We study a chain of N+1 phase oscillators with asymmetric but uniform coupling. This
type of chain possesses 2N ways to synchronize in so-called traveling wave states,
i.e., states where the phases of the single oscillators are in relative equilibrium.
We show that the number of unstable dimensions of a traveling wave equals the number
of oscillators with relative phase close to π. This implies that only the relative
equilibrium corresponding to approximate in-phase synchronization is locally stable.
Despite the presence of a Lyapunov-type functional, periodic or chaotic phase slipping
occurs. For chains of lengths 3 and 4 we locate the region in parameter space where
rotations (corresponding to phase slipping) are present.