  The Blazhko effect is a long term, generally irregular modulation of the
light curves that occurs in a sizeable number of RR Lyrae stars. The physical
origin of the effect has been a puzzle ever since its discovery over a hundred
years ago. We build here upon the recent observational and theoretical work of
Szabo et al. on RRab stars who found with hydrodynamical simulations that the
fundamental pulsation mode can get destabilized by a 9:2 resonant interaction
with the 9th overtone. Alternating pulsation cycles arise, although these
remain periodic, i.e. not modulated as in the observations.
  Here we use the amplitude equation formalism to study this nonlinear,
resonant interaction between the two modes. We show that not only does the
fundamental pulsation mode break up into a period two cycle through the
nonlinear, resonant interaction with the overtone, but that the amplitudes are
modulated, and that in a broad range of parameters the modulations are
irregular as in the observations. This irregular behavior is in fact chaotic
and arises from a strange attractor in the dynamics.
