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Evolutionary Kuramoto Dynamics

Common models of synchronizable oscillatory systems consist of a collection of coupled oscillators governed by a collection of differential equations. The ubiquitous Kuramoto models rely on an {\em a priori} fixed connectivity pattern facilitates mutual communication and influence between oscillators. In biological synchronizable systems, like the mammalian suprachaismatic nucleus, enabling communication comes at a cost -- the organism expends energy creating and maintaining the system -- linking their development to evolutionary selection. Here, we introduce and analyze a new evolutionary game theoretic framework modeling the behavior and evolution of systems of coupled oscillators. Each oscillator in our model is characterized by a pair of dynamic behavioral traits: an oscillatory phase and whether they connect and communicate to other oscillators or not. Evolution of the system occurs along these dimensions, allowing oscillators to change their phases and/or their communication strategies. We measure success of mutations by comparing the benefit of phase synchronization to the organism balanced against the cost of creating and maintaining connections between the oscillators. Despite such a simple setup, this system exhibits a wealth of nontrivial behaviors, mimicking different classical games -- the Prisoner's Dilemma, the snowdrift game, and coordination games -- as the landscape of the oscillators changes over time. Despite such complexity, we find a surprisingly simple characterization of synchronization through connectivity and communication: if the benefit of synchronization $B(0)$ is greater than twice the cost $c$, $B(0) > 2c$, the organism will evolve towards complete communication and phase synchronization. Taken together, our model demonstrates possible evolutionary constraints on both the existence of a synchronized oscillatory system and its overall connectivity.