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## Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results

## Abstract

In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott–Antonsen (OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. In conclusion, wemore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Ramakrishna Mission Vivekananda Univ., Howrah (India)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1512742

- Report Number(s):
- LA-UR-18-28300

Journal ID: ISSN 0022-4715

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Statistical Physics

- Additional Journal Information:
- Journal Name: Journal of Statistical Physics; Journal ID: ISSN 0022-4715

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Nonlinear dynamics and chaos; Synchronization; Coupled oscillators; Bifurcation analysis

### Citation Formats

```
Métivier, David, and Gupta, Shamik. Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results. United States: N. p., 2019.
Web. doi:10.1007/s10955-019-02299-z.
```

```
Métivier, David, & Gupta, Shamik. Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results. United States. doi:10.1007/s10955-019-02299-z.
```

```
Métivier, David, and Gupta, Shamik. Tue .
"Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results". United States. doi:10.1007/s10955-019-02299-z.
```

```
@article{osti_1512742,
```

title = {Bifurcations in the Time-Delayed Kuramoto Model of Coupled Oscillators: Exact Results},

author = {Métivier, David and Gupta, Shamik},

abstractNote = {In the context of the Kuramoto model of coupled oscillators with distributed natural frequencies interacting through a time-delayed mean-field, we derive as a function of the delay exact results for the stability boundary between the incoherent and the synchronized state and the nature in which the latter bifurcates from the former at the critical point. Our results are based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to both the kinetic equation for the single-oscillator distribution function in the case of a generic frequency distribution and the corresponding Ott–Antonsen (OA)-reduced dynamics in the special case of a Lorentzian distribution. Besides elucidating the effects of delay on the nature of bifurcation, we show that the approach due to Ott and Antonsen, although an ansatz, gives an amplitude dynamics of the unstable modes close to the bifurcation that remarkably coincides with the one derived from the kinetic equation. Further more, quite interestingly and remarkably, we show that close to the bifurcation, the unstable manifold derived from the kinetic equation has the same form as the OA manifold, implying thereby that the OA-ansatz form follows also as a result of the unstable manifold expansion. In conclusion, we illustrate our results by showing how delay can affect dramatically the bifurcation of a bimodal distribution.},

doi = {10.1007/s10955-019-02299-z},

journal = {Journal of Statistical Physics},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {4}

}