Simulations of pilot-wave dynamics in a simple harmonic potential

We present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states.

Figure 1

(a) Dependence of simulated trajectories on vibrational forcing $\gamma /{\gamma }_F$ and dimensional spring constant $k$ over the range $0.93\le \gamma /{\gamma }_F\le 0.98$ and $0.1\le k\le 5.8\mu \mathrm{N}/\mathrm{m}$ with resolution $\mathrm{\Delta }(\gamma /{\gamma }_F)=0.0016$ and $\mathrm{\Delta }k=0.3\mu \mathrm{N}/\mathrm{m}$. Open circles correspond to stable circular orbits and black closed circles to chaotic trajectories. Colored circles denote the various periodic and quasiperiodic orbits categorized in (b). The numbers in (a) correspond to the specific trajectories in (b). The dashed line denotes the locus of the curve $\sqrt{m/k}=T_d/(1-\gamma /{\gamma }_F)$, above which the memory time exceeds the characteristic crossing or orbital time $T_M>T_s$.

Figure 2

(a) Example of one period of a quasiperiodic trefoil trajectory for $\gamma /{\gamma }_F=0.9448$ under the influence of a harmonic potential with $k=2.5\mu \mathrm{N}/\mathrm{m}$, corresponding to $\mathrm{\Lambda }=0.5459$. (b) Example of a purely periodic lemniscate trajectory arising for $\gamma /{\gamma }_F=0.9516$ and $k=2.2\mu \mathrm{N}/\mathrm{m}$, corresponding to $\mathrm{\Lambda }=0.5893$. Accompanying panels show the time evolution of the dimensionless distance from the origin $R$ and the dimensionless angular momentum $L_z$, as defined in Eqs. (3) and (4). Average values are indicated by red lines and $u_0$ denotes the free-walking speed. Here $T$ is one orbital period. Trajectories are color coded according to walker speed nondimensionalized by the free speed $u/u_0$. Arrows indicate direction of motion.

Figure 3

Periodic and quasiperiodic trajectories found numerically. Trajectories are color coded according to the walker speed $u\left(t\right)$. Accompanying panels show the evolution of the dimensionless orbital radius $R$ and angular momentum $L_z$ of each trajectory; $T$ is the orbital period for each trajectory. The horizontal red lines denote mean values of each quantity and $u_0$ denotes the free-walking speed of a straight-line walker. The relevant parameter values for each trajectory are as follows: (a) $\gamma /{\gamma }_F=0.9618, k=1.9\mu \text{N}/\text{m}$, and $\mathrm{\Lambda }=0.6962$; (b) $\gamma /{\gamma }_F=0.9595, k=2.2\mu \text{N}/\text{m}$, and $\mathrm{\Lambda }=0.6227$; (c) $\gamma /{\gamma }_F=0.9664, k=2.2\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=0.6413$; and (d) $\gamma /{\gamma }_F=0.9459, k=2.8\mu \text{N}/\text{m}$, and $\mathrm{\Lambda }=0.4983$. Arrows indicate direction of motion.

Figure 4

Dimensionless mean radius $\overline{R}$ and mean angular momentum ${\overline{L}}_z$ for the periodic and quasiperiodic trajectories observed for $\gamma /{\gamma }_F>0.95$. The linear stability analysis of Labousse et al. [14] demonstrates that quantization is largely absent for lower values of $\gamma /{\gamma }_F$. Data have been symmetrized about ${\overline{L}}_z=0$.

Figure 5

Decomposition of chaotic trajectories into quasiperiodic components. The trajectories arise for (a) $\gamma /{\gamma }_F=0.9470, k=3\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=0.4717$ and (b) $\gamma /{\gamma }_F=0.9675, k=3\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=0.5733$. In (a), a lemniscate (red) and period-doubled lemniscate (blue) are apparent, while a drifting loop (green) is apparent in (b).

Figure 6

Time series of (a) the dimensionless orbital radius $R$ and (b) the dimensionless angular momentum $L_z$ for the chaotic trajectory in Fig. 5 with $\gamma /{\gamma }_F=0.9470, k=3\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=0.4717$. The red and blue portions in each panel correspond to the red and blue path segments highlighted in Fig. 5. The yellow boxes are expanded below.

Figure 7

Time series of (a) the dimensionless orbital radius $R$ and (b) the dimensionless angular momentum $L_z$ for the chaotic trajectory in Fig. 5 with $\gamma /{\gamma }_F=0.9675, k=3\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=0.5733$. The green portion in each panel corresponds to the green section highlighted in Fig. 5. The yellow boxes are expanded below.

Figure 8

(a) A chaotic trajectory is sliced into successive red and green subtrajectories. The color along the trajectory changes from red to green or vice versa whenever a local maximum in radial extent $|{\mathit{x}}_p\left(t\right)|$ is achieved. Here $\gamma /{\gamma }_F=0.971, k=0.74\mu \mathrm{N}/\mathrm{m}$, and $\mathrm{\Lambda }=1.132$. (b) Example of clustered data from subtrajectories along the chaotic trajectory in (a). Each blue circle, an $({\overline{L}}_z,\overline{R})$ pair, corresponds to a particular subtrajectory. Each red circle denotes the centroid of a cluster determined using the $K$-means algorithm. The overlying trajectories in black correspond to the periodic and quasiperiodic states shown in Fig. 4. The grid lines are the same as those used in Fig. 4.

Figure 9

Shown in green, red, and blue are centroids of clusters for chaotic trajectories at $\gamma /{\gamma }_F=0.971$, for spring constants in the range $0.1\le k\le 5.8\mu \mathrm{N}/\mathrm{m}$, with corresponding $\mathrm{\Lambda }$ in the range $0.3986\le \mathrm{\Lambda }\le 3.254$. The blue, red, and green markers denote spring constants in the ranges $k<0.57\mu \mathrm{N}/\mathrm{m}, 0.57\le k<1.36\mu \mathrm{N}/\mathrm{m}$, and $k\ge 1.36\mu \mathrm{N}/\mathrm{m}$, respectively.