Bistability in orbital trajectories of a chiral self-propelled particle interacting with an external field

In this paper, the dynamics of a self-propelled stochastic particle under the influence of an axisymmetric light field is experimentally studied. The particle under consideration has the main characteristic of carrying a light sensor in an eccentric location. For the chosen experimental conditions, the emerging trajectories are orbital, and, more interestingly, they suggest the existence of bistability. A mathematical model incorporating the key experimental components is introduced. By means of numerical simulations and theoretical analysis, it is found that, in addition to the orbiting behavior, the sensor location could produce trapped or diffusive behaviors. Furthermore, the study reveals that stochastic perturbation and the eccentric location of the sensor are responsible for inducing bistability in the orbital trajectories, supporting experimental observations.

Figure 1

Experimental setup. (a) The Kilobot is placed over a squared table of length $L=1.2\mathrm{m}$, illuminated by a flashlight located at height $H=1.4\mathrm{m}$. Robot top view: The sensor is located at an angle $\beta$ with respect to orientation $\mathit{n}$. (b) Photograph of the experiment. (c) Intensity profile measured at the surface of the table. The line stands for the fit of a super-Gaussian function.

Figure 2

Experimental results. (a) and (b) Deterministic behavior with $T=1\mathrm{s}$ and $T=2\mathrm{s}$, respectively. The Kilobot orbits around the spotlight with the sensor facing towards the center of the field. (c) and (d) Stochastic behavior with $T=1\mathrm{s}$ and $T=2\mathrm{s}$, respectively. The Kilobot performs more complex trajectories but is still biased by the position of the sensor. (e) and (f) Radial position as a function of time for the stochastic behavior with $T=1\mathrm{s}$ and $T=2\mathrm{s}$, respectively. Probability distribution (PDF) of (g), (h) the radial position and (i), (j) the azimuthal velocity. The black lines stand for the kernel density estimate.

Figure 3

Numerical results. (a) and (b) Estimated PDF for ${\tau }_{\alpha }={10}^{-3}$ and ${\tau }_{\alpha }={10}^{-2}$, respectively. (c)–(f) Radial position as a function of time and the corresponding probability distribution (PDF). The black lines stand for the kernel density estimate. The parameters are ${\tau }_{\eta }=1$, $a=4.5$, $\beta =1.4$, $d={10}^{-2}$, and (c), (d) ${\tau }_{\alpha }={10}^{-3}$ and (e), (f) ${\tau }_{\alpha }={10}^{-2}$.

Figure 4

Numerical results. (a)–(f) Translational (MSD), rotational (MSRD), and azimuthal mean square displacement (MSAD) for different $\beta$ values. The parameters are ${\tau }_{\eta }=1$, $a=4.5$, $d={10}^{-2}$, and (a)–(c) ${\tau }_{\alpha }={10}^{-3}$ and (d)–(f) ${\tau }_{\alpha }={10}^{-2}$.

Figure 5

Theoretical approximation. (a) Stationary probability ($P_s$) as a function of ${\tau }_{\alpha }$. The line stands for the deterministic equilibrium position given by Eq. (9). (b) and (c) Stability diagrams as a function of ${\tau }_{\alpha }$ and ${\tau }_{\eta }$, and ${\tau }_{\alpha }$ and $\alpha$, respectively. The solid lines stand for the condition for orbital trajectories given by Eq. (10). The parameters are $a=4.5$, $d={10}^{-5}$, and (b) $\alpha =1.4$ and (c) ${\tau }_{\eta }=1$.