Multistable Free States of an Active Particle from a Coherent Memory Dynamics

We investigate the dynamics of a deterministic self-propelled particle endowed with coherent memory. We evidence experimentally and numerically that it exhibits several stable free states. The system is composed of a self-propelled drop bouncing on a vibrated liquid driven by the waves it emits at each bounce. This object possesses a propulsion memory resulting from the coherent interference of the waves accumulated along its path. We investigate here the transitory regime of the buildup of the dynamics which leads to velocity modulations. Experiments and numerical simulations enable us to explore unchartered areas of the phase space and reveal the existence of a self-sustained oscillatory regime. Finally, we show the coexistence of several free states. This feature emerges both from the spatiotemporal nonlocality of this path memory dynamics as well as the wave nature of the driving mechanism.

Figure 1

Experimental setup and walker’s starting up. (a) Schematics of the experimental setup. A bath of silicon oil is vertically vibrated at 50 Hz with a tunable acceleration amplitude ${\gamma }_m$. It is set to ${\gamma }_m=4.5\mathrm{g}$ to allow a submillimetric drop of the same liquid to bounce in a period doubling regime. (b) Time evolution of the bath acceleration. For $t<0$, ${\gamma }_m=4.5\mathrm{g}$ the bouncing drop has no horizontal motion. At time $t=0$, ${\gamma }_m$ is suddenly increased above the walking threshold ${\gamma }_W\approx 4.8\mathrm{g}$ for which the drop self-propels on the bath surface, typically ${\gamma }_W<{\gamma }_m<{\gamma }_F\approx 5\mathrm{g}$. (c) Snapshots of the starting up of the walker taken with a high-speed camera (here time step is $\mathrm{\Delta }t=320\mathrm{ms}$ with an image width $\approx 45\mathrm{mm}$). The wave profile is initially circular. The motion breaks this symmetry and accumulating secondary sources changes the form of the wave profile.

Figure 2

Starting up characterization of the walker. (a) Time evolution of the drop speed for short [$\mathrm{Me}=7$ (red)] and long memory [blue line ($\mathrm{Me}=36$)]. The two cases correspond physically to two different drops. Experimental data (dashed lines) fitted by the model Eq. (2) (solid lines) with fitting parameters $C=0.25\mathrm{m}/{\mathrm{s}}^3$, $\eta =3.12{\mathrm{s}}^{-1}$ and $C=1.1\mathrm{m}/{\mathrm{s}}^3$, $\eta =4.72{\mathrm{s}}^{-1}$, respectively. The walker converges to the final constant speed $v_f=6.5\pm 0.2$ and $v_f=14.6\pm 0.1\mathrm{mm}/\mathrm{s}$, respectively. (b) Experimental frequency $\nu$ of the damped speed oscillation as a function of the final speed for various drops sizes and linear fit (slope 0.164. pink area: 95% confidence interval). (c) Time evolution of the experimental density of secondary sources $\mathrm{\Delta }\rho$ along the trajectory (measured in $0.1\times {\mathrm{mm}}^{-1}$, black) and of the force exerted by the waves on the drop ${\mathcal{F}}_{\text{wave}}$ (per unit mass, $\mathrm{m}/{\mathrm{s}}^{-2}$, red) deduced from Eq. (3) for $\mathrm{Me}=36$. The reference $\mathrm{\Delta }\rho =0$ is taken in the final constant speed limit.

Figure 3

Existence of multiple free states. (a) Numerical color map of the order parameter $\chi =\min \{\mathrm{\Delta }v/v_{\mathrm{f}},1\}$ for $C=1.1\mathrm{m}/{\mathrm{s}}^3$ with Me and of $\eta$. Starting from an immobile particle, three distinct regimes are observed: constant speed ($\chi =0$), stable oscillations ($0<\chi <1$) and chaotic regime ($\chi =1$). Depending on the initial conditions, the region of stable oscillations presents also solutions with a constant speed. The circles along the dashed line at $\eta =6.2{\mathrm{s}}^{-1}$ at $\mathrm{Me}=10$ (black), $\mathrm{Me}=80$ (blue), and $\mathrm{Me}=400$ (red) correspond to the regimes in Fig. 3. (b) Two-dimensional representation of the numerical dynamics in the phase space $(v,\dot{v})$. The dynamics converges to either fixed point [$\mathrm{Me}=10$ (black)], speed limit cycle [$\mathrm{Me}=80$ (blue)], or strange attractor [$\mathrm{Me}=400$ (red)]. (c) Experimental evidence of coexisting free states at $\mathrm{Me}=45$. Speed of a same drop with respect to its relative position: the drop is locked into a limit cycle (orange diamonds). After perturbations (not shown) it converges to a solution with constant speed (blue square). The motion are straight but initial directions are ill defined.