  We have performed a detailed systematic search for multiperiodicity in the
Population I Cepheids of the Large Magellanic Cloud. In this process we have
identified for the first time several new types of Cepheid pulsational
behaviour. We have found two triple-mode Cepheids pulsating simultaneously in
the first three radial overtones. In 9% of the first overtone Cepheids we have
detected weak, but well resolved secondary periodicities. They appear either
very close to the primary pulsation frequency or at a much higher frequency
with a characteristic period ratio of 0.60-0.64. In either case, the secondary
periodicities must correspond to nonradial modes of oscillation. This result
presents a major challenge to the theory of stellar pulsations, which predicts
that such modes should not be exited in Cepheid variables. Nonradial modes have
also been found in three of the fundamental/first overtone double-mode
Cepheids, but no such oscillations have been detected in single mode Cepheids
pulsating in the fundamental mode. In 19% of double-mode Cepheids pulsating in
the first two radial overtones (FO/SO type) we have detected a Blazhko-type
periodic modulation of amplitudes and phases. Both modes are modulated with a
common period, which is always longer than 700 days. Variations of the two
amplitudes are anticorrelated and maximum of one amplitude always coincides
with minimum of the other. We have compared observations of modulated FO/SO
Cepheids with predictions of theoretical models of the Blazhko effect, showing
that currently most popular models cannot account for properties of these
stars. We propose that Blazhko effect in FO/SO Cepheids can be explained by a
nonstationary resonant interaction of one of the radial modes with another,
perhaps nonradial, mode of oscillations.
