  Self-sustained oscillations, limit cycles, are a fundamental phenomenon
unique to nonlinear dynamic systems of high-dimensional phase space. They
enable understanding of a wide range of cyclic processes in natural, social and
engineering systems. Here we show that limit cycles form in coupled polariton
cavities following the breaking of Josephson coupling, leading to
frequency-comb emission. The limit cycles and destruction of Josephson coupling
both appear due to interplay between strong polariton-polariton interaction and
a dissipative contribution to the cavity coupling. The resulting nonlinear
dynamics of the condensates is characterized by asymmetric population
distribution and nontrivial average phase difference between the two
condensates, and by time-periodic modulation of their amplitudes and phases.
The latter is manifested by coherent emission of new equidistant frequency
components. The emission spectrum resembles that of a micro-frequency comb, but
originates from a fundamentally different mechanism than that of existing
frequency combs. It allows non-resonant excitation with a power input much
below the conventional semiconductor laser threshold. The comb line spacing is
determined by the interaction and coupling strengths, and is adjustable up to
multi-terahertz frequency. The work establishes coupled polariton cavities as
an experimental platform for rich nonlinear dynamic phenomena.
