Topological transitions in swarmalators systems

Abstract

After its development, the swarmalators model attracted a great deal of attention since it was found to be very suitable to reproduce several behaviors in collective dynamics. However, few works explain the transitions that are observed while varying system parameters. In this letter, we demonstrate that the changes observed in swarmalator dynamics are governed by changes in the system's topology. To provide a deeper understanding of these changes, we present a topological framework for the swarmalator system and determine the topological charge $Q$ and the helicity $γ$ of the corresponding topology. Investigations on synchronization and transition to synchronization are studied using this topological charge and the variance of the helicity.

Figures (4)

• Figure 1: Topological phase dynamics in the case of a second order transition corresponding to the regime of Fig. 10 in Ref. kongni2. After reaching the critical coupling value, the topological charge is $Q=0$, meaning that the particles start to be or already are in a coherent state. The progressive tendency towards zero of the helicity variance,$V(\gamma)$, is a hallmark of the second-order transition, and when it reaches a value of zero, all elements of the network are perfectly synchronized.
• Figure 2: Topological phase dynamics corresponding to the regime studied in Fig. 10 of Ref. kongni1. The two energy transitions, i.e., the drop at $\sigma_{a1} \simeq 0.125$, and the jump at $\sigma_{a2} \simeq 0.65$, find explanations in topological changes measured by the values of the winding number, as at the first transition it varies from $Q = 4$ (disorder state) at $\sigma_{a1}$ to $Q = 1$ (vortex state) at $\sigma_{a} = 0.15$, and in the second transition the charge varies from $Q = -1$ (anti-vortex) at $\sigma_{a2}$ to $Q = 0$ (trivial topology) at $\sigma_{a} = 0.675$.
• Figure 3: Topological phase dynamics in swarmalators with delayed interactions as considered in Ref. swarm6. Varying the delay $\tau$ leads the network from disorder to a ring static sync state through the boiling state, which is a non-topological chimera with $J = 0.1$ and $K = -0.5$.
• Figure 4: Topological transition in swarmalators with delayed interactions as considered in Ref. swarm6, while varying the phase coupling $K$ for $\tau = 4.5$. The system moves from RSS to a sync state through disorder followed by vortex-antivortex topologies that are non detectable using the order parameter $R$ (blue curve in panel a).