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

Sergey Kashchenko ()
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Sergey Kashchenko: Regional Scientific and Educational Mathematical Center “Centre of Integrable Systems”, P. G. Demidov Yaroslavl State University, 150003 Yaroslavl, Russia

Mathematics, 2023, vol. 11, issue 9, 1-34

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.

Keywords: bifurcations; stability; normal forms; singular perturbations; dynamics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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