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Fractal analysis of natural self-similar structures has been considered. Different approaches to the analysis of abstract mathematical fractals and natural fractals have been described. Numerical method of the fractal dimension calculation has been suggested. This method has been applied both for the model fractal (Sierpi´ nski carpet) and natural fractals (Saratov ravine network).

The cluster formation in the cyclic (4+1)-Lattice – Lotka–Volterra model is studied by Kinetic Monte Carlo simulations on a square lattice support. The features of cluster size distribution, spatial autocorrelation function and other dependences of the spatial dynamics of the system are under consideration. The role of cluster formation process and it effect on the systems dynamics is studied in this work. We show that the external mixing added to the initial scheme leads to the periodic self-oscillations appearance.

Considering model with bimodal dynamics we investigate the synchronization of different time scales. Transition between mode-locked and mode-unlocked chaotic attractors is investigated. It is shown that this transition involves a situation in which the synchronized chaotic attractor loses its band structure.

Spectral properties of the linear non-self-adjoint Perron – Frobenius operator of piece-wise linear chaotic maps having regular structure are investigated. Eigenfunctions of the operator are found in the form of Bernoulli and Euler polynomials. Corresponding eigenvalues are presented by negative powers of number of map brunches. The solution is obtained in general form by means of generating functions for eigenfunctions of the operator. Expressions for eigenfunctions and eigenvalues are different for original and inverse maps having even and odd number of branches.

The system of two coupled R¨ ossler oscillators is considered. Detailed investigation is carried out on the plane of parameters which control the period-doubling bifurcations in the subsystems. Dynamical regimes in different points of the control parameter plane are determined using the methods of the bifurcation plot and the highest nonzero Lyapunov exponent plot computation. The synchronization picture of two coupled R¨ ossler oscillators is compared with synchronization pictures of more simple systems: two coupled Van der Pol oscillators and coupled logistic maps.

An analytical Markov nonstationary model of electron emission is suggested. The model is determined by having limited life time of cathode. Some relations between probabilistic characteristics of emissivity and reliability of an emitter are obtained.

Methods of fullerene-coated tip field emitter creation were worked out and investigated. Basic rules and mechanisms of the microstructure origin on the fullerene surface during coverage formation and treatment (thermal and field) were determined. The emitters with fullerene coatings were made that secure stable currents from the single submicron tip up to 150 mA at static regime and up to 1.5–2 mA at pulsed one. Activation of the fullerene-coated field emitter by potassium atom and ion flows was studied.

Principal nonlinear characteristics of transmission lines on the basis of coupled structures with ferromagnetic films at excitation of different types of magnetostatic waves are described proceeding from theoretical and experimental researches. Following types of coupled systems are analyzed: «electrodynamic structure – ferromagnetic film»; the layered structure in the form of two coupled ferromagnetic films, each taken separately is considered as wave controlling structure for magnetostatic waves; and «electrodynamic structure – two coupled ferromagnetic films».

Magnetostatic surface wave (MSSW) bright solitons in a ferrite-dielectric-metal FDM) structure have been studied experimentally and numerically in the framework of he nonlinear Schr¨ odinger equation. The attention was focused on the influence of the  parametric instability on the soliton formation and propagation. We also discussed the contribution of the non-solitary (dispersive wave) part of the MSSW pulse on the soliton propagation, to show that their mutual interference leads to the levelling off or to the appearance of some peaks in the MSSW pulse output vs the input amplitude.

The introduction in the theory of discrete maps available for understanding to senior classes pupils and students of younger years is presented. It demonstrates correlation of discrete and continuous descriptions of dynamical systems. The physical examples facilitate a perception of a material. The computer tasks are presented, which decision gives a chance to generate enough capacious collection of programs, that can be used in research work.