Unsteady dynamics of a classical particle-wave entity

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We employ different spatial wave forms to understand the role played by both wave oscillations and spatial wave decay in the walking dynamics. We observe steady walking motion as well as unsteady motions such as oscillating walking, self-trapped oscillations, and irregular walking. We explore the dynamical and statistical aspects of irregular walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.

Figure 1

Schematic of the walking droplet particle-wave system, showing a droplet of dimensionless mass $\kappa$ located at $x_d$ and walking horizontally with velocity ${\dot{x}}_d$. The droplet experiences a wave force, ${-\beta \partial h/\partial x|}_{x=x_d}$, from the underlying wave field $h(x,t)$ (blue curve) and a drag force, $-{\dot{x}}_d$. The underlying wave field $h(x,t)$ is the superposition of the individual waves (black and gray curves), of the spatial form $A\left(x\right)$ and decaying exponentially in time, that are continuously generated by the droplet along its trajectory.

Figure 2

Comparison of the following three different (a) wave forms $A\left(x\right)$ and (b) their gradients: a Gaussian wave form $e^{-{(x/2)}^2}$ (yellow dash-dotted curve), a Bessel function wave form $J_0\left(x\right)$ (blue solid curve), and a sinusoidal wave form $cos\left(x\right)/2$ (red dotted curve).

Figure 3

Linear stability in the ($\kappa ,\beta$) parameter space for inline perturbations to the steady walking solution of a single droplet using (a) a Gaussian wave form $e^{-{(x/2)}^2}$, (b) a Bessel wave form $J_0\left(x\right)$, and (c) a sinusoidal wave form $cos\left(x\right)/2$. The gray dashed curve in (b) shows the path traversed in the parameter space for typical experimental parameters as the driving acceleration (or the memory) is increased. In each of (b) and (c), the inset shows the instability boundary using a logarithmic scale in both horizontal and vertical directions.

Figure 4

Instability boundary of the steady walking solution in the $(\kappa ,\beta )$ parameter space for a wave form $A\left(x\right)=cos\left(x\right)e^{-{(x/2l)}^2}/2$. Instability boundary (top panel) and the corresponding wave forms (bottom panel) are shown for $l=2.5$ (blue dash-dotted curve), $l=5$ (yellow dotted curve), $l=100$ (purple dashed curve), and $l\to \infty$ (green solid curve, wave form not shown).

Figure 5

Walking behaviors for a Bessel wave form. (a) Different dynamical behaviors observed in the $(\kappa ,\beta )$ parameter space at $t=1000$ from simulations initiated at $t=0$ with the droplet in the steady walking state for $t\le 0$. We explore the parameter space region $0.005\le \kappa \le 0.15$ and $0<\beta \le 500$ with resolution $\mathrm{\Delta }\kappa =0.005$ and $\mathrm{\Delta }\beta =1$. We observe steady walking (beige), oscillating walking (orange), self-trapped oscillations (navy blue), and irregular walking (blue). The solid black curve is the linear stability curve separating the steady walking and the unsteady walking regime. Typical trajectories of (b) irregular walking ($\kappa =0.10, \beta =101$), (c) oscillating walking ($\kappa =0.11, \beta =149$), and (e) self-trapped oscillations ($\kappa =0.12, \beta =139$) are shown along with the phase-space plots for (d) oscillating walking and (f) self-trapped oscillations.

Figure 6

Walking behaviors for a sinusoidal wave form. (a) Different dynamical behaviors observed in the $(\kappa ,\beta )$ parameter space at $t=1000$ from simulations initiated at $t=0$ with the droplet in the steady walking state for $t\le 0$. We explore the parameter space region $0.025\le \kappa \le 0.55$ and $0<\beta \le 500$ with resolution $\mathrm{\Delta }\kappa =0.025$ and $\mathrm{\Delta }\beta =1$. We observe steady walking (beige), oscillating walking (orange), self-trapped oscillations (navy blue), and irregular walking (blue). The solid red curve is the linear stability curve separating the steady walking and the unsteady walking regime. Typical trajectories of (b) irregular walking ($\kappa =0.30, \beta =71$), (c) oscillating walking ($\kappa =0.30, \beta =221$), and two different kinds of self-trapped oscillations (d) ($\kappa =0.30, \beta =171$) and (f) ($\kappa =0.30, \beta =401$) are shown. Phase space trajectories of self-trapped oscillations in (d) and (f) are shown in (e) and (g) respectively.

Figure 7

Comparison of the chaotic behavior in the droplet's dynamics using (a) a sinusoidal wave field ($\kappa =0.2, \beta =35$) and (b) a Bessel wave field ($\kappa =0.1, \beta =90$). For the droplet's dynamics, the time series of velocity $v_d$ is shown in the left panel, the projection of the chaotic attractor in the $(v_d,{\dot{v}}_d)$ phase space in the middle panel, and the 1D return map for the maximum speed in each oscillation in the right panel.

Figure 8

(a) Velocity bifurcation diagram for the sinusoidal wave form showing the maxima and minima $v_n$ of the velocity time series as a function of the parameter $\beta$ and a fixed $\kappa =0.2$. The time series of velocity and the corresponding map for the maximum consecutive absolute values are shown for (b) $\beta =35$, (c) $\beta =220$, (d) $\beta =240$, and (e) $\beta =320$.

Figure 9

Switching dynamics in the irregular walking regime for the sinusoidal wave form at $\kappa =0.2$ and $\beta =35$. (a) The velocity time series along with circles at the extrema of the oscillations are shown. The red filled circles indicate the extreme values of velocity before and after the flip. (b) Return map of absolute value of consecutive extrema in the time series, i.e., the absolute value of the consecutive circles in the left panel. The red and black branch corresponds to the red and black circles in the time series. (c) The same time series as in (a) but the markers now highlight the extreme values after a flip with a fixed number $N$ of oscillations between them. The $N=0$ (red circles), $N=1$ (yellow circles), and $N=2$ (purple circles) points are shown. (d) The map showing consecutive absolute values of the extrema after a flip with the $N=0, N=1$, and $N=2$ branches highlighted.

Figure 10

Statistics of the flip-flop process for $\kappa =0.2$ and $\beta =65$ using the sinusoidal wave field. (a) Schematic showing that a typical velocity time series of the droplet in the irregular walking regime can be thought of as a sum of a flip-flop process and an exponentially increasing sinusoid which dictates the flip after the amplitude reaches some threshold value [56]. (b) Time series of velocity for a typical droplet's trajectory in the irregular walking regime and (c) the corresponding projection of the chaotic attractor. The attractor has two basins that are labeled left “$L$” and right “$R$”. (d) Probability distribution for having $m$ oscillations between flips. (e) Probability distribution for the number of jumps $J$ in a given sequence of $M=30$ steps. In both (d) and (e), the histogram is from the numerical simulations while the red curves are best fits obtained using Eqs. (21) and (22) respectively.

Figure 11

Stationary probability distribution and autocorrelation for the velocity flip-flop process at $\kappa =0.2$ and $\beta =65$ using the sinusoidal wave field. (a) Stationary probability distribution for velocity. (b) Velocity autocorrelation function. In both panels, the blue curve is from numerical simulations of a walker while the green curve is the fit obtained by using the Langevin model with dichotomous noise. The steady walking speed $u=\sqrt{\beta /2-1}=5.6125$ is very close to the boundary of the Langevin model (vertical green line) in (a).

Figure 12

Diffusion-like motion of the droplet in the irregular walking regime for the sinusoidal wave form. Typical trajectories at (a) $\beta =65$, (b) $\beta =165$, (c) $\beta =200$, and (d) $\beta =300$ are shown for a fixed $\kappa =0.2$. (e) Time dependent diffusion exponent $\alpha \left(t\right)$ as a function of time for the trajectories shown in (a)–(d). For the statistical analysis, an ensemble of 5000 trajectories were simulated with each trajectory run for $t={10}^5$ from which we sampled 1000 logarithmically spaced points. The 5000 trajectories were generated by adding a small random perturbation in the range [0,0.1] in the droplet's steady walking speed at the start of the simulation.