## 4 From the Equations of Adler and [PERSON] to the Kuramoto Model of Synchronization  
The relationship between the Adler equation and the Kuramoto model of synchronization, and thus the possibility of building a bridge between their concepts and gyrotron physics, has been briefly pointed out in [6]. In this section, we discuss such a connection in more detail. Since its formulation, a seemingly simple model of synchronization proposed by [PERSON] [85] in 1974 has demonstrated its potential to describe various phenomena (e.g., synchronous rhythmic flashing of fireflies, pacemaker cells, neurons in the brain, voltage oscillations at a common frequency in an array of current-biased Josephson junctions, phase synchronization in electrical power distribution networks, rhythmic applause and many others [86; 87]) in various fields (neuroscience, physics and earth sciences, engineering, etc.) and to introduce a new paradigm in their studies [88; 89]. The Kuramoto model is a nonlinear dynamical system of coupled oscillators that initially have random natural frequencies and phases. It shows that if the coupling is strong enough, the system evolves to a state where all oscillators are in phase. The Kuramoto model is a system of \(N\) ordinary differential equations:  
\[\frac{d\theta_{i}}{dt}=\omega_{i}+\frac{K}{N}\sum_{j=1}^{N}\sin(\theta_{j}- \theta_{i}),\;j=1,\ldots,N. \tag{11}\]  
Here \(\theta_{i}(t)\) is a real-valued function of time \(t\) describing the state (phase) of the \(i\)-th oscillator, \(\omega_{i}\) is the natural (intrinsic) frequency of the \(i\)-th oscillator, and \(K\) is the strength of the coupling between the oscillators (when \(K=0\) the equations are linear and the oscillators are independent, ie. \(\theta_{i}(t)=\omega_{i}t\)). The coupling term \(K\) is a constant for uniform all-to-all coupling, but in general it can be defined as an N \(\times\) N adjacency matrix corresponding to other types of coupling, such as uniform, non-uniform, local, unidirectional, self-coupled, etc. It is easy to draw a parallel between this system and the Adler's equation (1) considered in Section 2, as they have similar structures, although their terms have different (albeit analogous) meanings. Note that the first term in Equation (1), \(\Delta\omega_{0}\), is the unperturbed beat frequency, while in the Kuramoto equation the first term is the natural frequency \(\omega\) of a single oscillator. Similarly, the second term of the Adler's equation is proportional to the strength \(E_{1}\) of the priming signal, while in Equation (11) it is proportional to the coupling strength \(K\). An illustration of the dynamics and synchronization of phase-coupled Kuramoto oscillators, created with the open-source software [90], is shown in Figure 7. Graphically the oscillators are represented by points with different angular positions (phases) on a circle. In this example, the selected parameters are the following: the number of oscillators \(N=100\) and the coupling strength \(K=0.5\) (in the figure designated by \(n\) and \(kappa\), respectively). Initially, the phases are distributed uniformly in the interval \([0,2\pi]\) and the natural frequencies are distributed uniformly in the interval \([1-b,1+b]\), where b is a parameter called breadth (see Figure 7a). Another parameter is the \"width\". It has no influence on the dynamics and is simply introduced to improve the visibility of the oscillators, because if it is zero, all oscillators are confined to the circle and it is more difficult to see what is happening due to overlapping of the points representing them. The angular frequencies \(\omega_{i}\), which indicate how fast the oscillator moves around the circle, are color-coded, with blue for the fastest, yellow for the slowest, and green in between. When \(K\) is increased beyond a critical value (in our example \(K=0.50\)), the nonlinear term forces the phases of all oscillators to approach a common limit, as shown in Figure 7b).