A system of equations describing two van der Pol
oscillators coupled to a one-dimensional distributed environment
(lossless transmission line for electronic oscillators or elastic
rod for mechanical oscillators) is considered. The resulting
boundary value problem with derivative boundary conditions for
hyperbolic partial differential equations has its solutions in a
one-to-one correspondence with the solutions of a system of
functional differential equations of neutral type. The location of
the eigenvalues of some matrix accounting for the so called
difference operator of the system of functional differential
equations introduces two cases of stability analysis: the stable
case (when the aforementioned eigenvalues are located inside the
unit disk of
![$\mathbb{C}$](img2.png)
) and the critical case (when they are
located on the unit circle). For the stable case a result of
stability by the first approximation is obtained, accounting for
oscillation quenching ("synchronization to zero"). The critical
case remains the "big unknown" of the theory of neutral
functional differential equations.