The two-dimensional lattice Boltzmann method was used to numerically study a sedimentation system with two particles having different densities in a vertical channel for Galileo numbers in the range of 5≤Ga≤15 (resulting in a Reynolds number, based on the settling velocity, approximately ranging between 0.6 and 7). Two types of periodic motion, differing from each other in terms of the size of the limit cycle, the magnitude of the time period, and their changes upon increasing the density difference between particles, are identified depending on whether there is a wake effect. The most prominent features of this system are discontinuous changes in the settling velocity (6.7≤Ga<9.7) and time period of oscillation (10.5≤Ga≤15) at a critical value of the density difference between particles. The first discontinuity results in an abrupt increase in the Reynolds number, associated with a Hopf bifurcation without the presence of vortex shedding. The second discontinuity is accompanied by the disappearance of "abnormal rotation" (referring to the situation in which a particle appears to roll up a wall when settling) of the heavy particle, which directly results from a sharp increase in the amplitude of oscillation induced by the enhanced wake effect at another critical density difference between particles. The wall effects on these discontinuous changes were also examined.