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One of the most fascinating and puzzling aspects of non-Hermitian systems is their spectral degeneracies, i.e., exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce to form a defective state space. While coupled magnetic systems are natural hosts of EPs, the relation between the linear and nonlinear spin dynamics in the proximity of EPs remains relatively unexplored. Here we theoretically investigate the spin dynamics of easy-plane magnetic bilayers in the proximity of exceptional points. We show that the interplay between the intrinsically dissipative spin dynamics and external drives can yield a rich dynamical phase diagram. In particular, we find that, in antiferromagnetically coupled bilayers, a periodic oscillating dynamical phase emerges in the region enclosed by EPs. Our results not only offer a pathway for probing magnetic EPs and engineering magnetic nano-oscillators with large-amplitude oscillations, but also uncover the relation between exceptional points and dynamical phase transitions in systems displaying non-linearities.