## 3 Van der Pol Equation and the Analysis of Gyrotron Oscillators
For example, in [71], a more general equation, namely the Van der Pol-Duffing equation, has been used to study the multistability and complex oscillatory regimes in a microwave generator with delayed reflection from the load. A detailed study of the phase dynamics and computer simulation of coupled Van der Pol and Van der Pol-Duffing oscillators is presented in [72]. \[\dot{F}=y, \tag{8a}\]2 Examining the existence of an invariant of the system. The solution is calculated for the parameters and initial conditions of \[\dot{F}=y, \tag{8a}\]3  
The Duffing equation, which is used to model both damped and driven oscillators, is a second-order nonlinear differential equation with cubic nonlinearity of the following type (compare with the Van der Pol equation (Equation (7))):  
\[\dot{F}=y, \tag{8a}\]4  
where the signal is a function of time \(F=F(t)\), \(\delta,\alpha,\beta\) and \(\gamma\) being constants (parameters of the model). More precisely, \(\delta\) controls the amplitude of the damping, while \(\alpha\) and \(\beta\) control the linear stiffness and the nonlinearity of the restoring force. Here, \(\gamma\) is the amplitude of the periodic driving force with an angular frequency \(\omega\). An excellent introduction to Duffing oscillators and their use for modeling a variety of physical processes can be found, for example, in [73]. Here, in Figure 6, we show the results of the reachability analysis of the Duffing equation [4]. Comparing them with those of the Van der Pol equation, it becomes clear that in the latter case the pattern visualizing the set of states is more complex due to the much richer dynamics of the Duffing oscillator. It is known also that the theory of nonlinear oscillators predicts a double-well potential for certain parameters of the Duffing equation, which allows the simultaneous existence of two steady states. This leads to a hysteresis in which two different amplitudes of the forced oscillation are possible. Depending on the initial state of the system (which can be either at rest or in strong oscillation), the oscillator spontaneously chooses one of the amplitudes when tuning the parameters adiabatically into the hysteretic regime. In addition, small  
\[\dot{F}=y, \tag{8a}\]5 Reachability analysis of two coupled Van der Pol oscillators. The set of initial states for \(\mu=1.0\) are \(F_{1,2}(0)\in[1.25,1.55]\) and \(dF_{1,2}(0)/dt\in[2.35,2.45]\). The limit cycle with \(\mu=2\) (blue color) is calculated for the initial conditions \(F_{1,2}(0)\in[1.55,1.85]\) and \(dF_{1,2}(0)/dt\in[2.35,2.45]\). fluctuations can cause unpredictable jumps between the two potential wells and lead to bistability of the oscillation amplitude. A similar behavior has been observed in gyrotron operation, e.g., hysteresis [74; 75; 76; 77], chaotic regimes and bifurcations [78; 79; 80; 81]. If we compare the [PERSON]'s equation (which was considered in the previous section) and the Van der Pol equation, it becomes clear that there are both differences and similarities between them. The Van der Pol oscillators are characterized by the following characteristics: (i) _Nonlinear damping_, and the damping force depends on the oscillation amplitude (it increases at larger displacements); (ii) _Limit cycle_ behavior; and (iii) _Self-excitation_. The main features of the [PERSON]'s equation, on the other hand, are the following: (i) The focus is on _phase dynamics_ rather than amplitude and describes how oscillators synchronize with each other; (ii) _Coupling mechanism_, which is characteristic for the [PERSON]'s oscillator, incorporates interactions between multiple oscillators or external influences that affect their phases; (iii) _Synchronization phenomena_: It provides insights into how different oscillatory systems can synchronize under certain conditions.