Features of the problem on synchronization of two van der Pol–Duffing oscillators in the case of a direct connection and the presence of symmetry

Two coupled van der Pol–Duffing oscillators are considered in the case of direct symmetric coupling. We show that synchronization of oscillations (i.e., the presence of stable limit cycles) is typical for self-oscillating systems. Asymptotic formulas for the corresponding solutions are obtained. It is found that the behavior of solutions is not affected by the presence or absence of resonances of eigenfrequencies in the linearized problem.