# Explosive synchronization coexists with classical synchronization in the Kuramoto model

## Abstract

Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10{sup 6}) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.

- Authors:

- Department of Physics, Bar-Ilan University, Ramat Gan (Israel)
- Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012 (Russian Federation)
- (Russian Federation)
- Department of Physics, East China Normal University, Shanghai 200062 (China)
- CNR-Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence (Italy)
- (Israel)

- Publication Date:

- OSTI Identifier:
- 22596640

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; MEAN-FIELD THEORY; OSCILLATORS; SIMULATION; SYNCHRONIZATION

### Citation Formats

```
Danziger, Michael M., E-mail: michael.danziger@biu.ac.il, Havlin, Shlomo, Moskalenko, Olga I., Kurkin, Semen A., Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Zhang, Xiyun, Boccaletti, Stefano, and The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv.
```*Explosive synchronization coexists with classical synchronization in the Kuramoto model*. United States: N. p., 2016.
Web. doi:10.1063/1.4953345.

```
Danziger, Michael M., E-mail: michael.danziger@biu.ac.il, Havlin, Shlomo, Moskalenko, Olga I., Kurkin, Semen A., Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Zhang, Xiyun, Boccaletti, Stefano, & The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv.
```*Explosive synchronization coexists with classical synchronization in the Kuramoto model*. United States. doi:10.1063/1.4953345.

```
Danziger, Michael M., E-mail: michael.danziger@biu.ac.il, Havlin, Shlomo, Moskalenko, Olga I., Kurkin, Semen A., Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054, Zhang, Xiyun, Boccaletti, Stefano, and The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv. Wed .
"Explosive synchronization coexists with classical synchronization in the Kuramoto model". United States.
doi:10.1063/1.4953345.
```

```
@article{osti_22596640,
```

title = {Explosive synchronization coexists with classical synchronization in the Kuramoto model},

author = {Danziger, Michael M., E-mail: michael.danziger@biu.ac.il and Havlin, Shlomo and Moskalenko, Olga I. and Kurkin, Semen A. and Saratov State Technical University, Politehnicheskaya, 77, Saratov 410054 and Zhang, Xiyun and Boccaletti, Stefano and The Italian Embassy in Israel, 25 Hamered Street, 68125 Tel Aviv},

abstractNote = {Explosive synchronization has recently been reported in a system of adaptively coupled Kuramoto oscillators, without any conditions on the frequency or degree of the nodes. Here, we find that, in fact, the explosive phase coexists with the standard phase of the Kuramoto oscillators. We determine this by extending the mean-field theory of adaptively coupled oscillators with full coupling to the case with partial coupling of a fraction f. This analysis shows that a metastable region exists for all finite values of f > 0, and therefore explosive synchronization is expected for any perturbation of adaptively coupling added to the standard Kuramoto model. We verify this theory with GPU-accelerated simulations on very large networks (N ∼ 10{sup 6}) and find that, in fact, an explosive transition with hysteresis is observed for all finite couplings. By demonstrating that explosive transitions coexist with standard transitions in the limit of f → 0, we show that this behavior is far more likely to occur naturally than was previously believed.},

doi = {10.1063/1.4953345},

journal = {Chaos (Woodbury, N. Y.)},

number = 6,

volume = 26,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}