Dynamics of coupled Van der Pol oscillators

In this paper, we consider the problem on the synchronization of two and three weakly coupled Van der Pol oscillators in the case where the oscillators are identical and the connection between them is weak. The existence and stability of two types of periodic solutions are studied under the assumption that the connection between the oscillators is dissipative or active. The analysis of the problem is based on methods of the qualitative theory of differential equations, namely, the method of integral manifolds and the method of normal Poincaré–Dulac forms. The problem is reduced to the study of normal forms. We used a version of the Krylov–Bogolyubov algorithm, which allows one to obtain asymptotic formulas for periodic solutions.