Devil's staircase and the absence of chaos in the dc- and ac-driven overdamped Frenkel-Kontorova model

The devil's staircase structure arising from the complete mode locking of an entirely nonchaotic system, the overdamped dc+ac driven Frenkel-Kontorova model with deformable substrate potential, was observed. Even though no chaos was found, a hierarchical ordering of the Shapiro steps was made possible through the use of a previously introduced continued fraction formula. The absence of chaos, deduced here from Lyapunov exponent analyses, can be attributed to the overdamped character and the Middleton no-passing rule. A comparative analysis of a one-dimensional stack of Josephson junctions confirmed the disappearance of chaos with increasing dissipation. Other common dynamic features were also identified through this comparison. A detailed analysis of the amplitude dependence of the Shapiro steps revealed that only for the case of a purely sinusoidal substrate potential did the relative sizes of the steps follow a Farey sequence. For nonsinusoidal (deformed) potentials, the symmetry of the Stern-Brocot tree, depicting all members of particular Farey sequence, was seen to be increasingly broken, with certain steps being more prominent and their relative sizes not following the Farey rule.

Figure 1

The average velocity $\overline{v}$ as a function of the average driving force $\overline{F}$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}r=0.01$, and different values of the ac amplitude $F_{\mathrm{ac}}=0,0.1,0.2,0.5$, and 2 from right to left. The case $F_{\mathrm{ac}}=0$ corresponds to the dc-driven system represented by the dashed line while the dotted line represents linear dependence for a system of free particles. The numbers mark harmonic steps.

Figure 2

Number of subharmonic Shapiro steps $N$ detected between the first and the second harmonics at different values of the ac amplitude $F_{\mathrm{ac}}$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, the force step $\mathrm{\Delta }F={10}^{-5}$, and the three values of deformation parameter $r=0.01,0.25,$ and 0.5 in (a), (b), and (c) respectively.

Figure 3

The average velocity as a function of the average driving force $\overline{v}\left(\overline{F}\right)$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, $r=0.5$, and $F_{\mathrm{ac}}=0.2$. Numbers mark harmonic and subharmonic steps. The devil's staircase in (b), (c), and (d) represent the high-resolution views of the selected areas in (a), (b), and (c), respectively.

Figure 4

The average velocity as a function of the average driving force $\overline{v}\left(\overline{F}\right)$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, $r=0.5$, and $F_{\mathrm{ac}}=1.1$. Numbers mark harmonic and subharmonic steps. The devil's staircase in (b), (c), and (d) represent the high-resolution views of the selected areas in (a), (b), and (c), respectively.

Figure 5

Part of the Stern-Brocot tree depicting all the members of Farey sequence ${\mathcal{F}}_5$ and some members of higher Farey sequences. Fractions corresponding to the integers $\frac{1}{1}$ and $\frac{2}{1}$ are not shown in the figure. Fraction $\frac{3}{2}$ represents the root or starting node of the tree.

Figure 6

Fractal dimension $D$ as a function of the ac amplitude $F_{\mathrm{ac}}$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$; the force step $\mathrm{\Delta }F={10}^{-5}$; and $r=0.01,0.25$ and 0.5 in (a), (b), and (c) respectively. The dash line marks the value $D=0.87$.

Figure 7

Plot of ${log}_{10}N\left(l\right)$ vs ${log}_{10}\left(\frac{1}{l}\right)$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, $r=0.01$, and $F_{\mathrm{ac}}=0.25$. The slope determines the fractal dimension $D$. The inset shows the corresponding region between the first and the second harmonic steps.

Figure 8

The largest Lyapunov exponent as a function of the ac amplitude $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, and $F_{\mathrm{ac}}=1.5,1.8$, and 10 in (a), (b), and (c), respectively.

Figure 9

$I\text{−}V$ characteristics and corresponding maximal Lyapunov exponent for a stack of seven intrinsic Josephson junctions, simulated from Eqs. (9), with $N=7$, $\alpha =0.1$, $A=0.8$, ${\omega }_{\text{rf}}=0.5$, and at two different values of dissipation parameter $\beta =0.3$ and 0.8 in (a) and (b), respectively. The dashed lines correspond to the zero value of LE, while the arrows in (b) indicate the chaotic regions that appear between adjacent Shapiro steps.

Figure 10

Maximal Lyapunov exponent as a function of the bias current $I$ and damping parameter $\beta$, for the stack of seven junctions. All other parameters are the same as in Fig. 9.

Figure 11

Critical depinning force $F_c$ and the width $\mathrm{\Delta }F$ of first harmonic Shapiro step $\frac{1}{1}$ as a function of the ac amplitude $F_{\mathrm{ac}}$ for $K=4$, ${\nu }_0=0.2$, $\omega =\frac{1}{2}$, and different values of deformation parameter $r=0.01,0.25$, and 0.5 in (a), (b), and (c) respectively.

Figure 12

The width $\mathrm{\Delta }F$ of the halfinteger Shapiro step $\frac{3}{2}$ as a function of the ac amplitude $F_{\mathrm{ac}}$. The rest of parameters are the same as in Fig. 11.

Figure 13

The width $\mathrm{\Delta }F$ of the Shapiro step $\frac{4}{3}$ and $\frac{5}{3}$ as a function of the ac amplitude $F_{\mathrm{ac}}$. The rest of parameters are the same as in Fig. 11.

Figure 14

The width $\mathrm{\Delta }F$ of the Shapiro step $\frac{5}{4}$ and $\frac{7}{4}$ as a function of the ac amplitude $F_{\mathrm{ac}}$. The rest of parameters are the same as in Fig. 11.

Figure 15

The width $\mathrm{\Delta }F$ of the Shapiro steps of subharmonic Shapiro steps with denominator being equal to 5 as a functions of the ac amplitude $F_{\mathrm{ac}}$. The rest of parameters are the same as in 11.