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

Abstract

Local dynamics of a large number of identical oscillators is investigated. 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 quasinormal 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.