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.
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