  We study a chain of $N+1$ phase oscillators with asymmetric but uniform
coupling. This type of chain possesses $2^{N}$ ways to synchronize in so-called
travelling 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 travelling wave equals the number of oscillators with relative phase close to
$\pi$. 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 length 3 and 4 we locate the region in parameter space where
rotations (corresponding to phase slipping) are present.
