The extended master stability function approach to alternating synchronization modes on networked oscillator systems

Abstract The spontaneous emergence of cluster synchronization modes on networked oscillator systems is one of the most important topics in recent years. However, alternating synchronization, as the most perfect and symmetric cluster synchronization mode, has not received much attention in relating to the theoretical analysis. In this paper, two types of alternating synchronization modes self-organized to emerge on networked oscillator systems, namely, the generally alternating synchronization (GAS) and the inversely alternating synchronization (IAS), are reported. Importantly, an extended master stability function (EMSF) approach is proposed to analyze these two types of alternating synchronization modes. In terms of the EMSF approach, the critical coupling strengths in inducing the phase transitions from the non-synchronous oscillatory behaviors respectively to the GAS and the IAS modes on one-dimensional oscillator rings are predicted accurately. The universality of the EMSF approach is confirmed, i.e. the EMSF approach is also applicable to higher-dimensional lattices, and heterogeneous and asymmetric systems. The results presented in this paper pave available ways in understanding and analyzing miscellaneous complicated cluster synchronization modes in real cases and are expected to have beneficial impacts in related fields.