We investigate the dynamical formation of nonlinear patterns in one-dimensional ring condensates under bichromatic periodic modulation of the interaction strength. The stability phase diagram of the condensate's homogeneous density state is analytically derived by introducing a suitable biharmonic variant of the Mathieu equation and computing the associated Floquet spectrum. It reveals the complex interplay among the driving parameters, i.e., amplitude, frequencies, and the so-called frequencies' mixing angle, which dictate the instability onset and the selective enhancement of higher-order resonance tongues, thus offering precise control over the excited modes. These results are in agreement with time-dependent mean-field simulations evidencing the emergence of density wave modulations of specific momenta, while enabling a deeper understanding of the nonlinear stage of the relevant instability. Further insights on the ensuing unstable nonlinear dynamics are provided through a reduced five-mode model which captures the instability onset, the oscillatory behavior of the mode populations and the phase-space dynamics, in agreement with the mean-field predictions. Our study highlights the versatility of bichromatic driving to generate and control complex nonlinear patterns that are within reach in present day ultracold atom experiments.

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• Bose-Einstein condensates