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