Synchronization, symmetry and rotating periodic solutions in oscillators with Huygens’ coupling

Abstract

Synchronization is widespread in complex systems made by multiple individuals. And rotating waves are usually used to describe the synchronization in coupled oscillators systems. Many systems are observed to produce patterns of rotating waves, but it is difficult to predict the type of them or understand the conditions under which they form. Here we present the concept of the rotating-periodic solution and develop new bifurcation theory about rotating-periodic solutions to show the mechanisms for the existence of various rotating waves.We use a Huygens’ coupling model to study this topic. All kinds of rotating waves like in-phase, anti-phase, periodic, cluster synchronous are different types of rotating-periodic solutions with different rotating matrices. In the symmetric Huygens model, the rotating matrices Q satisfying the system rotating invariance just form a special symmetry group. By using the conjugate classes of symmetry groups and the diagonalization method, we can obtain all kinds of rotating waves of finite identical oscillator systems. We calculate all possible rotating waves in three and four identical oscillator systems, and get a general result that the phase difference of the oscillators in a rotating wave can only be kπn (n is the number of the oscillators, 0≤k≤n, k is even). Furthermore, in order to obtain the existence of rotating waves, we establish a new rotating periodic solution Hopf bifurcation theory. This particular bifurcation theory can be used to replace Hopf bifurcation, double Hopf bifurcation and other more complex periodic solution bifurcation in a unified way to study the rotating waves.