title: 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(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.affiliation: Department of Physics, Bar-Ilan University, Ramat Gan, Israel.
authors.forename: Michael M
authors.initials: MM
authors.lastname: Danziger
authors.affiliation: Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia.
authors.forename: Olga I
authors.initials: OI
authors.lastname: Moskalenko
authors.affiliation: Faculty of Nonlinear Processes, Saratov State University, Astrakhanskaya, 83, Saratov 410012, Russia.
authors.forename: Semen A
authors.initials: SA
authors.lastname: Kurkin
authors.affiliation: Department of Physics, East China Normal University, Shanghai 200062, China.
authors.forename: Xiyun
authors.initials: X
authors.lastname: Zhang
authors.affiliation: Department of Physics, Bar-Ilan University, Ramat Gan, Israel.
authors.forename: Shlomo
authors.initials: S
authors.lastname: Havlin
authors.affiliation: CNR-Institute of Complex Systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence, Italy.
authors.forename: Stefano
authors.initials: S
authors.lastname: Boccaletti