## 3 Van der Pol Equation and the Analysis of Gyrotron Oscillators  
Although the Van der Pol equation was initially developed to describe nonlinear oscillators with damping (also known as relaxation oscillators), it is now used in many scientific fields to study a variety of dynamical systems. Its fundamental nature makes it a building block for the development of the theory of various oscillators, including gyrotrons. This equation can be written in the following form [51]:  
\[\frac{d^{2}F}{dt^{2}}-\mu(1-F^{2})\frac{dF}{dt}+F=0, \tag{7}\]  
where \(F\) is the amplitude of the oscillating variable (signal), which is a function of time \(t\), and \(\mu\) is a parameter characterizing the nonlinearity (strength of the damping). For the case \(\mu=0\) (no damping), Equation (7) reduces to the equation of harmonic oscillator. When \(\mu>0\), all initial conditions converge to a globally unique limit cycle. For electrical limit-cycle circuits (oscillators), there exists a closed trajectory in phase space that has the property that at least one other trajectory spirals into it as time approaches either infinity or negative infinity. When these circuits are driven near the limit cycle, they become entrained, i.e., the driving signal pulls the current along with it. Near the origin \(dF/dt=0\), the system is unstable, and far from the origin the system is damped. The numerical integration of the Van der Pol equation is an example of a stiff problem for which certain numerical methods (solvers) are numerically unstable, unless the step size is taken to be extremely small. This complicates its analysis and requires more computing resources. The qualitative analysis of this equation reveals some general features of the possible solutions [60]. If the solution \(F>>1\), both the restoring and damping forces are large, so that \(|F(t)|\) should decrease with time. In this case the system behaves like a strongly damped oscillator and it disperses energy. On the contrary, if the solution is small \(|F|<<1\), the damping force becomes negative, which should make \(|F(t)|\) tend to increase with time. Between these extreme cases, for typical parameters of the Van der Pol oscillators, as already mentioned, there exists a limit cycle (a unique periodic solution) with a \"shape\" that strongly depend on \(\mu\). In order to plot the phase-space portrait of the Van der Pol oscillator and to perform the reachability analysis of particular cases with different values of the parameter \(\mu\), it is convenient to write this equation in the following two-dimensional form:  
\[\dot{F}=y, \tag{8a}\] \[\dot{y}=\mu(1-F^{2})y-F, \tag{8b}\]  
where the dot represents the differentiation with respect to time. Before continuing the analysis of the above system using an advanced method--reachability analysis--it is instructive to point out that a similar system of equations was derived in [61] (see Equations (58) and (59) in [61]). It describes the pure amplitude interaction between two modes with closely spaced frequencies at the fundamental in the gyrotron cavity by averaging the governing equations of the physical model over fast phase beating. They show that in the case of \"strong\" coupling between the modes (depending on the initial conditions), the single-mode oscillations of any mode are stable. In the opposite case (\"weak\" coupling of the modes), when the self-saturation is stronger than the cross-saturation, the two-mode equilibrium is stable. In [62] a model, which describes the excitation of a gyrotron cavity, was developed using the Van der Pol equation. Similar parallels between theory of gyrotrons and Van der Pol oscillators are abundant (see e.g., [5; 16; 63; 64]). Reachability analysis makes it possible to find out which states a dynamical system can reach starting from a set of initial states and under the influence of a set of input trajectories and parameter values. We apply this approach to the Van der Pol equation to illustrate the form of the data it provides.