We investigate numerically the dynamical behavior in a semiconductor ring laser (SRL) subject to a periodic modulation of the injection current. By varying the amplitude and frequency of the modulation at a fixed bias current, different dynamical states including periodic, quasi-periodic, and chaotic states are found. At frequencies comparable to the relaxation oscillation frequency, the intensities of the counterpropagating modes of the SRLs may exhibit in-phase chaotic motion similar to single mode semiconductor lasers. However, antiphase chaotic oscillations in the modal intensities are observed for modulation frequencies significantly lower than the relaxation oscillations frequency. We show that this antiphase chaotic regime does not involve carrier dynamics and is a result of the underlying symmetry of the SRL. We derive a two-dimensional asymptotic model valid on time-scales longer than the relaxation oscillations, which reproduces the observed dynamical behavior. In a further simplification, we can link this reduced set of equations to Duffing-type oscillators.