### Stability of a Family of Traveling Wave Solutions in a Feedforward
Chain of Phase Oscillators

Stan Mintchev

Abstract.
We consider the joint dynamics of a homogeneously-coupled long
feedforward chain of phase oscillators communicating via a
pulse-response interaction borrowed from mathematical neuroscience. We
focus on a specific parameter regime in which the evolution of the
downstream units is asymptotic to a traveling wave. We execute a
careful numerical calculation of the generator of this traveling wave,
and proceed to study numerically the global stability of a single
oscillator driven by this generator. These results are then taken as a
hypothesis for a theoretical study of the dynamics of longer driven
chains: we prove that the generated traveling wave is globally stable
with respect to perturbations to finitely many sites in the
corresponding infinite oscillator chain. We conclude with a numerical
study that shows that the solution is part of a one-parameter family,
and we illustrate the structural robustness of this family with
respect to changes in the coupling strength. This is joint work with
Oscar Lanford.