A theoretical study of delayed feedback in a spin-torque nano-oscillator model is presented. The feedback acts as a modulation of the supercriticality, which results in changes in the oscillator frequency through a strong nonlinearity, amplitude modulations, and a rich modulation sideband structure in the power spectrum at long delays. Modulation sidebands persist at finite temperatures but some of the complex structure is lost through the finite coherence time of the oscillations.
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