S.A. Kashchenko, "Local Dynamics of Chains of Van der Pol Coupled Systems," Mathematical Notes vol. 108, pp. 901--905 (2020)

https://doi.org/10.1134/S0001434620110334

Abstract: This note considers the local dynamics of circular chains of coupled van der Pol systems. It is assumed that the number of elements in the chain is sufficiently large. The passage to a nonlinear boundary-value problem with continuous spatial variable is performed. For relations of diffusion type, critical cases in the problem of the stability of an equilibrium state are distinguished. It is shown that all of them have infinite dimension. The main result is the construction of special nonlinear parabolic boundary-value problems, which play the role of the first approximation. Their nonlocal dynamics determines the local behavior of the solutions to the initial problem.