Investigation of the Chaotic Dynamics of an Electron Beam with a Virtual   Cathode in an External Magnetic Field
  The effect of the strength of the focusing magnetic field on chaotic dynamic processes occurring inan electron beam with a virtual cathode, as well as on the processes whereby the structures form in the beamand interact with each other, is studied by means of two-dimensional numerical simulations based on solving a self-consistent set of Vlasov-Maxwell equations. It is shown that, as the focusing magnetic field is decreased,the dynamics of an electron beam with a virtual cathode becomes more complicated due to the formation andinteraction of spatio-temporal longitudinal and transverse structures in the interaction region of a vircator. The optimum efficiency of the interaction of an electron beam with the electromagnetic field of the vircator isachieved at a comparatively weak external magnetic field and is determined by the fundamentally two-dimensional nature of the motion of the beam electrons near the virtual cathode. 
E. N. Egorov and A. E. Hramov