Recent experiments using a rapidly rotating and precessing cylinder have shown that for specific values of the precession rate, aspect ratio and tilt angle, sudden catastrophic transitions to turbulence occur. Even if the precessional forcing is not too strong, there can be intermittent recurrences between a laminar state and small-scale chaotic flow. The inviscid linearized Navier–Stokes equations have inertial-wave solutions called Kelvin eigenmodes. The precession forces the flow to have azimuthal wavenumber $m=1$ (spin-over mode). Depending on the cylinder aspect ratio and on the ratio of the rotating and precessing frequencies, additional Kelvin modes can be in resonance with the spin-over mode. This resonant flow would grow unbounded if not for the presence of viscous and nonlinear effects. In practice, one observes a rapid transition to turbulence, and the precise nature of the transition is not entirely clear. When both the precessional forcing and viscous effects are small, weakly nonlinear models and experimental observations suggest that triadic resonance is at play. Here, we used direct numerical simulations of the full Navier–Stokes equations in a narrow region of parameter space where triadic resonance has been previously predicted from a weakly nonlinear model and observed experimentally. The detailed parametric studies enabled by the numerics reveal the complex dynamics associated with weak precessional forcing, involving symmetry-breaking, hysteresis and heteroclinic cycles between states that are quasiperiodic, with two or three independent frequencies. The detailed analysis of these states leads to associations of physical mechanisms with the various time scales involved.