A complex-variable invariant-manifold approach is used to construct the normal modes of weakly nonlinear discrete systems with cubic geometric nonlinearities and either a one-to-one or a three-to-one internal resonance. The nonlinear mode shapes are assumed to be slightly curved four-dimensional manifolds tangent to the linear eigenspaces of the two modes involved in the internal resonance at the equilibrium position. The dynamics on these manifolds is governed by three first-order autonomous equations. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifurcations of the calculated nonlinear normal modes are investigated.

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