On Landau scenario of chaotization for beam distribution

We examine a problem in nonlinear dynamics in which both regular and chaotic motions are possible. Thus we deal with some of the fundamental theoretical problem of accelerator physics, the theory of dynamical systems and other fields of physics. The focus is on the appearance of chaos in a beam distribution. A study of the problem is based on two observation: first, using the Lyapunov method and its extension we obtain solutions of partial differential equations. Using this approach we discuss the problem of finding a solution of the Vlasov-Poisson equation, i.e., some stationary solution where we consider a magnetic field as some disturbance with a small parameter. Thus the solution of the Vlasov equation yields an asymptotic series such that the solution of the Vlasov-Poisson equation is the basis solution for one. The second observation is that physical chaos is a weak limit of well known Landau bifurcations. This is proved using ideas on the nature of turbulence