Eﬀects of frequency-gap on explosive synchronization
Condensed Matter Theory I

102112037 Toshiki Nagasaki

Introduction

one more group which has n nodes. Their natural frequencies are equal to the average frequency. We vary n andSynchronization is one of the central phenomena represent- observe the behavior of the synchronization transitions.
ing the emergence of collective behavior in natural and synthetic complex systems. The most successful attempt is 3
Results
due to Kuramoto [1]. It consists of N coupled phase oscillators θi (t) having natural frequencies ωi distributed with When we consider two groups, which means we only choosea given probability density g(ω), and whose dynamics are frequency-gap as a condition, and we compute Eqs.(1),(2).
governed byThe system shows an explosive synchronization (Fig.1). In
addition, we study analytically on backward. It shows goodN∑˙
θi = ωi + λsin (θj − θi ) ; i = 1, · · · , N(1) agreement with simulation results. Then, we add one moregroup. As increasing n, the area of hysteresis loop dej=1
creases (Fig.2). Especially, when n gets over 2N , the area
The parameter λ accounts for the strength of the coupling vanishes. Therefore, when n/N is small, we can observe an
among interconnected oscillators. When λ is larger than a explosive synchronization.
critical value, the system is synchronized. We measure astate of the system using with the following order parameter.
r(t) =N1 ∑ iθi (t)eN i=1(2)When the value reaches r = 1, the system is fully synchronized, while r = 0 for the incoherent solution. Typically,
the average value of r as a function of λ displays a secondorder phase transition from r = 0 to r > 0.
The studies of synchronization transition in complex networks have attracted lots of attention since the report of anabrupt percolation transition on networks, in which abrupt
synchronization transitions have been reported. This phenomenon is called as an explosive synchronization transition, refers to a ﬁrst-order transition where all nodes in a
network abruptly get to the synchronous state with hysteresis loop when the coupling strength is larger than acritical value.
Gómez − Gardeñes et al. [2] and many researchers suggest that when the local dynamics (natural frequency) are
positively correlated to the local heterogeneous structure,
an explosive synchronization has been shown. Moreover,the large frequency mismatch of a pair of oscillators andthe disassortativity in node degree also cause an explosive
synchronization [3]. However, whether these conditions areenough to induce an explosive synchronization and which
condition is the most important criterion for the emergence of an explosive synchronization transition are still
open questions.

2ModelsFirst, We prepare a network only taking in frequency gapand observe the behavior of the synchronization transitions, varying coupling strength λ. All nodes connect toall nodes belonging to the other groups, but don’t interact with the nodes whose belong to the same group. We
compare the diﬀerence of the behavior, starting at random state (forward) and full synchronize state (backward).
Then, those nodes are divided to two groups. Each group
has N nodes. The nodes which belong to the same group
have almost the same natural frequency. Second, we addFig. 1: Explosive synchroFig. 2: n/N vs the area ofnization with two groups
hysteresis loop
(N = 50)

4Conclusion

To summarize, we have explored the important factors
which induce an explosive synchronization with simulations and analytical calculations. In recent study, somepossible factors are proposed. Especially, it is widely believed the large frequency mismatch of a pair of oscillators
is a direct cause of an explosive synchronization. So, in
our study, we constructed a simple model with such factor
and observed a behavior of synchronization transition. Asa result, we can observe an explosive synchronization whenthe nodes which have average frequency are relatively fewer
than other nodes whose natural frequencies are not equal
to the average frequency (n/N < 2). However, when n getsover 2N , the hysteresis loop disappears and the transition
becomes continuous. This result suggests that, as long as
frequency gap exists, explosive synchronization transition
occurs.

References[1] Y.Kuramoto, Chemical Oscillations, Waves, and Turbulence (Berlin ; Tokyo : Springer-Verlag, 1984)[2] J. Gomez-Gardenes, S. Gomez, A. Arenas, and Y.
Moreno, Phys. Rev. Lett. 106, 128701 (2011).
[3] L. Zhu, L. Tian, and D. Shi, Phys. Rev. E 88, 042921(2013).