On the role of resonances in double-mode pulsation

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Abstract
Simultaneous effects of resonant coupling and non-linear saturation of linear driving mechanism on the finite amplitude solution of multi-modal pulsation problem and on its stability are investigated. Both effects are calculated in the lowest order of approximation in terms of amplitudes. It is shown that the 2:1 resonance between one of the two linearly unstable modes and a higher frequency mode causes double-mode (fundamental and first overtone) pulsation. In a certain range of parameters, such as the frequency mismatch, the linear growth and damping rates, it is the only stable solution of the problem. Three-mode resonance is shown to promote a single-mode rather than a double-mode pulsation. Non-resonant interaction occurring through the collective saturation of the driving rates may, under a certain condition, lead to multimodal variability, but we argue that this condition is unlikely to be met in Cepheids and RR Lyrae stars.

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