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We investigated period doubling, a well‐known phenomenon in dynamical systems, for the first time in RR Lyrae models. These studies provide theoretical background for the recent discovery of period doubling in some Blazhko RR Lyrae stars with the  Kepler  space telescope. Since period doubling has been observed only in Blazhko‐modulated stars so far, the phenomenon can help in understanding the modulation as well. Utilizing the Florida–Budapest turbulent convective hydrodynamical code, we have identified the phenomenon in both radiative and convective models. A period‐doubling cascade was also followed up to an eight‐period solution, confirming that destabilization of the limit cycle is indeed the underlying phenomenon.  Floquet stability roots were calculated to investigate the possible causes and occurrences of the phenomenon. A two‐dimensional diagnostic diagram was constructed to illustrate the various resonances between the fundamental mode and the different overtones. Combining the two tools, we confirmed that the period‐doubling instability is caused by a 9:2 resonance between the ninth overtone and the fundamental mode. Destabilization of the limit cycle by a resonance of a high‐order mode is possible because the overtone is a strange mode. The resonance is found to be strong enough to shift the period of overtone by up to 10 per cent. Our investigations suggest that a more complex interplay of radial (and presumably non‐radial) modes could happen in RR Lyrae stars that might have connections with the Blazhko effect as well.

Period‐doubling bifurcation and high‐order resonances in RR Lyrae hydrodynamical models