Review of Actual Problems of Nonlinear Dynamics

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.

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.

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 time­delay systems. Alternative approaches are reviewed outlined in literature, as well as the prospects of further researches.

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.

Download full version

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.

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. Thisis a new class of waveguide systems. Different forms of the waveguides-excitation theory are considered.

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.

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.

Theoretical background of the wavelet­analysis and a series of applications of the given method are considered including a study of clustering phenomena for synchronous dynamics in structural units if the kidney, tactile information encoding by neurons of the trigeminal complex and detection of information messages from the chaotic carrying signal.

The article is based on the lecture that was given by the first author at the school StatInfo­2009. In the first part the microcanonical variant of multifractal formalism is reviewed. Possibilities for digital image analysis and reconstruction are discussed at the level of technical strictness.