From order to chaos in a chip-scale Kerr parametric oscillator

Integrated photonics has enabled a wide class of chip-scale light sources and quantum technologies. Within this field, microresonator-based degenerate optical parametric oscillators (DOPOs) have gained prominence. Above a critical power threshold, these systems undergo spontaneous symmetry breaking to settle into one of two stable, {\pi}-phase-shifted states -- a mechanism successfully used for quantum random number generation and photonic Ising machines. Here, we show that DOPOs based on the Kerr nonlinearity host a significantly broader range of nonlinear dynamics than previously explored. Using a silicon nitride microring resonator, we experimentally identify Hopf bifurcations that trigger a transition from stationary operation to self-sustained oscillations at MHz frequencies. By adjusting pump detunings and powers, we achieve turnkey control over these oscillatory regimes, navigating the system between stable binary states and periodic limit cycles. Furthermore, we report the experimental observation of period-doubling bifurcations, which numerical simulations reveal as the precursor to a cascading instability culminating in chaos at elevated pump powers. Our results establish a framework for controlling nonlinear instabilities in chip-scale parametric oscillators, with applications in programmable photonic hardware and dynamical optical computing.