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In this work we study the bifurcation of 2π-periodic solutions of the equation (l;E,σ) x + g(x) = E cos t - σh(t)x near a fixed solution of (l;E₀,0), when (E,σ) varies in a neighborhood of (E₀,0), where E₀ is a critical value of the parameter E, g, is like a general restoring term and h is a 2π-periodic function. We obtain the bifurcation diagram in the E,σ-plane under some generic conditions on g and h. Those conditions are not fulfilled in the significant case where h(t) Ξ 1, whose analysis leads to an essentialy distinct result. We also explain how the bifurcation diagram modifies when h passes from the generic situtation to the case h(t) Ξ 1. The main tools we use are the Reduction of Liapunov-Schmidt end the Implicit Function Theorem. An example is contructed at the end of the work.