Applied Problems of Nonlinear Oscillation and Wave Theory

In the context of the Schrodinger–Langevin–Kostin equation, the quantum anharmonic oscillator with one-term 4-degree potential has been numerically investigated. The generated frequencies of the oscillator are defined by the non-equidistant energy spectra, the number of  discrete frequencies depends on the initial state energy. Due to increasing of initial state energy,  spectra are displaced in the direction of higher frequencies. Influence of friction on transition from  excited state into ground one is also investigated.

The dynamics is studying for rigid body rotating around fixed axis Oz being central but not principal. Therefore the inertial torques Mx and My arose depending both on mass geometry Jxz, Jyz  and on angular velocity ω and acceleration ". Dry friction acting on axis’s supports with coefficient  δ leads to that the value of " serves as the reason and result of the motion simultaneously. There  were integrated numerically and/or analytically the dynamical equations of free and forced motion  including rotational harmonic and inharmonic oscillations too.

Researched the structure of Very Low Frequency (VLF) spectrum of heart rate variability (HRV)  and its nonlinear behavior in a relationship with the energy of oscillations, baroreflex and parasympathetic activity at functional tests of low intensity in 100 subjects (seven-test, deep breathing), including active orthostatic test of 32 subjects with orthostatic tachycardia in comparison to the control group of 20 subjects. There were three stages of research. The first  stage: created the method of spectral analysis of separate components of VLF.

The dynamics of ensembles of oscillators containing a small number of bibitemlits is discussed. The possible types of regimes and pecularities of bifurcations of regular and quasi-periodic attractors are analyzed. By using the method of Lyapunov exponents charts the picture of  embedding of quasi-periodic regimes of different dimension in the parameter space is revealed. Dynamics of ensembles of van der Pol and phase oscillators are compared.

The paper presents the results of theoretical and experimental studies of multispeed electron beams. Theoretical and experimental studies of the structure of such beams are considered. It is shown that turbulence forms in such electron beams. It leads to the formation of multiple bunches of space charge, which are sources of broadband noise-like fluctuations. The results of  experimental studies of small-sized laboratory model of the generator based on the principles of  formation and use of multispeed turbulent electron beams are shown.

The method is proposed for delay time estimation in time-delay systems from their time series. The method is based on the nearest neighbor method. It can be applied to a wide class of time-delay systems and it is still efficient under very high levels of dynamical and measurement noise.

The main trend of some blow­up systems is disturbed by log­periodic oscillations infinitely accelerating when approaching the blow­up point. Explanation of such behavior typical e.g. for seismic and economic phenomena could give an insight into the nature of blow­up point rising in this case as the condensation of constant phase points of oscillations. This viewpoint is a particular case of the more general approach that treats not oscillations as a disturbance of the growing trend, but the trend itself as a result of oscillatory process.

The main trend of some blow­up systems is disturbed by log­periodic oscillations infinitely accelerating when approaching the blow­up point. Explanation of such behavior typical e.g. for seismic and economic phenomena could give an insight into the nature of blow­up point rising in this case as the condensation of constant phase points of oscillations. This viewpoint is a particular case of the more general approach that treats not oscillations as a disturbance of the growing trend, but the trend itself as a result of oscillatory process.

We study model of monomode semiconductor laser with optoelectronic feedback, based on balanced equations with delay. We built sets of quasinormal forms in neighboorghood of bifurcation values. The possibility of coexistence of large amount of stable oscillating solutions is shown

We produce the results of computer analysis of complex dynamics of non-relativistic electron beam being placed in crossed electric and magnetic fields, in the regime of a virtual cathod forming in additional braking field. The modeling has been made in the framework of 2D numerical model in the geometry of magnetron-injector gun.