# Review of Actual Problems of Nonlinear Dynamics

## Two-stream instability – linear and nonlinear microwave phenomena

This article is the first of two parts of the review devoted to the phenomenon of two-beam instability in microwave electronics. The main goal is to cover as much as possible the most complete list of papers on various models and methods of analysis of the two-beam instability. The first part contains papers which, in our view, most fully describe the development of ideas of two-stream instability in microwave electronics.

## Two-stream instability – linear and nonlinear microwave phenomena

This article is the second part of the review of works devoted to the phenomenon of two-stream instability in microwave electronics. As it is known, a problem of creating devices operating in the terahertz frequency range is a rather actual today. Although there are many devices that can generate or amplify signals in this range, most of them refer to extremely powerful relativistic devices. At the same time there is a lack of compact medium power devices.

## Ultrawideband wireless sensor networks based on chaotic radiopulses

Wireless sensor networks that is a fast emerging branch of modern telecommunications are considered in this paper. Particular attention is paid on ultrawideband sensor networks where chaotic radiopulses are used as an information carrier between sensor nodes. Development of such wireless sensor networks became possible after longterm investigations of chaotic oscillations and chaos control.

## Fractional diffusion equation for aging and equilibrated random walks

We consider continuous time random walks and discuss situations pertinent to aging. These correspond to the case when the initial state of the system is known not at preparation (at t = 0) but at the later instant of time t_{1} > 0 (intermediate-time initial condition). We derive the generalized aging diffusion equation for this case and express it through a single memory kernel. The results obtained are applied to the practically relevant case of the equilibrated random walks.

## Entropy and forecasting of time series in the theory of dynamical systems

A contemporary consideration of such concepts as dimension and entropy of dynamical systems is given. Description of these characteristics includes into the analysis the other notions and properties related to complicated behavior of nonlinear systems as embedding dimension, prediction horizon etc., which are used in the paper. A question concerning the application of these ideas to real observables of the economical origin, i.e. market prices of the companies Schlumberger, Deutsche Bank, Honda, Toyota, Starbucks, BP is studied.

## Self-sustained oscillations in quasiharmonic and chaotic oscillators in the presence of fluctuations

The paper presents the results of the classical theory of fluctuations in the quasiharmonic van der Pol oscillator. Stochastic equations for amplitude and phase of selfsustained oscillations are formulated and then their solutions are analyzed. The autocorrelation function and power spectrum of noisy self-sustained oscillations are studied. Then the spectral and correlations characteristics of chaotic self-sustained oscillations are numerically analyzed in spiral chaos oscillators.

## Wavelet transform of time series and atmosphere dynamics

Wavelet transform is described as a new tool for investigation of data generated by the chaotic dynamic systems. Its usage is illustrated by the analysis of the temporal oscillation of the atmosphere zonal circulation index.

## Генезис схемы чуа

Статья представляет собой систематическое изложение последовательности технических этапов, пройденных автором при разработке схемы, генерирующей хаос. Процедура разработки, хотя и ясна по своей природе, не могла быть изобретена без использования некоторых важных свойств нелинейных схем и их физических реализаций.

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## Polynomial eigenfuctions of the perron–frobenius operator

In the paper, we reveal the structure of polynomial functions of the eigenfunctions and the kernel of the Perron–Frobenius operator for one-dimensional chaotic maps that iterative functions have the following properties: they are piecewise-linear ones; they have full branches transforming the domain of its definition to the full range of the mapping; the have arbitrary slope of branches; they have not some gaps between the branches.