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The interaction between a three-level cascade atom and an initial coherent field with a time-varying frequency has been investigated, including the dynamic behavior of the atomic level occupation probabilities and the entanglement between the atom and the field. The frequency of the field is modulated in the forms of sine and rectangle. It is found that the field-frequency variation has a great influence on the properties of the generalized Jaynes-Cummings model. Appropriate frequency modulation can enhance and stabilize the degree of entanglement between the atom and the field.