Random perturbations applied in tandem to an ensemble of oscillating objects can synchronize their motion. We study multiple copies of an arbitrary dynamical system in a stable limit cycle, described via a standard phase reduction picture. The copies differ only in their arbitrary phases $phi$. Weak, randomly-timed external impulses applied to all the copies can synchronize these phases over time. Beyond a threshold strength there is no such convergence to a common phase. Instead, using statistical sampling, we find remarkable erratic synchronization: successive impulses produce stochastic fluctuations in the phase distribution $q(phi)$, ranging from near-perfect to more random synchronization. The sampled entropies of these phase distributions themselves form a steady-state ensemble, whose average can be made arbitrarily negative by tuning the impulse strength. A stochastic dynamics model for the entropys evolution accounts for the observed exponential distribution of entropies and for the stochastic synchronization phenomenon.
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