Comparative dynamics of chains of coupled van der Pol equations and coupled systems of van der Pol equations

Abstract

Abstract We consider chains of van der Pol equations closed into a ring and chains of systems of two first-order van der Pol equations. We assume that the couplings are homogeneous and the number of chain elements is sufficiently large. We naturally realize a transition to functions depending continuously on the spatial variable. As $$t\\to\\infty$$ , we study the behavior of all solutions of such chains with initial conditions sufficiently small in the norm. We identify critical cases in the stability problem and show that they all have an infinite dimension. We construct special nonlinear boundary value problems of parabolic type without small parameters, which play the role of normal forms. Their local dynamics determines the behavior of solutions of the original boundary value problems with two spatial variables. We formulate conditions under which the dynamical properties of both chains are close to each other. We establish that in several cases, the dynamics of chains of systems of van der Pol equations turns out to be essentially more complicated and diverse compared with the dynamics of chains of second-order van der Pol equations.