We explore in detail the nontrivial and chaotic behavior of the traffic model proposed by Toledo et al. [Phys. Rev. E 70, 016107 (2004)] due to the richness of behavior present in the model, in spite of the fact that it is a minimalistic representation of basic city traffic dynamics. The chaotic behavior, previously shown for a given lower bound in acceleration/brake ratio, is examined more carefully and the region in parameter space for which we observe this nontrivial behavior is found. This parameter region may be related to the high sensitivity of traffic flow that eventually leads to traffic jams. Approximate scaling laws are proposed.