  Synchrony is inevitable in many oscillating systems -- from the canonical
alignment of two ticking grandfather clocks, to the mutual entrainment of
beating flagella or spiking neurons. Yet both biological and manmade systems
provide striking examples of spontaneous desynchronization, such as failure
cascades in alternating current power grids or neuronal avalanches in the
mammalian brain. Here, we generalize classical models of synchronization among
heterogenous oscillators to include short-range phase repulsion among
individuals, a property that abets the emergence of a stable desynchronized
state. Surprisingly, we find that our model exhibits self-organized avalanches
at intermediate values of the repulsion strength, and that these avalanches
have similar statistical properties to cascades seen in real-world systems such
as neuronal avalanches. We find that these avalanches arise due to a critical
mechanism based on competition between mean field recruitment and local
displacement, a property that we replicate in a classical cellular automaton
model of traffic jams. We exactly solve our system in the many-oscillator
limit, and obtain analytical results relating the onset of avalanches or
partial synchrony to the relative heterogeneity of the oscillators, and their
degree of mutual repulsion. Our results provide a minimal
analytically-tractable example of complex dynamics in a driven critical system.
