  Breaking the reciprocity of wave propagation is a problem of fundamental
interest, and a mucht-sought functionality in practical applications, both in
photonics and phononics. Although it has been achieved using resonant linear
scattering from cavities with broken time-reversal symmetry, such realizations
have remained inescapably plagued by inherent passivity constraints, which make
absorption losses unavoidable, leading to stringent limitations in transmitted
power. In this work, we solve this problem by converting the cavity resonance
into a limit cycle, exploiting the uncharted interplay between non-linearity,
gain, and non-reciprocity. Remarkably, strong enough incident waves can
synchronize with these self-sustained oscillations and use their energy for
amplification. We theoretically and experimentally demonstrate that this
mechanism can simultaneously enhance non-reciprocity and compensate absorption.
Real-world acoustic scattering experiments allow us to observe perfect
non-reciprocal transmission of audible sound in a synchronisation-based 3-port
circulator with full immunity against losses.
