Kuramoto’s Model
One of the most important basic phenomena in sciences is synchronization which was discovered by C. Huyghens (Pecora & Carroll, 1990; Pikovsky, Rosenblum, & Kurths, 2001; Rulkov et al., 1995; Kuramoto, 1984). Synchronization is a phenomenon which means there is an adjustment of the frequencies of periodic self-sustained oscillators through a weak interaction. The weakness of the interaction is to some extent a requirement. The frequencies adjustment is also referred to as phase locking or frequency entrainment.
Many types of synchronizations can be distinguished depending on the dynamic property of the system under investigation. One might have identical synchronization, generalized synchronization and phase synchronization for chaotic and non-chaotic systems (i.e. regular and chaotic systems) Synchronization might also happen in a system of two or many identical or not chaotic systems. It also may be seen as the appearance of a relation relating the phases of interacting systems or between the phase of a system and that of an external force. The synchronization phenomenon is not to be confused with resonance phenomenon and synchronous variation of two variables which does not lead to synchronization. In many areas of science collective synchronization phenomena have been observed. These areas overlaps biology, chemistry, physical and social systems as examples. This phenomenon is mainly observed when two or many oscillators lock on to a common frequency despite differences in the frequency of the individual oscillators. A popular model known as Kuramoto model has been developed and can be found in many textbooks and articles in the literature. Many methods such as the stability analysis can be used to address the synchronization phenomenon. The Kuramoto model consists of N oscillators whose dynamics are governed by the following equations 
(1)where Өi is the phase of the i th oscillator, ωi is its natural frequency, K is positive and stands for the coupling gain.
The oscillators are said to synchronize if
as
(2)The oscillators can also be said to synchronize when 
becomes constant asymptotically as the time goes on. If we represent the oscillators on a circle by points that move with the same angular frequency, then the phase difference which means the angular distance between these points remain constant over the time. Thus, one can define the order parameter r that measures the phase coherence of the oscillators population and takes values between 0 and 1 inclusively as follow
(3) The order parameter will play a key role in describing how synchronous the system is. If the oscillators synchronize, then the parameter converges to a constant magnitude, r ≤ 1 after a long time behavior, but if the oscillators add incoherently then the order parameter r remains close to zero. At this point the problem is to characterize the coupling gain K so that the oscillators synchronize.
Kuramoto showed that there exists a critical value Krc for K such that for all K < Krc, the oscillators remain unsynchronized, but for K > Krc the incoherent state becomes unstable, the oscillators start synchronizing and eventually r(t) will settle at some r∞(K) < 1.
There still exist many open problems to address on the influence of the magnitude of the order parameter combined to that of the gain on the behavior of the system.