Skip to main page content
U.S. flag

An official website of the United States government

Dot gov

The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

Https

The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2013:3:1281.
doi: 10.1038/srep01281.

Explosive transitions to synchronization in networks of phase oscillators

I Leyva  1 , A Navas, I Sendiña-Nadal, J A Almendral, J M Buldú, M Zanin, D Papo, S Boccaletti
Affiliations

Explosive transitions to synchronization in networks of phase oscillators

I Leyva et al. Sci Rep. 2013.

Abstract

The emergence of dynamical abrupt transitions in the macroscopic state of a system is currently a subject of the utmost interest. The occurrence of a first-order phase transition to synchronization of an ensemble of networked phase oscillators was reported, so far, for very particular network architectures. Here, we show how a sharp, discontinuous transition can occur, instead, as a generic feature of networks of phase oscillators. Precisely, we set conditions for the transition from unsynchronized to synchronized states to be first-order, and demonstrate how these conditions can be attained in a very wide spectrum of situations. We then show how the occurrence of such transitions is always accompanied by the spontaneous setting of frequency-degree correlation features. Third, we show that the conditions for abrupt transitions can be even softened in several cases. Finally, we discuss, as a possible application, the use of this phenomenon to express magnetic-like states of synchronization.

PubMed Disclaimer

Figures

Figure 1
Figure 1. Explosive transition to synchronization.
Phase synchronization level S (see text for definition) vs. the coupling strength d, for different values of the gap γ at 〈k〉 = 40 (panel a), and for different values of the average degree 〈k〉 at γ = 0.4 (panel b). In both panels, the continuous (dashed) lines refer to the forward (backward) simulations. See the Methods section for the construction procedure of the networks. In panels c and d, the degree ki that each node achieves after the network construction is completed (upper plots) and the average of the natural frequencies 〈ωj〉 for formula image (bottom plots) are reported vs. the node's natural frequency ωi, for 〈k〉 = 100 and frequency gaps γ = 0.0 (panel c), and γ = 0.4 (panel d). The red solid line in panel d is a sketch of the theoretical prediction f(ω) (see text). Panels e and f show S (color coded according to the color bar) in the parameter space (d, γ) for (e) 〈k〉 = 20 and (f) 〈k〉 = 60. The horizontal dashed lines mark the separation between the region of the parameter space where a second-order transition occurs (below the line) and that in which the transition is instead of the first order type (above the line). The yellow striped area delimits the hysteresis region.
Figure 2
Figure 2. Extension to different frequency distributions and network construction rules.
(Top row) S vs. d resulting from the forward (continuous lines) and backward (dashed lines) simulations of system (1) for different frequency distributions or network construction rules, and (bottom row) the corresponding distribution of the final node degree ki vs. the corresponding oscillator's natural frequency. (a)-(b) Rayleigh distribution for γ = 0.3. In panel b, the red solid line depicts the theoretical prediction f(ω); (c)-(d) uniform frequency distribution, but network constructed accordingly to the local mean field condition (see text) for γ = 0, and γ = 0.4. In panel d γ = 0.4; (e)-(f) Gaussian distribution with Z = 0.7. The insets in panels a and e report the corresponding distribution p(ω). See the text and the Methods section for the details on the specific construction procedure used in each case. In all cases, 〈k〉 = 60.
Figure 3
Figure 3. Explosive synchronization in networks with evenly spaced natural frequencies.
S vs. d resulting from the forward (continuous lines) and backward (dashed lines) simulations of system (1), for different values of the average degree 〈k〉 (see legend in panel a) after performing a random pruning (a) and a preferential pruning (b) of links in an all-to-all connected network (see the text and the Methods section for the networks construction). Natural frequencies are evenly spaced in the interval [0, 1]. In panel c, the degree ki that each node achieves after the network pruning is completed (upper plots) and the average of the local natural frequency 〈ωj〉 for formula image (bottom plots) are reported vs. the node's natural frequency ωi for the preferential pruning process with 〈k〉 = 10.
Figure 4
Figure 4. Magnetic-like states of synchronization.
(a) Time evolution of the parameter r(t) under a pacemaker forcing for a value d = 0.004 outside the hysteresis region (upper plot), and a value d = 0.009 within the hysteresis region (bottom plot). ωp = 1.0 and Dp = 0.0005 (bottom red line), Dp = 0.005 (middle blue line), and Dp = 0.02 (top black line). The pacemaker is active from t = 50 to t = 350, as marked by the vertical dashed lines. (b) Colormap of S = 〈r(t)〉t>350 (coded as indicated in the color bar), showing the region of the parameter space Dp-ωp where the magnetic-like state of synchronization is maintained after removal of the pacemaker. The initial frequencies of the oscillators are taken from a uniform distribution in the interval [0, 1]. γ = 0.49 and 〈k〉 = 40.
See this image and copyright information in PMC

Similar articles

See all similar articles

Cited by

See all "Cited by" articles

References

    1. Dorogovtsev S. N., Goltsev A. V. & Mendes J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275 (2008).
    1. Cohen R., Ben-Avraham D. & Havlin S. Percolation critical exponents in scale-free networks. Phys. Rev. E 66, 036113 (2002). - PubMed
    1. Li G., Braunstein L. A., Buldyrev S. V., Havlin S. & Stanley H. E. Transport and percolation theory in weighted networks. Phys. Rev. E 75, 045103 (2007). - PubMed
    1. Boccaletti S., Kurths J., Osipov G., Valladares D. L. & Zhou C. S. The synchronization of chaotic systems. Phys. Rep. 366, 1 (2002).
    1. Boccaletti S., Latora V., Moreno Y., Chavez M. & Hwang D. U. Complex networks: Structure and dynamics. Phys. Rep. 424, 175 (2006).

Publication types

LinkOut - more resources