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In this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi-Pasta-Ulam problem, a highly oscillatory Hamiltonian system. We also showdoi:10.1017/s1446181109000042 fatcat:37kbx25qg5fzdmmhwv6gt4sk3u