Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

In this paper, we discuss the mechanism of synchronization or cluster synchronization in the coupled van der Pol oscillator networks with different topology types by using the theory of rotating periodic solutions. The synchronous solutions here are transformed into rotating periodic solutions of some dynamical systems. By analyzing the bifurcation of rotating periodic solutions, the critical conditions of synchronous solutions are given in three different networks. We use the rotating periodic matrix in the rotating periodic theory to judge various types of synchronization phenomena, such as complete synchronization, anti-phase synchronization, periodic synchronization, or cluster synchronization. All rotating periodic matrices which satisfy the exchange invariance of multiple oscillators form special groups in these networks. By using the conjugate classes of these groups, we obtain various possible synchronization solutions in three networks. In particular, we find symmetry has different effects on synchronization in different networks. The network with more types of symmetry has more elements in the corresponding group, which may have more types of synchronous solutions. However, different types of symmetry may get the same type of synchronous solutions or different types of synchronous solutions, depending on whether their corresponding rotating periodic matrices are similar.

Automated summary

This paper discusses the mechanism of synchronization or cluster synchronization in coupled van der Pol oscillator networks using the theory of rotating periodic solutions. Synchronous solutions are transformed into rotating periodic solutions, and by analyzing bifurcations, critical conditions for synchronization are determined in different networks. The use of rotating periodic matrices helps identify various types of synchronization phenomena and the impact of symmetry on synchronization outcomes across networks.