the corresponding response function $\overline{v}\left(\overline{F}\right)$ in the small region between the second and the third harmonics for $m=0.2$ are presented. On this scale of $y$ axis only the largest Lyapunov exponent is visible. The rest of the parameters are the same as described in the caption of Fig. 7.

Figure 9

The average velocity as a function of the average driving force $\overline{v}\left(\overline{F}\right)$ for $m=1.25,K=4,F_{\text{ac}}=0.2,{\nu }_0=0.2$, and $\omega =\frac{1}{2}$. Numbers mark harmonic and subharmonic steps. The staircase in (b) and (c) represent the high-resolution views of the selected areas in (a).

Figure 10

The Lyapunov exponents ${\lambda }_i$ as a function of the average driving force $\overline{F}$ and the corresponding response function $\overline{v}\left(\overline{F}\right)$ in the region between the zero and the first harmonicstep for $m=1.25$ are presented. Different colors of ${\lambda }_i$ corresponds to different Lyapunov exponents. The rest of the parameters are the same as described in the caption of Fig. 9.

Figure 11

The average velocity as a function of the average driving force $\overline{v}\left(\overline{F}\right)$ for $K=4,F_{\text{ac}}=0.2,{\nu }_0=0.2,\omega =\frac{1}{2}$, and $m=0.2,2$, and 20 in (a)–(c), respectively. The dc curve presented by the red dashed line corresponds to the dc driven system where $F_{\text{ac}}=0$, while $K=0$ is linear response of the system of free particles. $F_c$ and $F_{c0}$ mark critical depinning force and dynamical dc threshold, respectively.

Figure 12

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $K=4,\omega =\frac{1}{2},F_{\text{ac}}=0.2,{\nu }_0=0.2$, and $m=1.25$. The dashed lines corresponds to the dc driven case where $F_{\text{ac}}=0$. $F_{c0}$ marks dynamical dc threshold.

Figure 13

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $m=5$, $K=4,\omega =\frac{1}{2},F_{\text{ac}}=0.2$, and ${\nu }_0=0.2$. The high-resolution view of the selected area in (a) is presented in (b) where the numbers mark Shapiro steps. $F_{c0}=F_c$ marks critical depinning force or dynamical dc threshold while $F_p$ represents critical pinning force in downforce direction.

Figure 14

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $K=4,\omega =\frac{1}{2},F_{\text{ac}}=0.2,{\nu }_0=0.2$ and $m=10$ and ${10}^2$ in (a) and (b), respectively. $F_{c0}=F_c$ and $F_p$ mark the critical depinning and pinning force, respectively. Inset shows the third harmonic step which appears in the downforce direction.