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Problems of history of the Hamiltonian chaos discovery are considered. The example of Hamiltonian systems are free-moving particles with elastic collisions called mathematical billiards. The contribution from Russian scientists to chaos discovery in conservative systems (billiards are particular case of such systems) is especially large. Demonstration of billiard’s chaotic behaviour is one of the milestones in chaos history.

Equations of this type describe a number of real-life processes like wave motion under ice mantle or propagation of waves of longitudinal deformation in thin cylinder shell. Using «Simplest Equation Method» exact solitary-wave solutions of the two-dimensional Kawahara Equation were obtained. On the basis of implicit pseudospectral method the numerical investigation is carried out. Regimes of two-dimensional deformation waves with classic solitary behavior were discovered.

Some results of the X-band magnetron research are presented. Its active conductivity and the reactive one are measured in dependence both on anode voltage and RF voltage of the oscillatory systems. Approximating formulas are built for these dependences. This is necessary for transient processes calculation in microwave systems of X-band electron accelerators. The comparison is carried out with the known relations for the S-band magnetron.

Spatial self-organization structures in the metallic materials after irradiation by laser are studied. General method of the computer analysis of such structures using multifractals approaches is described. Founded consistent patterns of the changes of multifractals sets in irradiated surface of solids are used for the modeling of the system at the surface of two-dimensional lattice.

It is shown that the structure of the invariant density for R´ enyi map xn+1 = bxn mod 1, (1 < b < 2) may be clarified by action of the Perron–Frobenius operator on the uniform distribution. The invariant density is presented by finite linear combination of characteristic functions defined on the unit interval according to special rule. Some algebraic equations with entire coefficients are formulated for parameter b corresponding values definition.

The results of measurements of ear-drum longitudinal displacement against vibrations excited by periodical sound pressure have been results. The existence of ear-drum longitudinal displacement with the increase of sound pressure intensity has been txperimentally proved. By spectrum of autodyne signal of semiconductor laser, using the window method of semiconductor laser analysis, by changes of phase of autodyne signal in time the values of such displacement for different levels of sound pressure has been determined.

The paper represents the results of numerical study of dynamical regimes and bifurcation transitions in oscillatory system with frequency-phase control. The study was carried out on the base of mathematical model with three degrees or freedom in cylindrical phase space. Rich variety of various attractors of oscillatory and rotatory type corresponding to modulating modes of the system has been detected. Various scenarios of transition from regular dynamical regimes to chaotic ones under variation of the control loops parameters are analyzed.

An approach to extract transmitted messages from the chaotic carrying signal is proposed based on the combination of dynamical systems reconstruction and the discrete wavelet-transform. It is shown that discrete wavelets allow one to increase the stability to noise of the detecting algorithm that deals with the reconstruction technique.