Delay induced double explosive transition in a swarmalator system

Synchronization and aggregation are important phenomena in the world, and many systems exhibit them. The swarmalator system is a good example, arousing interest from many scientists recently. Despite its rising interest, many do not consider delay of any form. Knowing that delay makes the system more realistic, surely containing rich dynamics, we use a delay term in the internal phase dynamics. Spanning through the various parameters and domains of this model, with the order parameter in its local and complex form, shows very interesting dynamics like the boiling chimera state in various domains and the ring static sync, which are primer discoveries in the field of delayed swarmalators. Also, the existence of phase delay helps us highlight a double explosive transition-first from synchronous to asynchronous and, secondly, from asynchronous to synchronous states, which have been used to understand some brain diseases. In this study, we lay down an additional base to the understanding of delay where environmental factors account for it, and so, we pave the way for more studies considering the reality of the environment on mathematical models.