Summary: Work on mode locking and mode jumping in the stable post-buckling of an elastic structure is extended to encompass the full range of flexural boundary conditions from simply supported to clamped. The two imaginary eigenvalues of the linearized equation suggest the possibility of doubly periodic responses. The parameter space of Arnold tongues is used to describe the extent to which the system locks onto a single wave, the mode to which it subsequently jumps, and the associated role played by the ratio of the two competing wavelengths. At compound bifurcation points, simply supported and clamped conditions both give interactions for which this ratio is rational, but in-between it is irrational and the associated response is quasi-periodic.