Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Abstract

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

Chains with Connections of Diffusion and Advective Types

“…The structure of solutions in the case a 2 < 2 is much more complicated than in the case a 2 > 2, because quasinormal forms at a 2 < 2 are complex boundary value problems of the Ginzburg-Landau type, and the solutions contain rapidly oscillating t components. Explicit formulas are obtained that allow us to trace the role of the parameter c, included in the delay coefficient (13).…”

“…it is natural to call (7) a diffusion-type coupling, since the right part of this equality resembles the expression for the standard difference approximation of the diffusion operator ∂ 2 u/∂x 2 . Such couplings were used, for example, in [8,[12][13][14]. Let us also note the work in [15], where chains of systems of laser equations were considered.…”