Aim. The main stages of scientific activity of V.A. Solntsev from the student's bench to the leader of the scientific school on radiophysics and microwave electronics are investigated. Method. Based on the primary publications, the main directions of the creative path of the scientist, whom determined the electronic age of the time, are analyzed. Results. It is shown how timely V.A. Solntsev found a substitute for research and calculation methods that ceased to satisfy the demands of science and production, and resolutely went on to develop new principles of amplification and generation, accurately determining the expiration time of previous methods and principles and giving way only to the routine apparatus. Discussion. Great educational work and scientific and organizational role of the outstanding scientist were noted. © 2018 Saratov State University. All rights reserved.
The aim of our work was to study the influence of the different brain rhythms (i.e. theta, beta, gamma ranges with frequencies from 5 Hz to 80 Hz) on the ultra slow oscillations (USOs with frequency of 0.5 Hz and below), where high and low activity states alternate. The USOs is usually observed within neural activity in the human brain and in the prefrontal cortex in particular during rest. The USOs are considered to be generated by the local cortical circuitry together with pulse-like inputs and neuronal noise. Structure of the USOs shows specific statistics and their characteristics has been connected with cognitive abilities, such as working memory performance and capacity. In our study we used the previously constructed computational model describing activity of a cortical circuit consisting of the populations of pyramidal cells and interneurons. This model was developed to mimic global input impinging on the local PFC circuit from other cortical areas or subcortical structures. The studied the model dynamics numerically. We found that frequency increase deferentially lengthens the up states and therefore increases stability of self-sustained activity with oscillations in the gamma band. We argue that such effects would be beneficial to information processing and transfer in cortical networks with hierarchical inhibition.
Aim of the work was to study the influence of different brain rhythms (i.e. theta, beta, gamma ranges with frequencies from 5 to 80 Hz) on the ultraslow oscillations with frequency of 0.5 Hz and below, where high and low activity states alternate. Ultraslow oscillations are usually observed within neural activity in the human brain and in the prefrontal cortex in particular during rest. Ultraslow oscillations are considered to be generated by local cortical circuitry together with pulse-like inputs and neuronal noise. Structure of ultraslow oscillations shows specific statistics and their characteristics has been connected with cognitive abilities, such as working memory performance and capacity. Methods. In the study we used previously constructed computational model describing activity of a cortical circuit consisting of the populations of pyramidal cells and interneurons. This model was developed to mimic global input impinging on the local prefrontal cortex circuit from other cortical areas or subcortical structures. The model dynamics was studied numerically. Results. We found that frequency increase deferentially lengthens the up states and therefore increases stability of self-sustained activity with oscillations in the gamma band. Discussion. We argue that such effects would be beneficial to information processing and transfer in cortical networks with hierarchical inhibition.
A general idea of the qualitative study of dynamical systems, going back to the works by A. Andronov, E. Leontovich, A. Mayer, is a possibility to describe dynamics of a system using combinatorial invariants. So M. Peixoto proved that the structurally stable flows on surfaces are uniquely determined, up to topological equivalence, by the isomorphic class of a directed graph. Multidimensional structurally stable flows does not allow entering their classification into the framework of a general combinatorial invariant. However, for some subclasses of such systems it is possible to achieve the complet combinatorial description of their dynamics.
In the present paper, based on classification results by S. Pilyugin, A. Prishlyak, V. Grines, E. Gurevich, O. Pochinka, any connected bi-color tree implemented as gradient-like flow of $n$-sphere, $n > 2$ without heteroclinic intersections. This problem is solved using the appropriate gluing operations of the so-called Cherry boxes to the flow-shift. This result not only completes the topological classification for such flows, but also allows to model systems with a regular behavior. For such flows, the implementation is especially important because they model, for example, the reconnection processes in the solar corona.
The phenomenological model of an ensemble of three neurons which are coupled by chemical (synaptic) and electrical couplings is studies. A neuron is modeled by the oscillator of van der Pol. The aim of work is a study of the influence of coupling’s strength and frequency detuning between elements at regime of sequential activity that is observed in the ensemble of neuron-like elements with chemical inhibitory couplings. Method. The research is made with usage of analytical methods of nonlinear dynamics and computer modeling. Results. The study showed that the introduction of arbitrarily small electrical coupling into ensemble of Van der Pol oscillators with chemical synaptic inhibitory couplings leads to the destruction of a stable heteroclinic contour between saddle limit cycles. We showed also that nonidentity of elements (while electrical couplings is absent) does not lead to the destruction of heteroclinic contour, what, in general, is not typical for such systems. Discussion. We suggest to consider studied ensemble of elements as phenomenological model of neuronal network. Such approach has advantages: it is possible to study low-dimensional neuronal models and reproduce the main effects that are observed in more complex models, for example, in biologically realistic model of Hodgkin- Huxley and in real experiments also.
This article contains analysis of properties discrete electron-wave interaction in resonance slow-wave system (SWS). Equations of interaction are written in matrix form. Eigenvalues of transfer matrix with electron beam define spread constant of four electron waves. Features of electron waves in SWS with «smooth» energy stream and in SWS with «winding» flow are examined. Simulation is performed and amplification of multi-section TWT with passband and stopband sections of SWS is found. Field distribution along stopband section is found.
We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole phase space is a large «chaotic sea» with elliptical islands randomly disposed within it; and the mixed dynamics which is characterized by the principle inseparability, in the phase space, of attractors, repellers and orbits with conservative behavior. In the first part of this series of our works, we present some elements of the theory of pseudohyperbolic attractors of multidimensional maps. Such attractors, the same as hyperbolic ones, are genuine strange attractors, however, they allow homoclinic tangencies. We also give a description of phenological scenarios of the appearance of pseudohyperbolic attractors of various types for one parameter families of three-dimensional diffeomorphisms, and, moreover, consider some examples of such attractors in three-dimensional orientable and nonorientable Henon maps. ́ In the second part, we will give a review of the theory of spiral attractors. Such type of strange attractors are very important and are often observed type in dynamical systems. The third part will be dedicated to mixed dynamics – a new type of chaos which is typical, in particulary, for (time) reversible systems i.e. systems which are invariant with respect to some changes of coordinates and time reversing. It is well known that such systems occur in many problems of mechanics, electrodynamics, and other areas of natural sciences.
The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios of spiral chaos, i.e. those elements of the theory that remain valid for any models, regardless of their origin. The mathematical foundations of this theory were laid in the 60th in the famous works of L.P. Shilnikov, and on this subject to date, a lot of important and interesting results have been accumulated. However, these results, for the most part, were related to applications, and, perhaps for this reason, the theory of spiral chaos lacked internal unity – until recently it seemed to consist of separate parts. As it seems for us, the main results of our review allow to fill this gap. So, in the paper we present a fairly complete and illustrative proof of the famous theorem of Shilnikov (1965), describe the main elements of the phenomenological theory of universal scenarios for the emergence of spiral chaos, and also, from a unified point of view, consider a number of three-dimensional models which demonstrate this chaos. They are both the classical models (the systems of Rossler and Arneodo–Coullet–Tresser) and several models known from applications. We discuss advantages of such a new approach to the study of problems of dynamical chaos (including the spiral one), and our recent works devoted to the study of chaotic dynamics of multidimensional flows (with dimension N > 3) and three-dimensional maps show that it is also quite effective. In particular, the next, third, part of the review will be devoted to these results.
Aim. The main stages of scientiﬁc activity of V.A. Solntsev from the student’s bench to the leader of the scientiﬁc school on radiophysics and microwave electronics are investigated. Method. Based on the primary publications, the main directions of the creative path of the scientist, whom determined the electronic age of the time, are analyzed. Results. It is shown how timely V.A. Solntsev found a substitute for research and calculation methods that ceased to satisfy the demands of science and production, and resolutely went on to develop new principles of ampliﬁcation and generation, accurately determining the expiration time of previous methods and principles and giving way only to the routine apparatus. Discussion. Great educational work and scientiﬁc and organizational role of the outstanding scientist were noted.
The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge–Kutta method to solve a system of differential equations. Main analysis of the dynamics is carried out on the basis of calculated spectrum of Lyapunov exponents depending on parameters of systems, so-called charts of Lyapunov exponents. Bifurcation trees, winding numbers, phase portraits and Poincar´e maps were also visualized. Results. Study of the dynamics of two coupled quasi-periodic generators for two sets of operating parameters of the subsystems is carried out. Two cases were studied when a two-frequency torus or chaotic oscillations (destroyed torus) are observed in the first oscillator. The dynamics of the second oscillator demonstrates different types of dynamics with a variation of the frequency detuning: periodic, quasi-periodic and chaotic. It is shown that for all parameters, phase synchronization of generators, broadband synchronization, and the phenomenon of oscillator death are observed. Dynamical regimes picture of the parameter plane frequency detuning – the coupling strength has a universal structure. Exit from the broadband synchronization region with a decrease in the coupling strength is accompanied by a quasiperiodic Hopf bifurcation and the birth of a three-frequency torus. Conclusion. The interaction of the simplest generators of quasi-periodic oscillations gives a rich picture of dynamic regimes: multi-frequency quasi-periodic oscillations with a different numbers of frequencies, chaotic behavior characterized by a different spectrum of Lyapunov exponents. Despite the variety of dynamic regimes, the synchronization picture of two dissipatively coupled quasi-periodic generators has a universal structure. Quasi-periodic phase and broadband synchronization are observed. The destruction of a torus in a subsystem leads to the destruction of multi-frequency tori in the system of two coupled oscillators, as well as a decrease in the variety of types of chaotic attractors.
Sergei P. Kurdyumov (1928–2004) and his distinguished contribution in the development of the modern interdisciplinary theory and methodology of study of complex self- organizing systems, i.e. synergetics, is under consideration in the article. The matter of a mathematical model of evolutionary dynamics of complex systems elaborated by him is demonstrated. The nonlinear equation of heat conductivity serves as a basis of the model. Under certain conditions, it describes dynamics of development of structures of different complexity in the blow-up regime. Methods of calculation of two-dimentional structures which are described by automodel solutions are considered; and their classification is given. The automodel problem is a boundary problem aiming to find eigen-values and eigen-functions for a nonlinear equation of elliptical type on a plane. Proceeding from the analysis of the model, a principle of coevolution was formulated by S.P. Kurdyumov. This is the principle of integration of simple structures into a complex one. Three notions of great significance follow from the principle, and namely: the notion of connection of space and time, the notion of complexity and its nature and the notion evolutionary cycles and switching over different regimes as a necessary mechanism of maintenance of «life» of complex structures. Approaches of possible application of this model for understanding of dynamics of complex social, demographic and geopolitical systems are viewed as well.