Double vibrational resonance is reported for a driven oscillator in a periodic structure of the Josephson junction type with high-frequency phase modulation. We identify two distinct phase modulation effects, namely, resonant induction and resonant amplification, leading to the appearance of a double resonance. We analyse these vibrational resonance phenomenon theoretically and numerically, and we show that the origin of the induced resonance is traceable to a transition from periodicity to quasiperiodicity associated with an attractor-merging crisis.