ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

Review of Actual Problems of Nonlinear Dynamics

Modeling from time series and applications to processing of complex signals

Signals obtained from most of real-world systems, especially from living organisms, are irregular, often chaotic, non-stationary, and noise-corrupted. Since modern measuring devices usually realize digital processing of information, recordings of the signals take the form of a discrete sequence of samples (a time series). The present paper gives a brief overview of the possibilities of such experimental data processing based on reconstruction and usage of a predictive empirical model of a time realization under study.

Unique properties of open cavities and wavegiudes containing layered metamaterial

The review of works on studying artificial media with negative refractive index and their unique application is given. We make a point of negative refraction and lens effects in plan-parallel layers of such metamaterials. The properties and stability conditions of waveguide modes in layered structures and open cavities are considered in terms of the effective diffraction and dispersion lengths.

Development and improvement of field emitters containing carbon materials

Achievements and problems in creation of field emitters for vacuum microwave devices are described. The main attention is devoted to the emitters made of containing carbon materials for high-voltage devices operating at technical vacuum conditions 10−6 – 10−8 Torr. The brief review of existing works is presented. Results of investigations performed in SPbSPU are described.

Theory of waveguides excitation

The theory of waveguide excitation is presented, based on expansions of the electromagnetic field by proper waves of waveguide. Necessary properties of smooth and periodic waveguides, including the conditions of orthogonality of plane and the volume of the waveguide are given. Main properties of pseudo-periodic waveguides are described. This is a new class of waveguide systems. Different forms of the waveguides-excitation theory are considered.

Stochastic bifurcations

The modern knowledges of bifurcations of dynamical systems in the presence of noise are presenred. The main definitions are given and certain typical examples of the bifurcations in the presence of additive and multiplicative noise are considered.

Chaos and nonintegrability in hamiltonian systems

The article is devoted to historical development of one key aspect of Hamiltonian systems – nonintegrability, and its relation with chaotic behavior of the system. Evolution from the concept of quite integrable system to partly integrable one is shown. The relation of nonintegrability with such fundamental concepts as Kolmogorov stability, systems with divided phase space, Arnold diffusion, Zaslavsky web and others is discussed. 

Autonomous systems with quasiperiodic dynamics examples and their properties: review

The paper is a review of well-known in nonlinear dynamics models with low dimensional of phase space and quasiperiodic behavior. Also new results related to analysis of many-frequencies quasiperiodic oscillations for models with external action and coupled oscillators are discussed. 

Hyperbolic strange attractors of physically realizable systems

A review of studies aimed on revealing or constructing physical systems with hyperbolic strange attractors, like Plykin attractor and Smale–Williams solenoid, is presented. Examples of iterated maps, differential equations, and simple electronic devices with chaotic dynamics associated with such attractors are presented and discussed. A general principle is considered and illustrated basing on manipulation of phases in alternately excited oscillators and time-delay systems.

Spectral problems for the Perron–Frobenius operator

A method of solving the spectral problem for the Perron–Frobenius operator of onedimensional piece­wise linear chaotic maps is demonstrated. The method is based on introducing generating functions for the eigenfunctions of the operator. It is shown that the behavior of autocorrelation functions for chaotic maps depends on eigenvalues of the Perron­Frobenius operator.

Patterns in excitable dynamics driven by additive dichotomic noise

Pattern formation due the presence of additive dichotomous fluctuations is studied an extended system with diffusive coupling and a bistable FitzHugh–Nagumo kinetics. The fluctuations vary in space and/or time being noise or disorder, respectively. Without perturbations the dynamics does not support pattern formation. With proper dichotomous fluctuations, however, the homogeneous steady state is destabilized either via a Turing instability or the fluctuations create spatial nuclei of an inhomogeneous states.