Large deviation theory has emerged as the natural language with which to study transitions between dynamical phases, such as the glass transition. Here we use this approach to show that the entrainment of an oscillator to an external driving force can be described as a phase transition between two regions in a joint space-time representation. Specifically, we numerically obtain exact solutions of the large deviation function for a discrete, finite model of an oscillator under a periodically varying external force. We find that the first derivative of the expectation value of the current diverges in the limit of large system size. For weak forcing, perturbation theory allows us to relate the observed frequency of the oscillator to the spectrum of eigenvalues for the unforced system. We find that in the entrainment region where the oscillator exhibits the frequency of the driving force, there is a strong coupling between the ground state and excited states.
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