  We study the effect of adding to a directed chain of interconnected systems a
directed feedback from the last element in the chain to the first. The problem
is closely related to the fundamental question of how a change in network
topology may influence the behavior of coupled systems. We begin the analysis
by investigating a simple linear system. The matrix that specifies the system
dynamics is the transpose of the network Laplacian matrix, which codes the
connectivity of the network. Our analysis shows that for any nonzero complex
eigenvalue $\lambda$ of this matrix, the following inequality holds:
$\frac{|\Im \lambda |}{|\Re \lambda |} \leq \cot\frac{\pi}{n}$. This bound is
sharp, as it becomes an equality for an eigenvalue of a simple directed cycle
with uniform interaction weights. The latter has the slowest decay of
oscillations among all other network configurations with the same number of
states. The result is generalized to directed rings and chains of identical
nonlinear oscillators. For directed rings, a lower bound $\sigma_c$ for the
connection strengths that guarantees asymptotic synchronization is found to
follow a similar pattern: $\sigma_c=\frac{1}{1-\cos\left( 2\pi /n\right)} $.
Numerical analysis revealed that, depending on the network size $n$, multiple
dynamic regimes co-exist in the state space of the system. In addition to the
fully synchronous state a rotating wave solution occurs. The effect is observed
in networks exceeding a certain critical size. The emergence of a rotating wave
highlights the importance of long chains and loops in networks of oscillators:
the larger the size of chains and loops, the more sensitive the network
dynamics becomes to removal or addition of a single connection.
