  We describea theoretical analysis of the nonlinear dynamics of third-harmonic
generation ($\omega\to3\omega$) via Kerr ($\chithree$) nonlinearities in a
resonant cavity with resonances at both $\omega$ and $3\omega$. Such a doubly
resonant cavity greatly reduces the required power for efficient harmonic
generation, by a factor of $\sim V/Q^2$ where $V$ is the modal volume and $Q$
is the lifetime, and can 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/cross-phase modulation (which
otherwise shifts the cavity frequencies out of resonance), and how to excite
the different stable solutions (and especially the high-efficiency solutions)
by specially modulated input pulses.
