## 3 Van der Pol Equation and the Analysis of Gyrotron Oscillators
The following examples were reproduced by us using the excellent open-source Julia ReachabilityAnalysis.jl library [3] and the packages in the JuliaReach [65] ecosystem, which can be used to implement reachability methods for linear, nonlinear, and hybrid ODEs. The results of the reachability analysis of the Van der Pol equation are shown in Figure 2, where the flowpipe of the problem is plotted. Recall that the solution obtained by set propagation consists of a flowpipe, which is just an array of reachable sets (_reach-sets_) and behaves like their set union. Figure 2 proves that there is no solution of the model with a \(\dot{F}\) value which is greater than 2.75, for any initial condition from the domain selected for this illustrative example (see the caption of Figure 2). The time horizon \(T=7\) is chosen such that the oscillator can do at least one full cycle. Computing the reachable set allows one to verify that all the behaviors of the closed-loop system stay within a desired range of operation and do not reach a forbidden region of the state space [1]. The corresponding time variation of the state variables \(F(t)\) and \(\dot{F}(t)\) of the above solution is shown in Figure 3.  
Figure 2: Results of the reachability analysis of the Van der Pol equation for \(\mu=1\). Shown are the flowpipe and the region of the initial conditions \(F(0)\in[1.25,1.55],\dot{F}(0)\in[2.35,2.45]\). The unsafe set is given by \(\dot{F}\geq 2.75\) for a time span \([0,7]\). The results of the reachability analysis of the Van der Pol oscillator can also be used to examine the existence of an invariant of the system. This is demonstrated in Figure 4, which proves that the flowpipe re-enters from where it started after one loop iteration. It can be seen that the reach set at index 388, corresponding to the time span [6.74967, 6.7599], is included in the union of reach sets with indices from 1 to 13 computed previously. Since all future trajectories starting from the 388 th reach set are already covered by the flowpipe; this indicates the presence of an invariant set. In other words, such limit-cycle behavior means that the oscillator tends to converge to a stable limit cycle regardless of the initial conditions. Such a system can generate oscillations without an external periodic driving force due to its inherent nonlinear properties. Figure 3: State variables as functions of time that correspond to Figure 2: \(F(t)\) and \(\dot{F}(t)\). The Van der Pol equation can be used to study coupled oscillators. In the case of two such oscillators, the corresponding system is  
\[\dot{F}_{1}=y_{1}, \tag{9a}\] \[\dot{y}_{1}=\mu(1-F_{1}^{2})y_{1}-2F_{1}+F_{2},\] (9b) \[\dot{F}_{2}=y_{2},\] (9c) \[\dot{F}=y, \tag{8a}\]0  
The results of the reachability analysis are shown in \[\dot{F}=y, \tag{8a}\]1 The system has a stable limit cycle, which becomes increasingly sharp for higher values of \(\mu\). The initial conditions 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 unsafe set is given by \(dF_{1,2}/dt\geqslant 2.75\) in a time horizon of [0, 7]. In the second case ( limit cycle with \(\mu=2\)), the initial condition is \(F_{1,2}(0)\in[1.55,1.85]\) and \(dF_{1,2}(0)/dt\in[2.35,2.45]\). The unsafe set is given by \(dF_{1,2}/dt\geqslant 4.05\) for a time horizon of [0, 8]. In this case, the time horizon \(T=8.0\) is used because \(T=7.0\) is not sufficient for the oscillator to make a complete loop. Without going into detail (due to the limited space here), it is important to point out that the forced Van der Pol oscillator (in this case, the right-hand side of Equation (7) contains a term corresponding to an external periodic driving force, e.g., \(A\sin(\omega t)\)) exhibits a variety of behaviors, including synchronization, hysteresis, bifurcations, and chaotic dynamics (see e.g., [66,67]). A system of Van der Pol equations is often used for analyzing the interaction between coupled limit-cycle microwave oscillators taking into account the delay [68, 69, 70].