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A review of the current state of different theoretical approaches to the description of turbulence in electron beams and electronic devices at microwave frequencies is shown. A three types of turbulent (nonlaminar) electron beams were considered. The first type of turbulent electron beam is caused by the intersection of electronic trajectories (e.g., due to thermal velocity) and it is common to the flow of electrons at all.

The paper is devoted to some historical aspects of the rapidly developing field of modern mathematics – nonlinear functional analysis, which is presented as the basis of the mathematical apparatus of nonlinear dynamics. Its methods are demonstrated on the example of bifurcation. The first bifurcations problem – Euler problem on elastic instability rod under longitudinal compressive forces is considered.

We review the results of studies of spontaneous and stimulated emission of relativistic particles in natural and photonic crystals.We consider the diffraction of electromagnetic waves in a crystal, and the resonance and parametric (quasi-Cherenkov) X-ray radiation, the radiation in the channeling of relativistic particles in crystals, diffraction radiation in conditions of channeling, diffraction radiation of a relativistic oscillator, induced radiation in multidimensional space-periodic resonators (natural or artificial (electromagnetic, photonic) crystals).

Sergei P. Kurdyumov (1928–2004) and his distinguished contribution in the development of the modern interdisciplinary theory and methodology of study of complex self-organizing systems, i.e. synergetics, is under consideration in the article. The matter of a mathematical model of evolutionary dynamics of complex systems elaborated by him is demonstrated. The nonlinear equation of heat conductivity serves as a basis of the model.