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The dynamics of a harmonically excited Duffing oscillator with a low-frequency modulation of its natural frequency is considered. The appearance of secondary nonlinear resonances as a result of the high- and low frequency interaction is demonstrated. It is found that these resonances cause the onset of additional bistable states as compared to the case when the frequency modulation is not applied. A comparative analysis of different types of bistable states in the Duffing oscillator is carried out as well.