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Low-dimensional behavior of Kuramoto model with inertia in complex networks.

Ji P, Peron TK, Rodrigues FA, Kurths J - Sci Rep (2014)

Bottom Line: Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz.In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks.Numerical simulations are also conducted to verify our analytical results.

View Article: PubMed Central - PubMed

Affiliation: 1] Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany [2] Department of Physics, Humboldt University, 12489 Berlin, Germany.

ABSTRACT
Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.

No MeSH data available.


Order parameter r vs coupling strengths K in scale-free networks (see Methods for details).The red curves indicate the results from simulations on the same network as in Figure 1. For each coupling, initial values of θ randomly select from [−π, π] and we set . The green dots shows analytic prediction of the stationary r(t) based on the self-consistent Eq. (8).
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f5: Order parameter r vs coupling strengths K in scale-free networks (see Methods for details).The red curves indicate the results from simulations on the same network as in Figure 1. For each coupling, initial values of θ randomly select from [−π, π] and we set . The green dots shows analytic prediction of the stationary r(t) based on the self-consistent Eq. (8).

Mentions: Let us consider again the nonlinear evolution of the order parameter r in complex networks. From the above analysis, we get that . To check the validity of this assumption, we compare the stationary solution with simulation results in Fig. 5. The theoretical predictions (green lines derived from Eq. (8) with effective coupling and f(K k, r)) are in agreement with red lines of numerical simulations.


Low-dimensional behavior of Kuramoto model with inertia in complex networks.

Ji P, Peron TK, Rodrigues FA, Kurths J - Sci Rep (2014)

Order parameter r vs coupling strengths K in scale-free networks (see Methods for details).The red curves indicate the results from simulations on the same network as in Figure 1. For each coupling, initial values of θ randomly select from [−π, π] and we set . The green dots shows analytic prediction of the stationary r(t) based on the self-consistent Eq. (8).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4007097&req=5

f5: Order parameter r vs coupling strengths K in scale-free networks (see Methods for details).The red curves indicate the results from simulations on the same network as in Figure 1. For each coupling, initial values of θ randomly select from [−π, π] and we set . The green dots shows analytic prediction of the stationary r(t) based on the self-consistent Eq. (8).
Mentions: Let us consider again the nonlinear evolution of the order parameter r in complex networks. From the above analysis, we get that . To check the validity of this assumption, we compare the stationary solution with simulation results in Fig. 5. The theoretical predictions (green lines derived from Eq. (8) with effective coupling and f(K k, r)) are in agreement with red lines of numerical simulations.

Bottom Line: Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz.In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks.Numerical simulations are also conducted to verify our analytical results.

View Article: PubMed Central - PubMed

Affiliation: 1] Potsdam Institute for Climate Impact Research (PIK), 14473 Potsdam, Germany [2] Department of Physics, Humboldt University, 12489 Berlin, Germany.

ABSTRACT
Low-dimensional behavior of large systems of globally coupled oscillators has been intensively investigated since the introduction of the Ott-Antonsen ansatz. In this report, we generalize the Ott-Antonsen ansatz to second-order Kuramoto models in complex networks. With an additional inertia term, we find a low-dimensional behavior similar to the first-order Kuramoto model, derive a self-consistent equation and seek the time-dependent derivation of the order parameter. Numerical simulations are also conducted to verify our analytical results.

No MeSH data available.