Dynamics of advectively coupled Van der Pol equations chain

A ring chain of the coupled Van der Pol equations with two types of unidirectional advective couplings is considered. It is assumed that the 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 the zero solution stability have infinite dimension. As the main result, the special nonlinear partial differential equations are constructed that do not contain small and large parameters, which are the equations of the first approximation: their solutions determine the main part of the asymptotic behavior of the original model solutions. Thereby, the nonlocal dynamics of the constructed equations describes the local structure of the Van der Pol chain solutions. The asymptotic behavior of the solutions is carried out. Differences were revealed when using various unidirectional couplings. It is shown that these differences can be significant. In some of the most interesting cases, the obtained equations 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.