Keywords: 
• Kuramoto模型 /
• 同步 /
• 相振子 /
• 多定态

Abstract: A significant phenomenon in nature is that of collective synchronization, in which a large population of coupled oscillators spontaneously synchronizes at a common frequency. Nonlinearly coupled systems with local interactions are of special importance, in particular, the Kuramoto model in its nearest-neighbor version. In this paper the dynamics of a ring of Kuramoto phase oscillators with unidirectional couplings is investigated. We simulate numerically the bifurcation tree of average frequency observed and the multiple stable states in the synchronization region with the increase of the coupling strength for N > 4, which cannot be found for N 6 3. Oscillators synchronize at a common frequency ˉω = 0 when K is larger than a critical value of N = 3. Multiple branches with ? ?= 0 will appear besides the zero branch, and the number of branches increases with increasing oscillators for the system N > 3. We further present a theoretical analysis on the feature and stability of the multiple synchronous states and obtain the asymptotically stable solutions. When the system of N = 2 reaches synchronization, the dynamic equation has two solutions: one is stable and the other is unstable. And there is also one stable solution for N = 3 when the system is in global synchronization. For the larger system(N >3), we study the identical oscillators and can find all the multiple branches on the bifurcation tree. Our results show that the phase difference between neighboring oscillators has different fixed values corresponding to the numbers of different branches. The behaviors in the synchronization region computed by numerical simulation are consistent with theoretical calculation very well. The systems in which original states belong to different stable states will evolve to the same incoherent state with an adiabatic decreasing of coupling strength. Behaviors of synchronization of all oscillators are exactly the same in non-synchronous region whenever the system evolves from an arbitrary branch according to the bifurcation trees. This result suggests that the only incoherent state can be attributed to the movement ergodicity in the phase space of coupled oscillators in an asynchronous region. When the system achieves synchronization, the phenomenon of the coexistence of multiple stable states will emerge because of the broken ergodicity. All these analyses indicate that the multiple stable states of synchronization in nonlinear coupling systems are indeed generically observable, which can have potential engineering applications.