Fig. 5: Effects of time-varying interactions and lateral forces in swarmalators.

a The mean fraction of swarmalators in clusters 〈f_c〉 relaxes slower than the mean synchronization 〈σ〉 after a switch in Γ (the light colored areas show the SEM over multiple runs). We selected Γ = −05 and Γ = 0.2 here, because these values result in the same swarmalator motility (see Fig. S11; SI section 6). b Switching back and forth between two values of Γ results in a loop in the 〈f_c〉–〈σ〉 space, because the system visits different states depending on the direction of the switching. 〈σ〉 and 〈f_c〉 are averaged over multiple runs. c State loop in the 〈f_c〉–〈σ〉 space in simulations, for slow switching (every 400δt) of Γ (gray, as in (b)) as well as for fast switching (every 200δt) from different initial Γ (purple and orange). 〈σ〉 and 〈f_c〉 are averaged over 50 runs. d Illustration of the state loop with two smaller loops (purple and orange) that are a result of faster switching in Γ from different initial values of Γ (see inset). e Heat map of the averaged normal displacement \({\langle {d}_{\perp,ij}\rangle }_{ij}\) against the synchronization σ_ij and reference point distance \(\left\vert {{{{\bf{q}}}}}_{ij}\right\vert\) between neighboring swarmalators in experiment. f The normal (perpendicular) displacement d_⊥,ij is defined as the component of the displacement d_i orthogonal to the direction to the neighboring reference point q_ij, multiplied by the sign of the angular velocity of i. g Snapshots of a rotating swarmalator cluster in simulations at different times (colors yellow to green to blue). h Mean and standard deviation of the angular velocity ω of a rotating cluster as a function of the number N of swarmalators in the system and the coupling parameter Γ in simulations In (g, h), the swarmalators were initialized on a hexagonal grid with a separation of 8 μm, the system was allowed to relax for 3000δt, and the data was collected and averaged over 30,000δt. In (a, b, c), the initial density is around 0.003 μm⁻². Source data are provided as a Source data file.