  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.
