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The population dynamics of an assembly of globally coupled homogeneous phase oscillators is studied in presence of non-Gaussian fluctuations. The variance of the underlying stochastic process grows as t + beta2t2 (beta being a constant) and therefore exhibits a super-diffusive behavior. The cooperative evolution of the oscillators is represented by an order parameter which, due to the ballistic beta2t2 contribution, obeys to a surprisingly complex bifurcation diagram. The specific class of super-diffusive noise sources can be represented as a random superposition of two Brownian motions with opposite drift and this allows to derive exact analytic results. We observe that besides the existence of the well known incoherent to coherent phase transition already present for Gaussian noise, entirely new and purely noise induced temporal patterns of the order parameter are realized. Hence, the ballistic contributions of the fluctuating environment does structurally modify the bifurcation diagram obtained for Gaussian noise. To illustrate potential implications of the developed class of models, we explore the dynamic behavior of a swarm formed by a planar society of particles with coupled oscillator dynamics. For this collective dynamics, we discuss how noise-induced periodic orbits of the swarm's barycenter may emerge.