Narrow Results

2008 accomplished the 100^th anniversary from the birth of L.D. Landau (Nobel Prize 1962). In the paper a short description of Kharkov period of Landau's work from 1932 to 1937 is given. One of areas of his work is connected with investigation of bremsstrahlung by ultra relativistic electrons in matter. In this paper a short review of the results obtained studying both Landau-Pomeranchuk-Migdal effect and coherent radiation by high-energy electrons and positrons in crystals, is given. The main attention is paid to the general properties and difference of radiation processes in amorphous media and in a crystal and to advances in the coherent bremsstrahlung studies.

The features of processes of chaotic discrete map in reverse time are considered. The fractal distribution of reverse iteration points is indicated. The approach to investigating reverse points manifold based on special ordering methods is proposed.

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

The influence of white and colored noise on the dynamics of self-sustained oscillator is considered in the regime of spiral (phase-coherent) chaos. The effective phase diffusion coefficient and power spectra are analyzed for noisy chaotic self-sustained oscillations. We show that chaotic oscillations can be synchronized by external narrow-band noise. Effects of chaos synchronization are compared for narrow-band noise signals with similar spectra but distinct probability distributions.

It is shown in the present paper that the transition to diffusion or spatiotemporal chaos in the phase space of solutions of the Kuramoto–Tsuzuki diffusion type partial differential equation occurs through the subharmonic and homoclinic cascades of bifurcations of two-dimensional invariant tori with respect to internal as well as external frequencies, and hence this scenario also can be described by universal FSM (Feigenbaum-Sharkovskii-Magnitskii) theory of transition to dynamical chaos in dissipative systems of nonlinear differential equations.

2008 accomplished the 100^th anniversary from the birth of L.D. Landau (Nobel Prize 1962). In the paper a short description of Kharkov period of Landau's work from 1932 to 1937 is given. One of areas of his work is connected with investigation of bremsstrahlung by ultra relativistic electrons in matter. In this paper a short review of the results obtained studying both Landau-Pomeranchuk-Migdal effect and coherent radiation by high-energy electrons and positrons in crystals, is given. The main attention is paid to the general properties and difference of radiation processes in amorphous media and in a crystal and to advances in the coherent bremsstrahlung studies.