  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
was observed only in Blazhko-modulated stars so far, the phenomenon can help in
the understanding of the modulation as well. Utilising the Florida-Budapest
turbulent convective hydrodynamical code, we identified the phenomenon in
radiative and convective models as well. A period-doubling cascade was also
followed up to an eight-period solution confirming that the destabilisation 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 display 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 9th overtone and the fundamental mode.
Destabilisation 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
sufficiently strong enough to shift the period of overtone with up to 10
percent. 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.
