## 4 From the Equations of Adler and [PERSON] to the Kuramoto Model of Synchronization
In a quasi-linear approximation, if the nonlinear dependence of the gain functions of the modes on their amplitudes can be described by polynomials, the first model is reduced to the following system of equations:  
Figure 7.1 Figure 7.2  
where \(M_{1,2}\) are the intensities of the modes. The first equation in this system describes the excitation of the second-harmonic mode, which begins oscillating first, while the following equation represents the mode excitation at the fundamental resonance. The parameters \(\sigma_{1,2}\) characterize the linear increments of the modes and are proportional to the ratio of the beam current to their starting currents. The other coefficients are influenced by the transit angles, in particular how they relate to the frequency discrepancies with respect to the resonant harmonics of the electron cyclotron frequency multiplied by the time the electrons transit through the cavity. In the study of nonlinear excitation of parasitic modes in harmonic gyrotrons [101; 102], these transit angles are such that the first mode falls into a zone of soft excitation. Its saturation is described by the first nonlinear term, while the second mode falls in the region of hard self-excitation. A more complicated nonlinearity accounts for its saturation due to its own intensity. As can be seen, the nonlinear excitation of this mode by the first mode is characterized by the second term in Equation (19a). The coefficient \(\gamma_{21}\) representing the cross-saturation effects is negative, indicating that an increase in the intensity of the first mode can incite the excitation of the second mode even if \(\sigma_{2}<0\). The previously mentioned model closely resembles the following system represented by Lotka-Volterra equations, which are commonly used in the field of ecology (see, e.g., [103]) and describes the relationship between populations of predators \(y\) and preys \(x\):  
Figure 7.3 Figure 7.4  
Here the prey's parameters, \(\alpha\) and \(\beta\), describe the maximum prey per capita growth rate, and the effect of the presence of predators on the prey death rate, respectively, while the predator's parameters, \(\gamma\), \(\delta\) describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate. The quadratic term in Equation (20a) and its coefficient \(\epsilon\) are introduced to model more realistic population dynamics, often incorporating effects like Intrspecies competition (where a larger population of one species limits its own growth) or interspecies competition (where one species negatively affects the growth of another). As can be seen, Equation (20a) for the prey population is completely equivalent to Equation (19a) for the first mode's intensity. In contrast, Equation (20b) is less comprehensive than Equation (19b) because the latter ignores the saturation of predators by its own population. When analyzing the interactions of modes in gyrotrons, these equations, alongside other similar models applied in areas like ecology as well as in epidemiology, economics, neuroscience, and many other domains, result in limit-cycles of the amplitudes in the phase space, as noted by [104]. Without going into details, it should be noted that these equations, although simple, can explain very complex nonlinear dynamics leading to phenomena such as Hopf bifurcation, the transition to chaos, and so on. In order to illustrate the complex dynamics described by the Lotka-Volterra system of equations, some results of the reachability analysis are shown in Figure 7.5 Figure 9 presents the flowpipeof the solution obtained for given parameters (see the figure caption for details). In this setting, \(\epsilon=0\), i.e., the quadratic nonlinearity is removed. How the solutions change when the quadratic term is included together with the uncertainty of the parameters and the initial conditions, respectively, can be seen in Figure 9b-d.