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We propose near-field inductive power transfer with three coils in the frequency range of a few MHz. The relay coil is part of a non-linear resonator having a diode connected in parallel to the resonant capacitor. We achieve power transfer at multiple frequencies which can be tuned by the input power level and mutual coupling between the coils. Higher harmonic generation is found dominant at lower input power, whereas subharmonic generation (period doubling and period quadrupling) is observed at high levels. Power transfer at subharmonics reveals robustness against level perturbations due to pronounced amplitude hysteresis effect. A circuit simulation based on standard non-linear diode SPICE models confirms the experimental results which are obtained on a demonstration prototype.