Generation of aperiodic motion due to sporadic collisions of camphor ribbons

We present numerical and experimental results for the generation of aperiodic motion in coupled active rotators. The numerical analysis is presented for two point particles constrained to move on a unit circle under the Yukawa-like interaction. Simulations exhibit that the collision among the rotors results in chaotic motion of the rotating point particles. Furthermore, the numerical model predicts a route to chaotic motion. Subsequently, we explore the effect of separation between the rotors on their chaotic dynamics. The numerically calculated fraction of initial conditions which led to chaotic motion shed light on the observed effects. We reproduce a subset of the numerical observations with two self-propelled ribbons rotating at the air-water interface. A pinned camphor rotor moves at the interface due to the Marangoni forces generated by surface tension imbalance around it. The camphor layer present at the common water surface acts as chemical coupling between two ribbons. The separation distance of ribbons ($L$) determines the nature of coupled dynamics. Below a critical distance ($L_T$), rotors can potentially, by virtue of collisions, exhibit aperiodic oscillations characterized via a mixture of co- and counterrotating oscillations. These aperiodic dynamics qualitatively matched the chaotic motion observed in the numerical model.

Figure 1

Theoretical consideration of two point particles with unit mass, which are moving on a unit circle. $O_1$ and $O_2$ correspond to the pivots, and hence, $L$ corresponds to the pivot-to-pivot distance between the ribbons in the experiments. This schematic figure is for the counterrotating case; however, the Hamiltonian in Eq. (9) remains the same for the corotating case as well.

Figure 2

(a) The variation of spectrum of Lyapunov exponents $\left({\lambda }_i\right)$ as a function of separation, $L$, between the two pivots. The Poincaré sections (taken at ${\theta }_1=0$) for (b) chaotic and (c) quasiperiodic motions at $L=1.5$ with two different initial conditions (${\theta }_1=2.4, {\dot{\theta }}_1=2.7, {\theta }_2=0, {\dot{\theta }}_2=5.72$) and (${\theta }_1=1.02, {\dot{\theta }}_1=2.03, {\theta }_2=0, {\dot{\theta }}_2=6$), respectively. The time series of (b) and (c) are shown in (d) and (e), respectively. The solid (black) and dashed (red) lines correspond to the first and second rotors, respectively.

Figure 3

The time series of (a) $sin\left({\theta }_1\right)$ and $sin\left({\theta }_2\right)$ and (b) $\dot{{\theta }_1}$ (left $y$ axis) and $\dot{{\theta }_2}$ (right $y$ axis) as a function of time of a chaotic motion with the initial conditions ${\theta }_1=3.31465984, {\dot{\theta }}_1=4.44012880, {\theta }_2=0, {\dot{\theta }}_2=-5$ at $K=2$ and $L=1.5$. The collision points are indicated as $C1,C2,...,C5$. The velocities of rotors near collision points, $C3$ and $C4$, are expanded in (b) where left and right $y$ axes represent ${\dot{\theta }}_1$ and ${\dot{\theta }}_2$, respectively. The schematic illustration of collisions near $C3$ and $C4$ where counterrotating rotors remain counterrotating and counterrotating rotors become corotating are shown in (c) and (d), respectively. The dashed (black) and solid (red) lines correspond to the first and second rotors, respectively.

Figure 4

(a) The fraction of initial conditions leading to chaotic motion ($f_{\text{chaos}}$) for different values of energy as a function of separation $L$. The Poincaré section for different trajectories corresponding to several initial conditions at different separations (b) $L=4$, (c) $L=3.5$, (d) $L=2.4$, and (e) $L=1.5$ at fixed energy $E=20$. The generation of Poincaré sections in (b)–(e) is the same as that of Figs. 2 and 2.

Figure 5

The variation of (a) $f_{\text{chaos}}$ and (b) the largest Lyapunov exponent as a function of the energy of the systems at $L=1.5$.

Figure 6

Schematic drawing of the collisions for (a) counterrotating and (b) corotating rotors. The dotted red lines represent the tangential momentum vectors of the rotors at the point of collision, and when shifted to respective centers of the circles, they are represented by solid red lines. The vectors $r_1$ and $r_2$ are radial vectors from the centers of respective circles to the point of collision represented by a solid black line. The arrows on circles represent the direction of rotation of each rotor. The rotors remain in counterrotation after collision in (a) while they change to corotations in (b).

Figure 7

Schematic of the experimental setup.

Figure 8

Aperiodic dynamics of the rotators result from the sporadic collisions between them. The time series corresponds to the “$y$ positions” of the rotors, i.e., the white regions present at each rotator's tips. The black and red lines correspond to the first and second rotors. E1, E2, and E3 represent the first, second, and third consecutive collisions between the rotors. The dashed lines correspond to a corotating pair, while the solid lines correspond to a counterrotating pair.