## 4 From the Equations of Adler and [PERSON] to the Kuramoto Model of Synchronization
A measure of the synchronization between the phases of the oscillators is the order parameter, which is defined as follows:  
\[re^{i\psi}=\frac{1}{N}\sum_{j=1}^{N}e^{\beta_{i}}. \tag{12}\]  
The order parameter \(r=0\) corresponds to a completely incoherent state, and a finite \(r\) corresponds to a certain degree of synchronization. Complete synchronization (\(r=1\)) requires a coupling strength \(K>K_{c}\), where \(K_{c}\) is a certain critical value. Combining Equations (11) and (12) gives  
\[\frac{d\theta_{i}}{dt}=\omega_{i}+Kr\sin(\psi-\theta_{i}), \tag{13}\]  
where \(\psi\) could be set to 0 since the Kuramoto model is invariant to such a change. Moreover, the latter form of the equation is more suitable for analytical and numerical analysis. The dynamics and synchronization of Kuramoto oscillators have many features in common with the most essential mechanism of the beam-wave interaction in the gyrotron cavity, namely azimuthal bunching of beam electrons gyrating with cyclotron frequency:  
\[\Omega_{C}=\frac{e}{\gamma m_{0}}B. \tag{14}\]  
Here \(e/m_{0}\) is the charge to the rest mass ratio of an electron and \(\gamma=\left(1-\frac{v_{2}^{2}+v_{1}^{2}}{c^{2}}\right)^{-1/2}=\left(1-\frac {v_{2}^{2}+v_{1}^{2}}{c^{2}}\right)^{-1/2}\) is the relativistic Lorentz factor, where \(v_{2}\) and \(v_{\perp}\) are the axial and transverse components of the total velocity \(v\) of the electron, respectively. In light of the Kuramoto model, one could consider the gyrating electrons as oscillators with intrinsic frequencies \(\Omega_{Ci}\) and phases distributed along a circle with a Larmor radius \(R_{L}\). The azimuthal bunching is essential for efficient energy transfer during beam-wave interaction. This mechanism is due to the dependence of the angular frequency of the rotating electrons on their energy, as can be seen in Equation (13). Therefore, when electrons are accelerated by the microwave field, their energy increases and their cyclotron frequency slows down. In contrast, electrons that lose energy and transfer it to the wave increase their  
Figure 7: Dynamics and synchronization of Kuramoto oscillators illustrated: (**a**) initial distribution by phases; (**b**) final state of oscillators [90]. cyclotron frequency. As a result, a bunch of electrons is formed, as illustrated in Figure 8 If the cyclotron frequency is equal to the frequency of the cavity mode (\(\omega=\Omega_{C}\)), the net equilibrium is zero on average (see Figure 8b). However, when \(\omega>\Omega_{C}\), the bunch slips towards the decelerating phase (see Figure 8c) and most of the electrons are decelerated, thus transferring energy to the generated radiation. Such a mechanism is realized when the following synchronization condition is maintained:  
\[\omega=s\Omega_{C}+v_{z}k_{z}, \tag{15}\]  
where \(\omega\) is the Doppler-shifted circular frequency of the electromagnetic field, \(k_{z}\) is the axial wavenumber, and \(s=1,2,3,\dots\) is the harmonic number of the gyrotron operation. The operating point is given by the intersection of the electron beam line (Equation (14)) and the dispersion relation of a certain cavity mode (TE\({}_{m,n}\) or TM\({}_{m,n}\)) with an eigenvalue \(\chi_{m,n}\):  
\[\omega^{2}=\omega_{c}^{2}+c^{2}k_{z}^{2}, \tag{16}\]  
where \(\omega_{c}=c\frac{\chi_{m,n}}{R_{c}}\) is the cut-off frequency and \(R_{c}\) is the radius of the resonant cavity. Another possible application of the Kuramoto model is the study of the interaction of modes in the gyrotron cavity, as pointed out in [6]. In this case, the different modes are considered as phase oscillators with natural frequencies corresponding to the eigenfrequencies of the modes and the coupling strength, which is a function of their starting currents and coupling factors. This work is currently in progress, and its results will be published elsewhere.