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Results of experimental research of synchronization of two coupled almost identical resonance microwave active oscillators on multicavity klystrons in the modes of periodic and chaotic oscillations are presented. It is shown that depending on type of coupling it is possible to realize a mode of mutual frequency capture, synchronization by means of chaos full elimination by outer harmonic signal, and full synchronization mode. A possibility of using the chaos elimination effect for generation of sequence of chaotic radio pulses is shown.

In the work a new model of a continuous active medium with unidirectional coupling of active elements is proposed. The Anishchenko–Astakhov oscillator was selected as an active element. The model shows both regular and chaotic in time regimes. The results  obtained for the medium are compared with the results for a chain of Anishchenko–Astakhov oscillators. The problem of conformity between the discrete model and the continuous medium is analyzed.

A phase synchronization coefficient estimate, obtained from a time series, can take a high value even for uncoupled oscillators in the case of short signals and close basic frequencies. Since such situations are widespread in practice, it is necessary to detect them to avoid false conclusions about the presence of coupling. We investigate statistical properties of the estimator with the use of an exemplary system – uncoupled phase oscillators. Conditions leading to high probability to get big values of the estimator are determined quantitatively.

The numerical scheme for calculation the set of Lyapunov exponents in distributed systems with delayed feedback based on a modification of Benettine algorithm is described. The results of numerical simulation of two such systems (active oscillator with cubic nonlinearity and active oscillator of klystron type) are presented. The sets of Lyapunov exponents in different regimes, particularly in regimes of «weak» and «developed» chaos are analyzed. The calculation peculiarities of the set of Lyapunov exponents in the systems with delayed feedback are discussed.

In present paper we study the effect of synchronization of two-frequency quasiperiodic oscillations. We analyze both external and mutual synchronization. The peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus are established. It is shown that in general case, one and then another one of the basic frequencies is locked. The results of computer simulation are confirmed experimentally.

The solution of two problems of scientific Olympiad are discussed. At the school level the analysis of the solutions illustrates examples of systems, which can display qualitative change of a state and for which a method of discrete maps may be applied.

The new form of work with young researchers – Scientific Olympiad is introduced. The announcement of Scientific Olympiad and the problems are presented.

The dynamics of quantum wave packets in one-dimensional system with spatially confined quadratic potential, feedback and friction was numerically investigated in the context of the Schrodinger–Langevin–Kostin equation. The coherent oscillations are  possible in the system under determined values of the feedback force and friction coefficient. There are the critical values of these quantities when the packet oscillations become complicated, the uncertainty product increases sharply, oscillates, but the Fourier-spectrum is everywhere dense.

Two models of semiconductor laser with delayed optical feedback are studied. We consider singularly perturbed problem because of the large parameter presence. We construct and discuss quasinormal forms of models in trancritical cases.

In this paper we consider various networks of mutually coupled identical Hodgkin–Huxley systems. The peculiarities of transition to complete synchronization in networks subjected to suprathreshold periodic driving and common random forcing are examined both theoretically and through numerical simulation. The conditions for global stability of complete synchronization in networks of two «star»-coupled structures are obtained within the framework of connection graph stability method. Various scenarios determining the increase of the number of elements in such ensembles are considered.