S.A. Kashchenko, Dynamics of advectively coupled Van der Pol equations chain‎, Chaos 31, 033147 (2021).

ABSTRACT

A ring chain of the coupled Van der Pol equations with two types of unidirectional advective couplings is considered. It is assumed thatthe number of elements in the chain is sufficiently large. The transition to a distributed model with a continuous spatial variable is realized.We study the local—in the equilibrium neighborhood—dynamics of such a model. It is shown that the critical cases in the problem of thezero solution stability have infinite dimension. As the main result, the special nonlinear partial differential equations are constructed thatdo not contain small and large parameters, which are the equations of the first approximation: their solutions determine the main part ofthe asymptotic behavior of the original model solutions. Thereby, the nonlocal dynamics of the constructed equations describes the localstructure of the Van der Pol chain solutions. The asymptotic behavior of the solutions is carried out. Differences were revealed when usingvarious unidirectional couplings. It is shown that these differences can be significant. In some of the most interesting cases, the obtainedequations of the first approximation contain two spatial variables; therefore, it is natural to expect the appearance of complex dynamic effects.The studies are methodologically based on the method for constructing quasinormal forms developed by the author.