Hysteretic depinning of a particle in a periodic potential: Phase diagram and criticality

We consider a massive particle driven with a constant force in a periodic potential and subjected to a dissipative friction. As a function of the drive and damping, the phase diagram of this paradigmatic model is well known to present a pinned, a sliding, and a bistable regime separated by three distinct bifurcation lines. In physical terms, the average velocity $v$ of the particle is nonzero only if either (i) the driving force is large enough to remove any stable point, forcing the particle to slide or (ii) there are local minima but the damping is small enough, below a critical damping, for the inertia to allow the particle to cross barriers and follow a limit cycle; this regime is bistable and whether $v>0$ or $v=0$ depends on the initial state. In this paper, we focus on the asymptotes of the critical line separating the bistable and the pinned regimes. First, we study its behavior near the “triple point” where the pinned, the bistable, and the sliding dynamical regimes meet. Just below the critical damping we uncover a critical regime, where the line approaches the triple point following a power-law behavior. We show that its exponent is controlled by the normal form of the tilted potential close to its critical force. Second, in the opposite regime of very low damping, we revisit existing results by providing a simple method to determine analytically the exact behavior of the line in the case of a generic potential. The analytical estimates, accurately confirmed numerically, are obtained by exploiting exact soliton solutions describing the orbit in a modified tilted potential which can be mapped to the original tilted washboard potential. Our methods and results are particularly useful for an accurate description of underdamped nonuniform oscillators driven near their triple point.

Figure 1

Phase diagram of Eq. (1) for a driven particle in the washboard potential $V\left(x\right)=-cos\left(x\right)$ in the large-time asymptotics. Control parameters are the driving force $f>0$ and the damping parameter $\alpha \equiv \gamma /\sqrt{m}$. In the stable limit cycle domain, the mean velocity $v$ of the particle is finite (and the motion is periodic), while it is zero in the stable fixed point domain (the particle is pinned in a local minimum). In the bistable regime, whether $v=0$ of $v>0$ depends on the initial conditions. The magenta dash-dotted line $f_c\left(\alpha \right)$ represents a homoclinic bifurcation, the green dashed line a finite-period saddle-node bifurcation, while the solid light blue line is an infinite period bifurcation. The lines are obtained using standard numerical integration methods for Eq. (1). In this paper we obtain analytical expressions for $f_c\left(\alpha \right)$ in the $\alpha \to 0$ limit for an arbitrary pinning potential $V\left(x\right)$, and we describe the universal power-law behavior of $f_c\left(\alpha \right)$ in the $\alpha \to {\alpha }_c$ limit when approaching the triple point where the three bifurcation lines meet.

Figure 2

Hysteresis loops in the velocity-force characteristics for $V\left(x\right)=-cos\left(x\right)$ in Eq. (1). For high damping, corresponding to masses $m<m_c$ at fixed $\gamma$, with $\gamma =1$ in this case, the particle depins at $f_c^0\equiv 1$ (magenta lines with empty points), without displaying hysteresis. For masses $m>m_c$ (green lines with filled points), there is a hysteretic depinning: Slowly increasing the applied force from $f=0$, the particle depins at $f_c^0$, acquiring a finite steady-state velocity only above it. Then, slowly decreasing $f$ from above $f_c^0$, the particle gets pinned only at the critical force $f_c\left(m\right)<f_c^0$. In this paper we analytically calculate the hysteretic range $[f_c^0-f_c\left(m\right)]$ for masses close to $m_c$.

Figure 3

Phase-space trajectories obtained numerically from Eq. (1), using different initial conditions, for the weakly damped case $\alpha <{\alpha }_c$. In (a), $f<f_c\left(\alpha \right)$: All initial conditions starting at the left margin of the plot with different velocities are finally spirally attracted to one of the periodic images of the stable fixed points of the tilted washboard potential, such that $V^{\prime }\left(x\right)-f=0$ and $V^{″}\left(x\right)>0$. In (b), $f=f_c\left(\alpha \right)$: Low initial velocity trajectories are trapped but large velocity trajectories approach the homoclinic orbit (in purple dashed line). In (c), $f_c\left(\alpha \right)<f<f_c^0$: Trajectories are either trapped or acquire a finite time-averaged velocity converging to the stable limit cycle. In (d), $f=f_c^0$: Stable fixed points disappear, and all trajectories are attracted to the running periodic orbit with a finite average velocity. With dashed lines we show trajectories starting with infinitesimally positive velocity at one of the unstable fixed points $x^{*}$ of the tilted potential (open circles), such that $V^{\prime }\left(x^{*}\right)-f=0$ and $V^{″}\left(x^{*}\right)<0$. The homoclinic orbit corresponds exactly to the dash line in (b).

Figure 4

The nontilted (inset) and tilted (main) potentials ${\mathcal{V}}_0\left(x\right)$ and ${\mathcal{V}}_F\left(x\right)$ defined by Eqs. (8) and (9), respectively (parameters are $\mu =g=1$ and $F=\frac{1}{2}{F}_c^0$). A critical trajectory $x^{★}\left(t\right)$ of a massive particle, with the dynamics (10), is represented with a green dashed line: It joins the location $-x_0$ of a local maximum at time $t=-\infty$ to its periodic image $x_0$ at time $t=\infty$.

Figure 5

Landscape of force $F\left(x\right)=-V^{\prime }\left(x\right)+f$ at external force $f$ close to depinning ($f=f_c^0-\delta f$ with $\delta f>0$), presenting the critical points $x_c^{\pm }=x_c\pm \sqrt{\delta f/\kappa }$.

Figure 6

Steady-state time-averaged velocity as a function of the mass above the critical mass, $m-m_c$. With a continuous green line we show a fitted power law, where the exponent and the critical mass are fitting parameters. Then we find $m_c=0.70757\pm 0.00002$.

Figure 7

For the potential $V\left(x\right)=-cos\left(x\right)$, as the mass of the particle gets close to the critical mass from above, the difference between the critical forces tends to zero as given by Eq. (32), indicated by a continuous green line. Here we employ a numerical simulation using $\ell \equiv 2\pi$, $\gamma \equiv 1$, and ${\mathfrak{f}}_0\equiv 1$.

Figure 8

Critical force as a function of the damping parameter $\alpha$, defined as $\alpha =\gamma /\sqrt{m}$. The expected behavior, from Eq. (62), is correct on the limit $\alpha \ll {\alpha }_c$.

Figure 9

Critical mass $m_c$ vs. the normal form exponent $\mathrm{{\rm Y}}$ (a) and $\beta =1-1/\mathrm{{\rm Y}}$ (b). A fair linear fit is obtained in (b) in the full range of $\mathrm{{\rm Y}}$ shown in (a).

Figure 10

Critical exponent $\delta$, controlling the bistable driving force range $f_c^0-f_c\left(m\right)$, showing that $\delta \approx \mathrm{{\rm Y}}$ (main figure). Insets show two typical power-law fits $f_c^0-f_c\left(m\right)\approx {\left[m-m_c\left(\mathrm{{\rm Y}}\right)\right]}^{\delta }$ used to extract $\delta$ vs. $\mathrm{{\rm Y}}$.

Figure 11

Large mass behavior of the critical force $f_c\left(m\right)$ vs. $\mathrm{{\rm Y}}$. Insets: Fits to $f_c\left(m\right)\approx A\left(\mathrm{{\rm Y}}\right)/\sqrt{m}$. Main figure: $A\left(\mathrm{{\rm Y}}\right)$ for a range of $\mathrm{{\rm Y}}$ around the standard $\mathrm{{\rm Y}}=2$ [or $-cos\left(u\right)$ potential].

Figure 12

On the cosine potential, the steady-state time-averaged velocity, as a function of the force, undergoes different depinning transitions depending on the mass of the particle. For zero-mass, the black dashed line shows the analytical result, with $\beta =1/2$. For masses above $m_c$, we can only see a mild tendency toward the regime $\beta \to 0$ that describes the expected behavior $v\sim {\left\{ln[f-f_c\left(m\right)]\right\}}^{-1}$.

Figure 13

Using Eq. (60), we fit the exponent by proposing different values for the critical mass. Here we show the resulting reduced chi-square for each value, as a test for the goodness of fit. From the lowest value of the reduced reduced chi-square, we estimate $m_c=0.70757\pm 0.00002$.