Chains of Coupled Limit-Cycle Oscillators

In this chapter collective effects in chains of diffusively coupled limit-cycle oscillators with different individual frequencies are investigated by asymptotic and numerical methods. In the case of weak coupling using phase approximation (see Chap. 5), conditions for the onset and existence of a global synchronization regime are determined. Cluster synchronization regime and mechanisms of synchronization transitions are studied. Main results are (1) the existence of mono- and multistable regimes of cluster synchronization and (2) the soft respective hard transitions between these structures, which are consisting of a different number of clusters. Synchronization is observed in a broader range of parameters in a randomly formed chain than in the case of a regular arrangement of oscillators with monotonically varying individual frequencies along the chain.