We describe a theoretical analysis of the nonlinear dynamics of third-harmonic generation via Kerr nonlinearities in a resonant cavity with resonances at both w and 3w. Such a doubly resonant cavity greatly reduces the required power for efficient harmonic generation, by a factor of V/Q^2, where V is the modal volume and Q is the lifetime, and even exhibit 100% harmonic conversion efficiency at a critical input power. However, we show that it also exhibits a rich variety of nonlinear dynamics, such as multistable solutions and long-period limit cycles. We describe how to compensate for self- and cross-phase modulation (which shifts the frequencies out of resonance) and how to excite the different stable solutions (and especially the high-effficiency solutions) by spatially modulated input sources.