An archetypal zer<>:-dispersionsystem, the tilted Duffing oscillator (TDO), is studied for weak dissipation under the action of a periodic force of intermediate amplitude, small enough for resonances to be possible but large enough for chaos to be possible too. It is found that, as in Hamiltonian systems, an overlap of resonances of different orders is important for the onset of chaos. The onset of chaos is also facilitated by certain global bifurcations of the averaged system.