Hopf bifurcation theorem of rotating periodic solutions in a class of anti-synchronous systems.

It has been widely confirmed that symmetry can lead to synchronization. However, it is also evident that synchronization or cluster synchronization phenomena commonly occur in asymmetric networks. In this article, we study the Hopf bifurcation of anti-synchronous periodic solutions in a class of asymmetric coupled oscillator systems. In contrast to symmetric synchronous systems, e.g., pendulums with Huygens' coupling, oscillators in ring networks, neurons in mirror symmetric networks, the systems we investigated exhibit anti-synchronous periodic solutions that do not align with their symmetry. Due to the absence of symmetry in these systems, the simultaneous diagonalization between the Jacobi matrix of equilibrium and the rotating matrix representing the types of synchronous solutions is mathematically impossible. This introduces greater complexity in the analysis of dynamics. The main innovation of this paper is to establish the Hopf bifurcation of the rotating periodic solutions for such asymmetric systems. By demonstrating the bifurcation of anti-synchronous periodic solutions in two coupled oscillator systems with three and four oscillators, corresponding to whether the Jacobi matrix of the system is semi-simple, we illustrate the validity of our method. [ABSTRACT FROM AUTHOR]