The effect of a bilinear coupling in a system of degenerate modes is discussed. The normal mode frequencies are obtained for some simple couplings. If each mode is coupled to all the other degenerate modes, the maximum shift in normal mode frequency due to the coupling increases with the number of degenerate modes. Using the same methods, a many-boson problem and a truncated version of it can be compared. The energy eigenvalues have the same relationship as the exact and first-order perturbation excitation energies of the Bogoliubov model Hamiltonian for bosons.