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We discuss some mathematical effects that appear by motion of a closed traffic flows on a ring road. The quasi-linear equation of traffic flow with a concave fundamental diagram is studied. We consider weak and periodic (with respect to x) solutions that satisfies the standard Oleinik entropy condition. Under the additional assumption of a piecewise linear structure of the fundamental diagram, it is shown that any such solution stabilizes to a moving ordinary wave. We formulate the strict mathematical statement. The results of computer experiments are presented.