Fig. 3: Designated leaders induce periodic limit cycles.

a, b Features of DL beating explained with schematic (a) and micrograph sequence (b) of a 2-particle heterogeneous system. The leader particle is able to grow a large bubble promptly and subsume the smaller bubbles of neighbouring particles across several rounds of bubble coalescence. Scale bars, 1 mm. c, d, Phase portraits of homogeneous (c) and heterogeneous (d) systems of N = 2, 3, 6, and 8. Only the latter is able to maintain the closed-loop orbits at high particle counts. e Schematic of recurrence time calculation. The recurrence time is the time it takes to return from a given system configuration to the neighbourhood of said configuration (see “Methods”). f Recurrence histogram compiling all of the recurrence times observed across experiments of the 2-particle heterogeneous system (N = 1 + 1DL). g Recurrence entropy as a function of N for both homogeneous (yellow) and heterogeneous/DL (blue) systems. Low recurrence entropy is a quantitative indicator of periodic behaviour. The homogeneous system's recurrence entropy trends upward, suggesting a decay in periodicity, while the DL system's entropy remains low in accordance with its observed periodicity even at high N.