Phase transitions on a multiplex of swarmalators

Dynamics of bidirectionally coupled swarmalators subject to attractive and repulsive couplings is analyzed. The probability of two elements in different layers being connected strongly depends on a defined vision range rc which appears to lead both layers in different patterns while varying its values. Particularly, the interlayer static sync π has been found and its stability is proven. First-order transitions are observed when the repulsive coupling strength σr is very small for a fixed rc and, moreover, in the absence of the repulsive coupling, they also appear for sufficiently large values of rc. For σr=0 and for sufficiently small values of rc, both layers achieve a second-order transition in a surprising two steps that are characterized by the drop of the energy of the internal phases while increasing the value of the interlayer attractive coupling σa and later a smooth jump, up to high energy value where synchronization is achieved. During these transitions, the internal phases present rotating waves with counterclockwise and later clockwise directions until synchronization, as σa increases. These results are supported by simulations and animations added as supplemental materials