When a system is driven periodically in time it can display dynamical localization, i.e its energy grows extremely slowly and may saturate at a value smaller than the infinite temperature limit. We show that by making the period of the perturbation longer this phenomenon is destroyed and the energy of the system grows quickly and saturates at the infinite temperature limit. By performing analytical calculations and numerical simulations on a specific spin-chain model, we convincingly show that this phenomenon is related to the breaking down of a particular form of perturbation theory (in the duration in of driving) and that the system displays a phase transition as a function of the duration of the driving. These findings are valid for both classical and quantum systems. We finally discuss how realize this interesting non-equilibrium physics in cold atom experiments.
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