Dynamics of Large Systems of Identical Oscillators with Delayed Couplings of Advective Type

Local dynamics  of a large number of identical oscillators is
investigated.
%They arise, for example, in standard difference approximations of the advection (transfer) operator.
Instead of a system of coupled equations,   a single partial
integro-differential equation is proposed which can model the
chain with coupling lines of advective type and takes into account
large delay time in coupling lines. We show that asymptotically
infinite number of modes are excited at the critical conditions
when zero equilibrium state becomes unstable. The special
nonlinear boundary value problems of the parabolic type are derived
which play a role of the quasi-normal forms  in the cases of bi- and
unidirectional couplings, for weak and strong dissipation of
the oscillators, for distributed and discrete couplings, for odd and
even numbers of the oscillators. The nonlocal dynamics of each
quasi-normal form describes the behavior of the main terms of the
asymptotic expansions of nonlinear solutions of the original
problem.