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Explosive or Continuous: Incoherent state determines the route to synchronization.

Xu C, Gao J, Sun Y, Huang X, Zheng Z - Sci Rep (2015)

Bottom Line: The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous.The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation.Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China.

ABSTRACT
Abrupt and continuous spontaneous emergence of collective synchronization of coupled oscillators have attracted much attention. In this paper, we propose a dynamical ensemble order parameter equation that enables us to grasp the essential low-dimensional dynamical mechanism of synchronization in networks of coupled oscillators. Different solutions of the dynamical ensemble order parameter equation build correspondences with diverse collective states, and different bifurcations reveal various transitions among these collective states. The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous. The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation. Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

No MeSH data available.


(a) Synchronization boundary line Eq. (18) with the initial distribution of phase are randomly drawn from interval [−δ,δ]. (b) The measure of the synchronous state against the coupling strength for different K. (c) The ensemble order parameter phase space for 0 < α ≤ 0.5π, the limit cycle corresponding to the splay state, and the fixed point corresponding to the synchronous state (SS). (d) The order parameter against the corresponding coupling strength with different sizes.
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f2: (a) Synchronization boundary line Eq. (18) with the initial distribution of phase are randomly drawn from interval [−δ,δ]. (b) The measure of the synchronous state against the coupling strength for different K. (c) The ensemble order parameter phase space for 0 < α ≤ 0.5π, the limit cycle corresponding to the splay state, and the fixed point corresponding to the synchronous state (SS). (d) The order parameter against the corresponding coupling strength with different sizes.

Mentions: which is represented in Figs. 2(a).


Explosive or Continuous: Incoherent state determines the route to synchronization.

Xu C, Gao J, Sun Y, Huang X, Zheng Z - Sci Rep (2015)

(a) Synchronization boundary line Eq. (18) with the initial distribution of phase are randomly drawn from interval [−δ,δ]. (b) The measure of the synchronous state against the coupling strength for different K. (c) The ensemble order parameter phase space for 0 < α ≤ 0.5π, the limit cycle corresponding to the splay state, and the fixed point corresponding to the synchronous state (SS). (d) The order parameter against the corresponding coupling strength with different sizes.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4498219&req=5

f2: (a) Synchronization boundary line Eq. (18) with the initial distribution of phase are randomly drawn from interval [−δ,δ]. (b) The measure of the synchronous state against the coupling strength for different K. (c) The ensemble order parameter phase space for 0 < α ≤ 0.5π, the limit cycle corresponding to the splay state, and the fixed point corresponding to the synchronous state (SS). (d) The order parameter against the corresponding coupling strength with different sizes.
Mentions: which is represented in Figs. 2(a).

Bottom Line: The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous.The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation.Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and the Beijing-Hong Kong-Singapore Joint Centre for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China.

ABSTRACT
Abrupt and continuous spontaneous emergence of collective synchronization of coupled oscillators have attracted much attention. In this paper, we propose a dynamical ensemble order parameter equation that enables us to grasp the essential low-dimensional dynamical mechanism of synchronization in networks of coupled oscillators. Different solutions of the dynamical ensemble order parameter equation build correspondences with diverse collective states, and different bifurcations reveal various transitions among these collective states. The structural relationship between the incoherent state and the synchronous state leads to different routes of transitions to synchronization, either continuous or discontinuous. The explosive synchronization is determined by the bistable state where the measure of each state and the critical points are obtained analytically by using the dynamical ensemble order parameter equation. Our method and results hold for heterogeneous networks with star graph motifs such as scale-free networks, and hence, provide an effective approach in understanding the routes to synchronization in more general complex networks.

No MeSH data available.