Abstract.
Synchronization of a large collection of coupled, simple dynamical systems
is a problem that has applications from neuroscience to traffic modeling to
modeling of consensus formation.

Consider an array of identical linear oscillators, each coupled to its
front and rear neighbor. The coupling may be asymmetric. If we kick the
front oscillator (the leader), how does this signal propagate through the
system? In some isolated cases, for certain values of the parameters, the
answer is well-known, but until recently the only general results
applicable to large systems were very qualitative.

We developed a theory that gives the correct quantitative description. The
theory uses ideas from partial differential equations, but without taking a
continuum limit. We will describe the theory and the conjectures it is
based upon, as well as the quantitative results.