Order parameter analysis of synchronization transitions on star networks

Abstract

Collective behaviors of populations of coupled oscillators have attracted much attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations by adopting the star-topology of coupled oscillators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe-Strogatz transformation, Ott-Antonsen ansatz, and the ensemble order parameter approach. Different solutions of the order parameter equation correspond to diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions are revealed in the star-network model by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis.

Phase transition in coupled star networks

“…This reduced equation is consistent with the Ott-Antonsen ansatz for systems with an infinite size [28]. 4) with the transformation In Eq.…”

Threefold way to the dimension reduction of dynamics on networks: an application to synchronization

“…III to get analytical and numerical insights about the emergence of chimeras and explosive synchronization. Interestingly, we find that the reduced dynamics obtained with DART are similar to those deduced from the Ott-Antonsen Ansatz [41,61]. Yet, we provide a new perspective on the existence of chimera states for homogeneous and heterogeneous modular graphs.…”

“…( 59). This is explained by the fact that the equilibrium point is not generally stable for all σ > σ c and for α = 0 [61]. However, the coupling value σ b we obtain is not equal to the value…”

“…Dimension reductions are useful in predicting synchronization regimes, even the more exotic ones [41,[60][61][62]. In this section, we use the formalism proposed in Sec.…”

“…It is used to describe the occurrence of sudden catastrophes, such as epileptic seizures or electric breakdown [86][87][88]. A simple but instructive example of a system that can evolve towards explosive synchronization is the Kuramoto model on the star graph with degree-frequency correlation [23,61,[89][90][91]. Interestingly, the model's bifurcation diagram for the global synchronization observable exhibits hysteresis [89].…”