Double explosive transitions to synchronization and cooperation in intertwined dynamics and evolutionary games

Collective behavior, from murmurations to synchronized beating of heart cells, governs some of the most beautiful and important aspects of nature. Likewise, cooperation - the act of sacrificing personal benefits for the common good - is one of the pillars of social evolution, and it is the basis for the emergence of collective organized actions from single-cell organisms to modern human societies. Here we merge these two phenomena into a single model, considering an ensemble of networked oscillators, where each oscillator can be either a cooperator or a defector, and with only cooperators contributing to synchrony. At the same time, the value of the order parameter in the neighborhood of each oscillator is considered as an effective local temperature which determines the strategy updating procedure in the evolutionary game. The emergence of cooperation is thus intertwined with that of synchronization, producing a novel and fascinating dynamics which includes a double explosive transition.