Collective dynamics of random Janus oscillator networks

Open Access

Collective dynamics of random Janus oscillator networks

Thomas Peron, Deniz Eroglu, Francisco A. Rodrigues, and Yamir Moreno

Phys. Rev. Research 2, 013255 – Published 4 March 2020

Abstract

Janus oscillators have been recently introduced as a remarkably simple phase oscillator model that exhibits nontrivial dynamical patterns—such as chimeras, explosive transitions, and asymmetry-induced synchronization—that were once observed only in specifically tailored models. Here we study ensembles of Janus oscillators coupled on large homogeneous and heterogeneous networks. By virtue of the Ott-Antonsen reduction scheme, we find that the rich dynamics of Janus oscillators persists in the thermodynamic limit of random regular, Erdős-Rényi, and scale-free random networks. We uncover for all these networks the coexistence between partially synchronized states and a multitude of solutions of a collective state we denominate as a breathing standing wave, which displays global oscillations. Furthermore, abrupt transitions of the global and local order parameters are observed for all topologies considered. Interestingly, only for scale-free networks, it is found that states displaying global oscillations vanish in the thermodynamic limit.

Received 11 September 2019
Accepted 29 January 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.013255

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Collective behavior in networks Network phase transitions Synchronization transition

Networks

Authors & Affiliations

Thomas Peron^1,2,*, Deniz Eroglu^3,†, Francisco A. Rodrigues¹, and Yamir Moreno ^2,4,5

¹Institute of Mathematics and Computer Science, University of São Paulo, São Carlos 13566-590, São Paulo, Brazil
²Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50018, Spain
³Department of Bioinformatics and Genetics, Kadir Has University, 34083 Istanbul, Turkey
⁴Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain
⁵ISI Foundation, Torino 10126, Italy

^*thomaskaue@gmail.com
^†deniz.eroglu@khas.edu.tr

Article Text

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Issue

Vol. 2, Iss. 1 — March - May 2020

Subject Areas

Complex Systems

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Images

Figure 1
Illustration of heterogeneous networks of Janus oscillators. Janus faces in layer “ $+$ ” (“–”) oscillate with frequency ${\hat{ω}}_{1}$ ( ${\hat{ω}}_{2}$ ).
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Figure 2
Dynamics of a random regular network with $k = 40$ . Temporal evolution of (a) order parameters $r_{1, 2}$ measuring the level of synchronization within subpopulations, and total order parameter $R = \frac{1}{2} \sqrt{r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} cos δ}$ ; and (b) evolution of phase lag $δ$ . Initial conditions: $r_{1} (0) = 0.61, r_{2} (0) = 0.83, δ (0) = 2 π / 3$ . Parameters: $β = 0.015, σ = 0.005$ and $Δ = 1, N = 2500$ .
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Figure 3
(a) Stability diagram of system (5) for $k = 40$ and $Δ = 1$ . “Partial sync” refers to the state in which $r_{1, 2} = 1$ and $δ \neq 0$ . SW denotes parameter regions for each $r_{1, 2} = 1$ and for which the phase lag $δ$ rotates with a nonzero frequency. The regions where a multitude of solutions with $r_{1, 2} < 1$ and $\dot{δ} \neq 0$ are found are labeled as “breathing SW.” (b) Order parameter for $β = - 0.3$ . Solid and dashed lines of $R$ for $σ \in (- \infty, σ_{2}] \cup [σ_{1}, \infty)$ are obtained with Eq. (7). Line for $r_{1} = r_{2}$ denotes the symmetric solution of Eq. (5). Dots are obtained by numerically evolving the original system (1) with $N = 2500$ ; each dot is an average over $t \in [250, 500]$ , with time step $d t = 0.01$ . Gray lines for $σ \in [σ_{c_{4}}, σ_{c_{3}}]$ are generated numerically by evolving system (5) with random initial conditions. Gray dots depict the corresponding result yielded by the original system [Eq. (1)]. For the sake of clarity, we show only a small sample of the possible states attainable in the gray area.
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Figure 4
(a) Order parameters $R$ and $r_{1, 2}$ for ER networks with $〈 k 〉 = 40$ . Dots are numerical experiments with $N = 2500$ oscillators. To highlight the dynamical similarity between random regular and dense ER networks, critical conditions and branches in this panel are obtained as in Fig. 3. (b) Synchronization curves of an uncorrelated SF network with $p_{k} \sim k^{- γ}$ , where $γ = 2.25$ and $N = 10^{4}$ oscillators. Conditions $σ_{1, 2}^{(net)}$ are given by Eq. (10). (c) Evolution of mean-field frequency $\dot{Θ}$ associated with the order parameter $R (t) e^{i Θ (t)} = (r_{1} e^{i ψ_{1}} + r_{2} e^{i ψ_{2}}) / 2$ for the same SF network of panel (b). Internal coupling is set to $β = 0.25$ in all panels.
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Figure 5
(a) Stability diagram of system (5) considering a random regular network of Janus oscillators with $k = 40$ and frequency mismatch $Δ = 1$ . Dashed lines depict the regions in the parameter space considered in the following panels. Descriptions of panels (b) and (c) follow as in Fig. 2 of the main text. Panels (d) and (e) show the evolution of the total average frequency $Ω$ and average frequencies of each subpopulation $Ω_{1, 2}$ for $β = - 0.3$ and 0.25, respectively. For the total average frequency $Ω$ , we highlight two cases with different symbols, namely, “ $Ω$ (SW)” denotes the average frequency calculated in the SW state, while “ $Ω$ (bSW)” stands for the same quantity calculated now in the breathing-SW state. Panels (f) and (g) depict the locking frequency $\dot{Θ}$ as a function of coupling $σ$ .
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Figure 6
(a) Order parameters, (b) average frequencies, and (c) locking frequency $\dot{Θ}$ of ER networks with $N = 2500$ Janus oscillators, and average degree $〈 k 〉 = 40$ . Panels (d), (e), and (f) display the corresponding results for SF networks with $P (k) \sim k^{- γ}$ , where $γ = 2.25$ , and $N = 10^{4}$ Janus oscillators. Other parameters used in all panels: $β = 0.25$ and $Δ = 1$ .
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