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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Mathematical Biology

Background:

  • Synchronization and aggregation are ubiquitous phenomena observed in various natural and artificial systems.
  • Swarmalator systems, which model interacting agents, have garnered recent scientific attention.
  • The impact of time delays on swarmalator dynamics remains underexplored, despite their potential to introduce complex behaviors.

Purpose of the Study:

  • To investigate the effects of incorporating a phase delay into the internal dynamics of swarmalator systems.
  • To explore the novel synchronization patterns and transitions that emerge in delayed swarmalator models.
  • To establish a foundation for understanding how environmental factors influencing delay impact mathematical models.

Main Methods:

  • Development of a delayed swarmalator model with a focus on internal phase dynamics.
  • Analysis of system behavior across various parameter spaces and domains.
  • Utilizing the local and complex order parameter to characterize emergent dynamics.

Main Results:

  • Discovery of unique phenomena such as the 'boiling chimera' state and 'ring static synchronization'.
  • Identification of a double explosive transition: synchronous to asynchronous, then asynchronous to synchronous states.
  • Demonstration that phase delay significantly enriches the dynamic repertoire of swarmalator systems.

Conclusions:

  • The inclusion of phase delay in swarmalator models leads to fundamentally new collective behaviors and transitions.
  • These findings provide insights into phenomena relevant to understanding brain diseases characterized by altered synchronization.
  • This work underscores the importance of considering environmental delays in mathematical modeling for increased realism.