Nonlinear dynamics of electrons in a pump magnetic field of a free electron maser

We showed that in the vicinity of the magnetoresonance (the undulator frequency is close to the cyclotron one), one can efficiently transfer kinetic energy of longitudinal constant motion of an electron to its transversal-oscillatory degrees of freedom, thus providing for a successful amplification of a seed microwave. An analytical condition of the dynamics chaos onset caused by coupling between electron undulations and cyclotron rotation in the pump magnetic field is presented; Lyapunov exponent maps are built. The closed non-involutive set of first integrals is found. Using Nehoroshev theorem, it is determined that the dimension of invariant tori is 2. We establish that interaction between an electron beam and microwave does not lead to a noticeable broadening of the dynamics chaos zone, if this interaction is sufficiently small.