## 4 From the Equations of Adler and [PERSON] to the Kuramoto Model of Synchronization
Figure 8: Azimuthal bunching of electrons gyrating in the gyrotron cavity at cyclotron frequency and interacting with the high-frequency electromagnetic field \(E_{x}\) of an operating mode with a frequency of \(\omega\). (**a**) Initial uniform distribution of electrons by phase. (**b**) Bunching of electrons when \(\omega=\Omega_{C}\). (**c**) Slippage of the formed bunch in the direction of the decelerating phase when \(\omega>\Omega_{C}\). A similar approach has been applied in a computer simulation of the dynamic effects of gyrating electrons in a magnetoplasma that lead to to synchronization and phase coherence among the electrons [91]. Recall that in gyrotrons two different regimes of mode interaction are possible, namely amplitude and phase-amplitude interaction [5,92], while the Kuramoto model only describes the phase interaction of coupled oscillators. Therefore, the Kuramoto model is not sufficient in the case of amplitude-phase interaction and should be supplemented by the Stuart-Landau equation [69], which is a generic model that describes the behavior of a nonlinear limit-cycle oscillator near the Hopf (or Andronov-Hopf [46,58,93]) bifurcation. For the amplitude and the phase it gives [94],  
\[\frac{d|A|^{2}}{dt}=\sigma_{r}|A|-\frac{l_{r}}{2}|A|^{3}, \tag{17}\]  
and  
Figure 7.0  
where \(A=|A|e^{i\phi}\) is a complex amplitude of the perturbation, \(\sigma=\sigma_{r}+i\sigma_{i}\) is the complex growth rate, \(l=l_{r}+il_{i}\) is a complex number, and \(l_{r}\) is the Landau constant. The synchronization of two generic oscillators coupled with a delay, namely the Kuramoto oscillator, which only describes the phase dynamics, and the Stuart-Landau oscillator, which includes a variable amplitude, has been investigated in [95]. The models of mode interaction in gyrotrons [5,6,96,97] and population dynamics in ecological systems [98,99] share surprising parallels, particularly in how multiple interacting entities can lead to stable or unstable equilibria. Both systems exhibit phenomena like mode competition, multi-mode equilibria, and frequency locking, often governed by similar nonlinear equations. These parallels provide insights into understanding complex systems by drawing analogies between seemingly different domains. In gyrotrons, multiple modes with similar frequencies can compete for the energy of the electron beam. Similarly, in ecological systems, different species compete for resources such as food, space, and sunlight. Both models describe how the state of the system evolves over time. In gyrotrons, the amplitude and frequency of the different modes change due to interactions. In ecological systems, the population size fluctuates over time due to births, deaths, and interactions with other species. Both the one and the other system can exhibit stable states (single-mode operation in gyrotrons; a stable ecosystem in nature) or chaotic behavior (multimode operation, population collapse). Complex interactions can lead to unpredictable dynamics and chaos. Furthermore, both systems are influenced by positive and negative feedback loops. In gyrotrons, the interaction between the electron beam and the electromagnetic fields can create feedback loops that affect the mode dynamics. In ecological systems, interactions like predation or competition create feedback loops that regulate population sizes. Either of these systems can exhibit emergent properties, meaning that the overall behavior of the system is more than the sum of its individual parts. For example, in a gyrotron, the interaction of multiple modes can lead to new types of instabilities. In ecological systems, the interactions between multiple species can lead to complex ecosystems with unexpected properties. As an example, following [5,6], we will consider the parallel between the model proposed by [PERSON] [100], which describes the nonlinear interaction of the amplitudes of two modes, and the Lotka-Volterra equation that describes the relationship between predator and prey populations.